Algebras of Non-Local Screenings and Diagonal Nichols Algebras
In a vertex algebra setting, we consider non-local screening operators associated with the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated with a d...
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| description | In a vertex algebra setting, we consider non-local screening operators associated with the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated with a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are unique or come as one- or two-parameter families. Examples include realizations of Lie superalgebras. We then study the associated algebra of screenings with improved methods. Typically, for positive definite lattices we obtain the Nichols algebra, such as the positive part of the quantum group, and for negative definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite quantum group with a large center.
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| first_indexed | 2026-03-21T08:48:00Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 018, 81 pages
Algebras of Non-Local Screenings
and Diagonal Nichols Algebras
Ilaria FLANDOLI and Simon D. LENTNER
Department of Mathematics, University of Hamburg,
Bundesstraße 55, 20146 Hamburg, Germany
E-mail: ilaria.flandoli@gmail.com, simon.lentner@uni-hamburg.de
URL: http://simon.lentner.net
Received November 02, 2020, in final form February 14, 2022; Published online March 11, 2022
https://doi.org/10.3842/SIGMA.2022.018
Abstract. In a vertex algebra setting, we consider non-local screening operators associated
to the basis of any non-integral lattice. We have previously shown that, under certain
restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or
Nichols algebra associated to a diagonal braiding, which encodes the non-locality and non-
integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as
classified by Heckenberger, and find all lattice realizations of the braiding that are compatible
with reflections. Usually, the realizations are unique or come as one- or two-parameter
families. Examples include realizations of Lie superalgebras. We then study the associated
algebra of screenings with improved methods. Typically, for positive definite lattices we
obtain the Nichols algebra, such as the positive part of the quantum group, and for negative
definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite
quantum group with a large center.
Key words: Nichols algebras; quantum groups, screening operators; conformal field theory
2020 Mathematics Subject Classification: 16T05; 17B69
Contents
1 Introduction 2
2 Preliminaries on Nichols algebras 7
2.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Generalized root system and Weyl groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Preliminaries on screening operators 15
3.1 Vertex algebras and their representation theory . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Screening operators and Nichols algebra relations . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Analytical continuation of screening relations 21
4.1 Commutativity relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Truncation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Analytical continuation by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Serre relations for Cartan matrix entry −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Formulation of the classification problem 31
mailto:ilaria.flandoli@gmail.com
mailto:simon.lentner@uni-hamburg.de
http://simon.lentner.net
https://doi.org/10.3842/SIGMA.2022.018
2 I. Flandoli and S.D. Lentner
6 Cartan type 34
6.1 q diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2 Construction of (mij) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.3 Central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.4 Algebra relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.5 Examples: Cartan type realizations in rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . 37
7 Super Lie type 40
7.1 q diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.2 Construction of (mij) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.3 Central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.4 Algebra relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.5 Examples in rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.6 Arbitrary rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.7 Sporadic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.7.1 G(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8 Other cases in rank 2 54
8.1 Construction of (mij) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9 Rank 3 65
10 Rank ≥ 4 72
11 Tables: realizing lattices of Nichols algebras in rank 2 and 3 74
References 78
1 Introduction
Let Λ be a lattice with basis a1, . . . , ar and inner product (−,−), not necessarily integral. We
consider the Gram matrix mij = (ai, aj) and a braiding matrix qij = eπi(ai,aj). In the theory of
Nichols algebras, we can associate to the data (qij) the Nichols algebra B(qij) of diagonal type.
In the theory of vertex algebras, we can associate to Λ, (mij) an abelian intertwining algebra
and to each ai a non-local screening operator Zi on the Heisenberg vertex algebra Hr.
It was proven by the second author in [49] that the screening operators Zi obey the relations
of the Nichols algebra B(qij), provided that (mij) is subpolar (Definition 3.3). This condition
imposes a certain lower bound for the sums of mij over any subset of indices and it ensures
convergence of integrals such as∫ 1
0
· · ·
∫ 1
0
∏
i<j
(zi − zj)
mijdz1 · · · dzn.
The goal of this article is to find all lattices Λ, (mij), such that the associated diagonal
braiding (qij) gives a finite-dimensional Nichols algebra in the classification of [34], and such
that the Weyl reflections on (qij) in the theory of Nichols algebras lift to reflections on (mij)
in a suitable sense (Definition 5.1). In this case we say that Λ, (mij) realizes the braiding
matrix (qij). This provides an interesting zoo of examples to extend the screening method and
all its related questions, such as the logarithmic Kazhdan Lusztig conjecture to cases beyond
quantum groups. This main motivation for our article will be discussed further below.
As a second goal, for each realizing lattice Λ we continue to study the algebra of screening
operators. First we determine which data (mij) are subpolar, in which case the screening algebra
is a surjective image of the Nichols algebra. We find that all Nichols algebras that do not come
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 3
as q-parametrized families have at most one realizing lattice, and this realization is subpolar.
For the realizing lattices that come in q-parametrized families,1 we encounter m-parametrized
families of realizations with q = eπim and outside a certain range of parametersm the subpolarity
of (mij) fails. In these situations we study the products of screening operators and their relations
using analytic continuation. Since we have not yet found a general approach for this, we explicitly
treat the most common relations: the commutation relations [xi, xj ]q, the truncation relations
of simple root vectors xni = 0, and the Serre relations [xi, [xi, xj ]q]q for a Cartan matrix entry
cij = −1, where [xi, xj ]q is the braided commutator defined in formula (2.1). As it turns out,
each of these relations can be analytically continued and holds outside an explicit set of poles.
The method of analytic continuation we employ to this end are on the one hand explicit
expressions in terms of Gamma functions due to the Selberg integral formula and to the An-
Selberg integral formula [60, 62], and on the other hand a new less explicit analytic continuation
for general Selberg integrals using recursion in Section 4.3. We believe that the latter method
can be greatly improved and should settle a large class of relations. Moreover we believe that
our results suggest an explicit g-Selberg integral formula for all generalized root systems (e.g.,
for Lie superalgebras), but both questions are beyond the scope of the present article. From
a Nichols algebra perspective it is interesting, that we require the knowledge which order the
zero in the quantum symmetrizer of a relation has. For example, the quantum Serre relation
for a simply-laced quantum group at q2 ̸= −1 exhibits a simple zero. At q2 = −1 it exhibits
a double zero and at the same time the Selberg integral exhibits for m ∈ −1
2 −N0 a simple pole,
thus the quantum Serre relation holds even in these cases.
As a main example, let g be a complex finite-dimensional semisimple Lie algebra of rank r with
simple roots α1, . . . , αr and Killing form normalized to (αi, αi) ∈ {2, 2d} with lacity d ∈ {1, 2, 3}.
Let q ̸= ±1 be a root of unity and consider the braiding qij = q(αi,αj), whose associated Nichols
algebra B(qij) is the positive part of the small quantum group uq(g)
+ [8, 50, 55]. For every
rational number m ̸∈ Z with q = eiπm we obtain a realizing lattice by taking the root lattice
of g rescaled by m.
We find that the rescaled root lattices are the only realizations (mij) of (qij) if ord
(
q2
)
>
d+ 1 and ord
(
q2d
)
> 2. Otherwise, there exist additional families of realizations (mij), mostly
associated to Lie superalgebras, which incidentally have for these small orders of q the same
braiding matrices, but different realizing lattices. For example, for g of type An at q2 = −1 we
have additional realizations associated to A(n′|n′′), n′ + n′′ = n − 1 that contain root lattices
of A(n′), A(n′′) rescaled by m′, m′′ with m′ +m′′ = 1 and a fermionic root of length 1. Since
the screening operators and their relations depend on (mij), not just on (qij), these realizations
behave very different, more similar to Lie superalgebras.
Assume now that (mij) is the realization obtained by rescaling the root lattice of g by
m, assume further that g is simply-laced and q2 ̸= −1, then we can summarize our analytic
continuation results on the relations of the screening operators as follows:
� For 0 < m < 1 the parameters (mij) are subpolar and thus all Nichols algebra relations
hold. Differently spoken, the algebra of screenings is a surjective image of the Borel part
of the small quantum group uq(g).
� For m < 0 the Serre relations hold. The truncation relations of simple root vectors fail.
Differently spoken, for q2 ̸= 1 the algebra of screenings is a surjective image of the Borel
part of the Kac–DeConcini–Procesi quantum group UK
q (g).
� for m > 1 the Serre relations and the truncation relations of simple root vectors hold.
We conjecture that for m > 1 also the truncation relations of non-simple root vectors hold, so
that also in this case we get the Borel part of uq(g). For q
2 = −1 there is an additional relation
1Cartan type, super Lie type and the color Lie algebra [33, Table 1, row 6]. Some come with more than one
parameter q, m. The families [34] rows 5 and 6 have no realizations.
4 I. Flandoli and S.D. Lentner
[xj , [xi, [xj , xk]q]q]q = 0 in [3], which we conjecture to hold for m < 0, but we have no guess
whether it holds for m > 1. We conjecture that all surjections above are in fact isomorphisms by
the universal property of the Nichols algebra, since the screening operators have a Leibniz-type
product rule.
The similar statement for simply-laced quantum supergroups is more complicated and de-
pends on m′, m′′ and their relation in the specific case, see Corollary 7.19. We encounter again
the small quantum super group and the Kac–DeConcini–Procesi quantum super group. We
also encounter versions of the latter where only the truncation relations for simple root vectors
rescaled by m′ or m′′ fail, for example in A(n′, n′′) those in A(n′) or A(n′′), which are interesting
intermediate Hopf algebras. The truncation relations for the fermionic simple root x2f = 0 always
holds as expected. The additional relation has ranges for m in which we can prove it, but in
particular for large m′, m′′ we have no assertion on the additional relation or the Serre relation.
We now discuss the background and application on the vertex algebra side in more detail:
A vertex algebra V [25, 44] is, very roughly speaking, an algebra whose multiplication depends
analytically on a complex variable z. Screening operators Za are linear endomorphisms on V
constructed by taking the left-multiplication with a fixed element a ∈ V and integrating z over
a circle around the singularity at z = 0. Using the operator product expansion one can compute
the commutator of such operators, and in this way screening operators are important sources of
actions of Lie algebras on vertex algebras.
Vertex algebras have a natural notion of representations, and under certain finiteness-assump-
tions the category of representations becomes a braided tensor category [41]. The tensor product
is defined by a universal property involving so-called intertwining operators and the braiding
comes from analytically continuing these from z to −z around their singularity at z = 0. In
particular, nontrivial double braiding means the analytical continuation is multivalued.
We can now attempt to define screening operators Za for elements a which are not in the vertex
algebra V but in some vertex algebra module M, using intertwining operators instead of the
vertex algebra’s multiplication. We call these non-local screening operators, because we are now
dealing with integrals over multivalued functions and we encounter nontrivial double braidings.
Such screening operators are less well-behaved, e.g., with respect to the Virasoro action, and
they do not form a Lie algebra. The main result of [49] is that instead, under the assumption
of subpolarity mentioned above, any fixed set of non-local screenings on the Heisenberg vertex
algebra and lattice vertex algebras (and conjecturally on every suitable vertex algebra) generates
the Nichols algebra associated to the braiding on M. Instances of non-local screening operators
have appeared in the literature for a long time (see, e.g., [20, 22]). In the setting discussed next,
it was conjectured that they generate the Borel parts of quantum groups, which is now settled
by the results in [49]. Semikhatov and Tipunin proposed in [57, 58] to extend this program to
Nichols algebras of diagonal type. Our article builds on these ideas.
One major intention behind studying these non-local screening operators was and is the
logarithmic Kazhdan Lusztig conjecture [1, 21, 28], which roughly states the following: Fix
a semisimple complex finite-dimensional Lie algebra g and consider the vertex algebra VΛ asso-
ciated to the rescaled root lattice Λ =
√
pΛg, whose braided tensor category of representations
is the category of vector spaces graded by Λ∗/Λ. Consider the non-local screening operators Zai
associated to inversely rescaled coroots ai = α∨
i /
√
p. Then the kernel of all Zai defines a ver-
tex subalgebra W ⊂ V, whose category of representations is conjecturally a non-semisimple
modular tensor category equivalent to the category of representations of the small quantum
group uq(g), q = e
2πi
2p , more precisely to some quasi-Hopf algebra variant. In the smallest case
g = sl2 the conjecture was solved affirmatively, after about 20 years of research by several groups
[1, 12, 14, 28, 30, 53, 61]. For quantum groups associated to arbitrary g see [2, 21, 27, 49].
In a more general setting and in view of the mentioned results on Nichols algebras, the second
author has proposed the following problem, which is probably very hard:
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 5
Problem 1.1. Let V be a vertex algebra and Za1 , . . . ,Zar a set of non-local screenings generating
a Nichols algebra B (or some extension, due to poles). What is the category of representations
of the kernel of screenings W ⊂ V?
We expect that the finite-dimensionality of B implies finiteness conditions on W and its
category of representations. From a physics intuition, the problem can be interpreted as asking
for the representation theory of an orbifold of V by an action of the Nichols algebra B or some
extension. A generic guess for a braided tensor category would be the relative Drinfeld center of
the representations of the algebra B inside Rep(V), relative to Rep(V). In the original setting of
the logarithmic Kazhdan Lusztig conjecture, this correctly reproduces the expected quasi-Hopf
algebra variant of the small quantum group [14, 27, 54]. However, already for the so-called
p, p′-models, which would correspond to sl2, q = e
πi p
′
p , the result is slightly larger, see, e.g., [26].
A second question is how general are the modular tensor categories obtained by this procedure:
Problem 1.2. Classifying semisimple modular tensor categories is hard, let alone non-semi-
simple modular tensor categories. Can we classify modular tensor categories U whose semisimple
part (roughly speaking) is a fixed semisimple modular tensor category C? Again, the generic
choice for such U is a relative Drinfeld center of representations of some Nichols algebra B ∈ C.
A very small step towards such a classification for uq(sl2) was recently obtained in work of the
second author [12], which finally settled the smallest case of the logarithmic Kazhdan Lusztig
conjecture. The overall idea behind this second proposed problem is a categorical version of the
Andruskiewitsch–Schneider program [8], which aims to classify finite-dimensional Hopf algebras
for a given semisimple coradical, and additionally to ask for the existence of a nondegenerate
braiding. For example, the main result in cit. loc. is a classification of all finite-dimensional
Hopf algebras with coradical a finite abelian group algebra C[A] (of order excluding certain
small prime divisors) in terms of one ore more Nichols algebras uq(g)
+ in the modular tensor
category of A-Yetter–Drinfeld modules, plus so-called lifting data. In the present context, we
would equivalently speak about the modular tensor category C = VectA and different choices
of quadratic forms on A, and we want to ask further which of the lifting data admits a non-
degenerate braiding. A probable answer would be, that only the quantum doubles U(χ) of
Nichols algebras of diagonal type defined in [35] (respectively again quasi-Hopf algebra versions,
depending on the quadratic form) give rise to such modular tensor categories with semisimple
part C. Of course there might be further examples not coming from quasi-Hopf algebras.
The intend behind the present article is to conduct a sweep for all possible examples, in
which V is still the lattice vertex algebra and the Nichols algebra involved is not necessarily
associated to a quantum group, but finite-dimensional. Thereby we encounter most (but not
all) finite-dimensional Nichols algebras of diagonal type. Our guess would be that the kernel
of screenings has the same modular tensor category of representations as the associated U(χ).
Where we have parametrized families of solutions (in particular in Lie and super-Lie type) we
have ranges of parameters in which some or all truncation relations in the Nichols algebras do
not hold. One would expect the associated W to have a representation category related to
the relative Drinfeld center of this algebra, or, more algebraically spoken, to the category of
representations of a mixed quantum group, see, e.g., [29, 31] and [10, 48]. A further step would
be to consider the folded Nichols algebras in [47] over central extensions of abelian groups, for
which V should be taken to be the respective orbifold of a lattice vertex algebra.
The paper is organized as follows: In Section 2 we present preliminaries on Nichols algebras
and in particular on the Weyl groupoid of reflections for a braiding of diagonal type.
In Section 3 we briefly present the notion of a vertex algebra VΛ, its representation theory
and the action of screening operators on it. Then we review the main result and its proof
in [49] that non-local screenings Zi = Zai for the top elements eai in modules of the Heisenberg
6 I. Flandoli and S.D. Lentner
vertex algebra fulfill the Nichols algebra relations, if the symmetric matrix mij = (ai, aj) is
subpolar (Definition 3.3), which puts lower bounds on the sums of mij over all subsets J of
indices and ensures the convergence of the relevant integrals and related hypergeometric series.
As we show, subpolarity of (mij) and of all matrices (mι
ij) obtained from (mij) by repeating
some rows and columns follows, if the matrix (mij) is positive definite and all |mii| ≤ 1. Under
the assumption of subpolarity, the Nichols algebra relations for screening operators follow by
expanding monomials of screening operators in terms of certain integrals F((mi), (mij)), which
in turn can be rewritten as a quantum symmetrizer of another integral F̃((mi), (mij)), which
exists for subpolar parameters.
In Section 4 we want to improve the results reviewed in the last section by analytically
continuing F̃((mi), (mij)) to parameters (mij) beyond subpolarity and study which of the most
common Nichols algebra relations still hold for screening operators: In Section 4.1 we take
an explicit analytic continuation in terms of Beta functions and show that the commutativity
relations always continue to hold (see however Example 4.6 for a counterexample). In Section 4.2
we take an explicit analytic continuation in terms of Gamma functions using the Selberg integral
formula [56] and thereby we show that the truncation relations of simple root vectors continue to
hold iff mii > 0. In Section 4.3 we derive (somewhat similar to the Gamma function) a recursive
formula for generalized Selberg integrals for n = 3 and thereby we obtain a non-explicit analytic
continuation with explicit simple poles in the region m12 + m23 + m13 > −2 and m12 > −1.
We would expect this ansatz to work in general if the subpolarity condition holds merely for
J = I. This result implies our result on the Serre relations for realizations (mij) of Cartan type
for m > 0, while for m < 0 one would have to consider additional boundary terms, and for the
realization (mij) of super Lie type for m < 0, while for m > 0 the subpolarity condition fails
even for J = I. On the other hand, we use the An-Selberg integral formula in [60, 62] which
produces an explicit expression in terms of Gamma functions for realizations of type An, but
only for a certain linear combination of generalized Selberg integrals. For this reason we can
only put this to use for Serre relations of Cartan type for m < 0. On the other hand this second
approach has the potential to give explicit answers for, say, truncation relations of non-simple
root vectors in Cartan type An.
In Section 5 we formulate the precise conditions under which we call (mij) (resp. the associ-
ated lattices Λ) a realization of a given braiding (qij). We also introduce the notions of a pair of
vertices (i, j) resp. a root α being m-Cartan or m-truncation, according to the respective notions
of being q-Cartan and q-truncation in Section 2, and we discuss the relations between these no-
tions. We visualize a realization by adding the rational numbers mii, mij +mji = 2mij below
a q-diagram with decorations qii, qijqji. For example, the Cartan type realization of uq(g)
+ in
type C3 in the next section with q = eπim is depicted as
q2
2m
q2
2m
q4
4m
q−2
−2m
q−4
−4m
In Section 6 we address braidings and Nichols algebras of Cartan type, which give the Borel
parts uq(g)
+ of the small quantum groups. We rescale the root lattice of g by a parameter m
and prove that this lattice Λ with mij = (αi, αj)gm always realizes this braiding. Then we aim
to calculate the algebra of screening operators depending on m. We establish that subpolarity
holds for 1
2d ≥ m > 0. Beyond these values, the truncation relations of simple root vectors hold
for m > 0 and the Serre relations (for g simply laced) hold regardless of m. Regrettably, we
are neiter able to check the truncation relations for non-simple root vectors (although we would
assume them to hold accordingly) nor the additional relations that appear for certain small
orders of q [3]. For both we would either require analytic continuation of more complicated
integrals (extending the techniques discussed in Section 4) or a reflection theory of screening
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 7
operators, which links the screening operator expression associated to a non-simple root to the
corresponding screening operator of a simple root after a change of base in Λ by reflection. On
the other hand, we regard the possibly failing additional relations to be interesting candidates
for highly nontrivial extensions or liftings of Nichols algebra appearing as algebras of screenings.
In Section 7 we proceed as in the previous section, but for Lie superalgebras. In Definition 7.5
we consider the standard chamber, in which there is a single fermionic simple root αf and make
an ansatz for a matrix (mij) depending in general on 2 parameters m′, m′′ corresponding to
a rescaling of the root lattices of the subsystems generated by α1, . . . , αf−1 and αf+1, . . . , αr and
mff = 1. To prove that this is indeed a realization, we formulate in Corollary 7.8 a condition
for all roots, which we prove in Corollary 7.12 to hold automatically except in four explicit
situation. When we go through all root systems in Sections 7.5 and 7.6, we determine these
open situations and compute additional conditions relatingm′, m′′ such that also these situations
hold. We then verify that the subsequent conditions relating q′, q′′ are indeed the conditions
given in Heckenberger’s list; one could say that we derive the logarithmic versions of these
conditions. For example for A(n′, n′′) we find m′ +m′′ = 1 and correspondingly the condition
q′q′′ = −1.
Proving that (mij) is the unique realization for large order of q in Lemma 7.14 is on the other
hand a simple application of our result in Cartan type. Again, for small order of q we can have
roots that are both q-Cartan and q-truncation, and correspondingly multiple solutions (mij),
for which these roots are possibly either m-Cartan or m-truncation.
In Section 8 we construct realizing lattices for all other finite-dimensional diagonal Nichols al-
gebras in rank 2. In Section 8.1 we construct realizations (mij) and check that they are com-
patible under reflection. The first case (row 6 in [33, Table 1]) corresponds to a Z3-(color-)Lie
algebra [4, 64], it has a free parameter q and is accordingly again realized by a 1-parameter
family of lattices. As for the Lie and Lie super type we get a Nichols algebra for m > 0,
while for m < 0 some truncation relations fail. For q = −1 there are again additional relations
in [3] that we cannot account for. For all other Nichols algebras in rank 2 we find a unique
realization, subpolarity holds and the screening algebra is the Nichols algebra. In Section 8.2
we show that the examples presented in the previous sections exhaust all realizing lattices for
rank 2 finite-dimensional diagonal Nichols algebras braiding and that the classification is thus
complete.
Finally in Sections 9 and 10 we generalize the construction and classification to rank 3 and
explain how the answer in rank ≥ 4 can be obtained in each instance. We conclude by a table
of all realizations in rank 2 and 3.
2 Preliminaries on Nichols algebras
We start by giving the basic definitions and examples regarding Nichols algebras of diagonal
type. For a thorough account, we refer the reader to the text book [38] and the survey [3], which
includes generators and relations for each Nichols algebra of diagonal type.
2.1 Definition and properties
In this article we work over the field of complex numbers C. A braided vector space (M, c) is
a vector space together with a braiding c : M ⊗M → M ⊗M , which is a linear map that fulfills
the braid relation or Yang–Baxter equation
(id⊗ c)(c⊗ id)(id⊗ c) = (c⊗ id)(id⊗ c)(c⊗ id).
8 I. Flandoli and S.D. Lentner
Hence, a braided vector space comes with an action ρn of the braid group Bn on M⊗n by acting
on the i-th and (i+ 1)-th tensor factor:
ci,i+1 := id⊗ · · · ⊗ c⊗ · · · ⊗ id.
In this article we restrict ourselves to braidings of the following type:
Definition 2.1. Let M be a vector space of dimension r, later called rank, and with a fixed
basis x1, . . . , xr. Let (qij) for i, j = 1, . . . , r be an arbitrary matrix with qij ∈ C×. With this
data we define a braiding of diagonal type on M via
c : c(xi ⊗ xj) = qijxj ⊗ xi.
Definition 2.2. Let (M, c) be a braided vector space. There is a canonical projection Bn ↠ Sn,
which sends the braiding ci,i+1 to the transposition (i, i+1). The set-theoreticMatsumoto section
s : Sn → Bn is defined by (i, i+ 1) 7→ ci,i+1 and by the property s(xy) = s(x)s(y) whenever the
length of xy (as a shortest word in a finitely presented group) is the sum of the lengths of x
and y. Then we define the quantum symmetrizer as a linear endomorphism of M⊗n by
Xq,n :=
∑
τ∈Sn
ρn(s(τ)),
where ρn is the representation of Bn on M⊗n induced by the braiding c. Then the Nichols
algebra or quantum shuffle algebra generated by (M, c) is defined by
B(M) :=
⊕
n
M⊗n/ ker(Xq,n).
This particular definition of Nichols algebras is due to Woronowicz [63] and Rosso [55], and
in this way Nichols algebras will appear in the present article. It enables one in principle to
compute B(M) in each degree, but it is very difficult to find generators and relations for B(M),
since in general the kernel of the map Xq,n is hard to calculate in explicit terms. In fact B(M)
is a Hopf algebra in a braided sense and as such it enjoys several equivalent universal properties,
see, e.g., [38, Chapters 1 and 7].
2.2 Examples
We now present some examples.
Lemma 2.3. For a diagonal braiding (qij) we have explicitly
Xq,n :=
∑
σ∈Sn
q(σ)σ, q(σ) =
∏
i<j, σ(i)>σ(j)
qij .
As an immediate consequence:
Lemma 2.4 (Rank 1). Let M = xC be a 1-dimensional vector space with braiding given by
q11 = q ∈ C×, then
C ∋ Xq,n =
∑
τ∈Sn
q
|τ |
11 =
n∏
k=1
1− qk
1− q
=: [n]q!.
This polynomial has zeros at all q ̸= 1 of order ≤ n, so the Nichols algebra is
B(M) =
{
C[x]/
(
xℓ
)
, for q ∈ Gℓ, ℓ > 1,
C[x], else,
where we denote by Gℓ the set of primitive ℓth root of unity.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 9
Example 2.5 (quadratic relations). Let xi, xj ∈ M . Then we have a quadratic relation
xixj − qijxjxi = 0
in the Nichols algebra iff the double braiding is trivial qijqji = 1. For example, the braiding
matrices with all entries +1 or −1 have as Nichols algebra the polynomial algebra or the exterior
algebra, respectively.
We introduce the notation of the q-commutator
[xi1 · · ·xim , xj1 · · ·xjn ]q
:= (xi1 · · ·xim)(xj1 · · ·xjn)−
( ∏
1≤a≤m
1≤b≤n
qiajb
)
(xj1 · · ·xjn)(xi1 · · ·xim). (2.1)
Definition 2.6. Starting with [34], a braiding of diagonal type (qij) can be encoded into a graph
decorated with complex numbers: Each node corresponds to an element xi, 1 ≤ i ≤ r in the
basis of M and is decorated with the complex number qii (self braiding). The edge between
any xi and xj is decorated with the complex number qijqji (double braiding):
q11 q22q12q21
For qijqji = 1 we do not draw the edge; as discussed above, in this case xi, xj commute up to the
factor qij . For qijqji ̸= 1 we call the vertices connected i ∼ j. The authors call this a q-diagram
to distinguish it from the notion of a Dynkin diagram in the next subsection.
A deeper reason behind this definition is that the q-diagram captures the information that
essentially determines the structure of the Nichols algebra. Nichols algebras with the same q-
diagram are not isomorphic in general, but they have, e.g., the same dimension. In a suitable
sense, they are equivalent up to a 2-cocycle twist.
Example 2.7 (quantum group, [8, 50, 55]). Let g be a finite-dimensional complex semisimple
Lie algebra of rank r with root system Φ and simple roots α1, . . . , αr and Killing form (αi, αj).
Let q be a primitive ℓ-th root of unity. Consider the r-dimensional vector space M with diagonal
braiding qij := q(αi,αj). Then the Nichols algebra B(M) is isomorphic to the positive part uq(g)
+
of the small quantum group uq(g), which is a quotient of the deformation of the universal
enveloping of a Lie algebra U(g).
As an example, the q-diagrams for g of type A1 ×A1, A2 and B2 are
q2 q2 q2 q2q−2 q2 q4q−4
2.3 Generalized root system and Weyl groupoid
Every Nichols algebra, which is finite-dimensional and of diagonal type, comes with a generalized
root system, a Cartan graph, a Weyl groupoid and a PBW-type basis [34, 36, 39]. For the same
statement beyond diagonal type see [7]. These structures play in many respects a similar role
as root system, Cartan matrix and Weyl group play for Lie algebras.
Before giving the definitions relevant to this article, we summarize how Weyl groupoids and
generalized root systems are explained geometrically in [15]: A root system in the usual sense is
an arrangement of hyperplanes in some Euclidean vector space Rr and a choice of normal vectors
thereof, called roots, which is stable under the reflections on these hyperplanes and fulfills some
10 I. Flandoli and S.D. Lentner
integrality condition. The connected components of the complement of all hyperplanes are called
Weyl chambers, and the normal vectors of the walls of any fixed Weyl chamber give a basis of
the vector space, a set of simple roots. The reflections act transitively on the Weyl chambers
and every roots has integer coefficients with respect to every fixed set of simple roots.
In a generalized setting, called a crystallographic arrangement, we drop the euclidean product
of the ambient vector space (so the roots are a choice of linear functions), but we keep demanding
that every root has integer coefficients with respect to every set of simple roots. The reflection σi
on the i-th wall is the associated linear map that sends the simple root αi to −αi and αj , i ̸= j
to αj − cijαi, where −cij is the maximal value for which this is a root. In contrast to usual
root systems, the set of all roots, written in different bases of simple roots, may not always
give the same set of coordinate tuples. Consequently the reflections generate a Weyl groupoid,
whose objects are different types of Weyl chambers, and each object has attached its own Cartan
matrix cij depicted by a Dynkin diagram. Such data has been axiomatized and classified under
the name Cartan graph [7, 35, 36, 39], which we now discuss. Actually, this behaviour is already
familiar from Lie superalgebras, where different sets of simple roots contain different parities,
and in extreme cases such as D(2, 1;α) even different Dynkin diagrams, see Example 2.18.
In the following account we follow [35, Section 3.2] or [38, Chapters 9, 10 and 15], where more
details can be found:
Definition 2.8 (generalized Cartan matrix). Let I := {1, . . . , r}, where r is called rank, and
{αi | i ∈ I} the standard basis of ZI . A generalized Cartan matrix C = (cij)i,j∈I is a matrix in
ZI×I such that
(M1) cii = 2 and cjk ≤ 0 for all i, j, k ∈ I with j ̸= k,
(M2) if i, j ∈ I and cij = 0, then cji = 0.
In the following definition we think on A as a set of Weyl chambers.
Definition 2.9 (Cartan graph). Let A be a non-empty set, ρi : A → A a map for all i ∈ I, and
Ca = (cajk)j,k∈I a generalized Cartan matrix in ZI×I for all a ∈ A. The quadruple
C = C
(
I, A, (ρi)i∈I , (C
a)a∈A
)
is called a Cartan graph if
(C1) ρ2i = id for all i ∈ I,
(C2) caij = c
ρi(a)
ij for all a ∈ A and i, j ∈ I.
Definition 2.10 (Weyl groupoid). Let C = C
(
I, A, (ρi)i∈I , (C
a)a∈A
)
be a Cartan graph. For all
i ∈ I and a ∈ A define σa
i ∈ Aut(ZI) by
σa
i (αj) = αj − caijαi for all j ∈ I.
The Weyl groupoid of C is the category W(C) such that Ob(W(C)) = A and the morphisms
are compositions of maps σa
i with i ∈ I and a ∈ A, where σa
i is considered as an element in
Hom(a, ρi(a)). The cardinality of I is called the rank of W(C).
A Cartan graph axiomatizes a set of Cartan matrices, one for every Weyl chamber (or every
type of Weyl chamber) a ∈ A, and reflections σa
i on simple roots αi in the Weyl chamber a, which
are linear maps between a space ZI attached to a and to the Weyl chamber after reflection ρi(a).
Let C be a Cartan graph. For all a ∈ A define the set of real roots at a by
(Rre)a =
{
σi1 · · ·σik(αj) | k ∈ N0, i1, . . . , ik, j ∈ I
}
⊆ ZI .
A real root α ∈ (Rre)a is called positive if α ∈ NI
0.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 11
Definition 2.11 (root system). Let C = C
(
I, A, (ρi)i∈I , (C
a)a∈A
)
be a Cartan graph. For all
a ∈ A let Ra ⊆ ZI , and define ma
i,j = |Ra ∩ (N0αi + N0αj)| for all i, j ∈ I and a ∈ A. We say
that
R = R(C, (Ra)a∈A)
is a root system of type C, if it satisfies the following axioms.
(R1) Ra = Ra
+ ∪ −Ra
+, where Ra
+ = Ra ∩ NI
0, for all a ∈ A.
(R2) Ra ∩ Zαi = {αi,−αi} for all i ∈ I, a ∈ A.
(R3) σa
i (R
a) = Rρi(a) for all i ∈ I, a ∈ A.
(R4) If i, j ∈ I and a ∈ A such that i ̸= j and ma
i,j finite, then (ρiρj)
ma
i,j (a) = a.
Lemma 2.12. Let C be a Cartan graph and R a root system of type C. Let a ∈ A. Then for all
i ̸= j
−caij = max
{
m ∈ N0 |αj +mαi ∈ Ra
+
}
.
As a convention, we name a positive root α by indicating with which multiplicity each simple
root αi appears in α (in some fixed Weyl chamber), e.g.,
α12 = α1 + α2, α123 = α1 + α2 + α3, α122 = α112 = 2α1 + α2, . . . .
For roots α, β ∈ Ra we can define a Cartan matrix entry independent of a
−cα,β = max{m ∈ N0 |α+mβ ∈ R+}.
The root system R is called finite iff for all a ∈ A the set Ra is finite. By [16] if R is a finite
root system of type C, then R = Rre, and hence Rre is a root system of type C in that case.
Example 2.13 (Cartan type). Let g be a semisimple finite-dimensional complex Lie algebra.
It is well-known that is uniquely determined (up to isomorphisms) by its root system, which is
the root system of a finite Weyl group W . The corresponding Cartan graph has exactly one
object a and Ca is the Cartan matrix of W . The set Ra is the root system of W . Alternatively,
we can consider the Cartan graph with one object a for each Weyl chamber and the Cartan
matrices attached to all objects are equal.
The finite Weyl groupoids are classified in [16, 17]; apart from the finite Weyl groups there are
an infinite family in rank 2, parametrized by triangulations of n-gons, an additional series Dn,m
and 74 sporadic examples. It is proven in [15, Theorem 1.1] that the crystallographic arrange-
ment mentioned in this section’s introduction are in bijection to connected simply connected
Cartan graphs for which the set of real roots is finite.
We now discuss how Cartan graphs are attached to Nichols algebras of diagonal type, see
[35, Section 3.3] respectively [38, Chapter 15], and we introduce some additional notation for
our purpose:
Lemma 2.14. Suppose for some m ∈ N0 holds q−m
ii = qijqji or q
(1+m)
ii = 1, then in the Nichols
algebra B(qij) the quantum Serre relation holds
(adc xi)
1+mxj = 0
with (adc xi) := [xi, y]c the braided commutator defined above.
12 I. Flandoli and S.D. Lentner
Definition 2.15. To every braiding matrix (qij) we define the associated Cartan matrix (cij)
for all i ̸= j by
cii = 2 and cij := −min
{
m ∈ N0 | q−m
ii = qijqji or q
(1+m)
ii = 1, qii ̸= 1
}
.
We assume from now on that cij is finite. We call two vertices i ̸= j connected i ∼ j iff cij ̸= 0.
We need technical terms to refer to these two conditions. More global terms will be discussed
in Definition 2.19. We add a prefix q- to stress the dependence on the braiding (qij) and to
distinguish them from the respective notions with prefixm- depending on the inner product (mij)
of the lattice.
Definition 2.16. For a braiding matrix (qij), we call an ordered pair of indices (i, j) with i ̸= j
to be q-Cartan if the first condition holds in the minimum m, and we call it q-truncation if the
second condition holds in the minimal m, i.e., if
q
cij
ii = qijqji, or q
1−cij
ii = 1.
A pair (i, j) can be both q-Cartan and q-truncation, namely iff qijqji = qii. Otherwise we will
call a pair only q-Cartan, respectively only q-truncation.
We remark that if qijqji is a power of qii at all, then (i, j) is already q-Cartan.
Pairs (i, j) with i ̸∼ j are always q-Cartan and never q-truncation for qii ̸= 1.
The matrix (cij) associated to (qij) is a generalized Cartan matrix. The braiding matrix (qij)
can be extended uniquely to a bicharacter q : Zr ×Zr → C× with q(αi, αj) = qij . A base change
by precomposing with a reflection σk gives a new bicharacter and braiding matrix (q′ij) = rk(qij)
defined by
q′ij := q(σk(αi), σk(αj))
A short but important calculation [32, equation (3)] gives
q′kk = qkk, q′ii = qii · pckiki , q′kiq
′
ik = qkiqik · p−2
ki , q′ijq
′
ji = qijqji · p
ckj
ki p
cki
kj ,
where, in our previous wording,
pki :=
{
1, if (k, i) is q-Cartan,
q−1
ii qijqji if (k, i) is q-truncation.
Corollary 2.17. If (k, i) is q-Cartan for all i ̸= k, then rk(qij) = (qij).
The set A of all braiding matrices arising in this way together with the maps rk defines
a Cartan graph. The Nichols algebras associated to these different braiding matrices are not
isomorphic, but they have the same dimension (if finite), an isomorphic Drinfeld double, and
are closely related by [13, 35, 37]. By [7, Remark 2.8] and [35, Theorem 3.13] we can write as
Nr
0-graded vector spaces
B(qij) =
⊗
α∈R+
C[xα]/
(
xord(q(α,α))α
)
and then the set of degrees R+ is a root system for this Cartan graph. In particular, B(q) has
finite dimension if the root system R+ is finite and all ord(q(α, α)) are finite.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 13
Example 2.18 (D(2, 1;α), [39]). We consider, as an example, the finite-dimensional diagonal
Nichols algebra of rank 3 with a braiding (qij) in an initial Weyl chamber, which has the following
properties
qii = −1, qijqji = ζ,
with i ̸= j and ζ ∈ G3 a primitive third root of unity. As q-diagram
−1
−1 −1
ζ ζ
ζ
As it turns out, the overall root system has seven positive roots. If {α1, α2, α3} are the simple
roots in the Weyl chamber shown above, then the positive roots in this basis are
{α1, α2, α3, α12, α23, α13, α123}
and the Cartan matrix of this Weyl chamber is
(
cIij
)
=
2 −1 −1
−1 2 −1
−1 −1 2
.
We now reflect around α2. Then the new simple roots are {α12,−α2, α23}. We compute the
new q-diagram r2(qij) = (q′ij) using this basis transformation and the extension of (qij) to
a bicharacter q:
q′22 = q(−α2,−α2) = q22 = −1,
q′11 = q(α12, α12) = (q12q21)q11q22 = ζ,
q′33 = q(α23, α23) = (q23q32)q33q22 = ζ,
q′12q
′
21 = q(α12,−α2)q(α12,−α2) = (q12q21)q
−2
22 = ζ−1,
q′32q
′
23 = q(α23,−α2)q(α23,−α2) = (q23q32)q
−2
22 = ζ−1,
q′13q
′
31 = q(α12, α23)q(α23, α12) = (q12q21)(q23q32)(q13q31)q
2
22 = 1.
ζ −1 ζζ−1 ζ−1
In this new basis the positive roots are
{α12,−α2, α23, α1, α3, α123, α13}
and the new Cartan matrix is hence
(
cIIij
)
=
2 −1 0
−1 2 −1
0 −1 2
.
Even though this particular Cartan matrix is of type A3, the root system has one root more
than the root system A3. In the following figure we show the hyperplane arrangement of the
root system in an affine resp. projective picture:
14 I. Flandoli and S.D. Lentner
Each of the seven projective lines corresponds to the hyperplane through the origin orthogonal
to one root. Each triangle is a Weyl chamber with the three adjacent hyperplanes corresponding
to the three simple roots. Equilateral triangles (white) correspond to the Cartan matrix I and
right triangles (grey) to the Cartan matrix II.
We now introduce some more properties of roots in a root system associated to a braiding
matrix (qij), which are relevant to this article.
Definition 2.19.
1. A root α is called q-Cartan root, if in every Weyl chamber containing α = αi as a simple
root, all pairs of vertices (i, j) with i ∼ j are q-Cartan.
2. A root α is called q-truncation, if in every Weyl chamber containing α = αi as a simple
root, all pairs of vertices (i, j) with i ∼ j are q-truncation.
3. A root α is called only q-Cartan root, if in every Weyl chamber containing α = αi as
a simple root, all pairs of vertices (i, j) with i ∼ j are only q-Cartan.
The term Cartan- was coined in this context by [9] as Cartan vertex αi (meaning that all
pairs of vertices (i, j) are q-Cartan). The term Cartan root α appears first in [10]. In [5] initially
the independence of this notion from the basis of simple roots was proven implicitly. The authors
thank I. Angiono for explaining the following much simpler proof that uses a base-independent
characterization of the term:
Proposition 2.20. A root α is a q-Cartan root iff the values q(α, β)q(β, α) for all β ∈ Zr lie
in the multiplicative group generated by q(α, α).
As a consequence, suppose in some Weyl chamber, which containing α = αi as a simple root,
that all pairs of vertices (i, j) with i ∼ j are q-Cartan. Then the same is already true in all such
Weyl chambers and α is a Cartan root.
The other two notions we defined above do not have such nice characterizations (and are
probably less fundamental) and require the knowledge of the entire root system. The following
special case is frequent and easy to recognize:
Example 2.21 (fermionic root). A root α with q(α, α) = −1 is q-truncation.
Remark 2.22. Only very few root systems admit roots α that are neither q-Cartan nor q-
truncation. In such a case, there has to exist a Weyl chamber containing α = αi as a simple
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 15
root and the following type of rank 3 subdiagram
q
αi
q−a q′
where q−a ̸= q (in particular q ̸= −1), so the pair (2, 1) is q-Cartan and not q-truncation and
where q′ is no power of q, so the pair (2, 3) is q-truncation and not q-Cartan. Such a chamber
must exist, otherwise α is q-truncation, and then such a neighbor must exist in this chamber,
otherwise α is q-Cartan in all chambers.
A quick inspection of all q-diagrams of rank 3 thus shows that roots being neither q-Cartan
nor q-truncation appear for [33, rows 16 and 17], and consequently for q-diagrams of rank ≥ 4
containing either of these.
Roots α that are only q-Cartan are q-Cartan roots such that ord(q(α, α)) > 1−cij in all Weyl
chambers containing α = αi as a simple root, and all j ∼ i. For example for Nichols algebras of
Cartan type uq(g)
+ we have (see Proposition 6.4)
g q both q-Cartan and q-truncation
An q2 = −1 all roots
Bn, Cn, F4 q4 = −1 long roots
Bn, Cn, F4 q2 ∈ G3 short roots
G2 q6 = −1 long roots
G2 q2 ∈ G4 short roots
It is interesting, that these are precisely the cases, in which the Nichols algebras need additional
relations, see [3].
3 Preliminaries on screening operators
3.1 Vertex algebras and their representation theory
A vertex operator algebra (VOA) is, roughly speaking, a commutative algebra that depends
analytically on a complex variable z. More precisely, a vertex operator algebra V is an infinite-
dimensional graded vector space with a linear map
Y: V ⊗C V → V
[[
z, z−1
]]
,
where V
[[
z, z−1
]]
denotes Laurent series in a formal variable z with coefficients in V. The
axioms of a vertex operator algebra include a version of commutativity or locality, which relates
Y(a, z)Y(b, w) and Y(b, w)Y(a, z) for z, w, z − w ̸= 0. As an implication, one also has a version
of associativity, which relates these two expressions to Y(Y(a, z−w)b, w). An additional axiom
requires that conformal transformations of the variable z in Y(a, z) are compatible with an action
of the Virasoro algebra on V, which is part of the data. Standard mathematical textbooks on
vertex operator algebras include [25, 44]. Vertex operator algebras are motivated by physics,
where they describe the holomorphic (chiral) part of a 2-dimensional quantum field theory with
conformal symmetry.
There is a straightforward notion of a module W over a vertex algebra V. Under certain
finiteness-assumptions on a vertex operator algebra V, the category of V-modules has a tensor
product ⊠ and a braiding [42].
16 I. Flandoli and S.D. Lentner
Example 3.1. The easiest example of a vertex operator algebra is the Heisenberg algebra Hr
based on an r-dimensional Euclidean vector space Rr. The irreducible modules Ha are para-
metrized by vectors a ∈ Rr, with H0 = Hr, tensor product Ha⊠Hb = Ha+b, and braiding given
by the scalar eπi(a,b).
From the perspective of our article, this is already an interesting vertex operator algebra: In
the next section we will define screening operators Zai , and the idea of this article is to analyze
the algebra generated by these screening operators, which will be largely determined by the
braidings qij = eπi(ai,aj).
We also introduce some more vertex algebras that eventually come into play, and which
motivate our work:
Example 3.2. For every even integral lattice Λ with inner product ( , ), it is possible to associate
a lattice vertex algebra VΛ. Its category of representations is equivalent to the category of Λ∗/Λ-
graded vector spaces, with associator ω and braiding σ associated to a quadratic form eπi(a,a),
a ∈ Λ∗. If we restrict these modules to the Heisenberg algebra Hr ⊂ VΛ, the modules decompose
and we have Va+Λ =
⊕
a′∈a+ΛHa′ .
Both examples are treated in [25, 44] at the level of vertex algebras and modules. The
standard reference for the tensor product and braiding in both examples is [18]. See also the
overview in [14, Section 2.3].
3.2 Screening operators and Nichols algebra relations
We are now going to define the main protagonists of this paper, the screening operators. They
go back to [19] and appear throughout vertex operator literature. Our main focus are screening
operators for elements in a module different from the vacuum module, and we call those non-local
screening operators. We introduce them from a slightly novel perspective:
Given V a VOA, W module of V and w ∈ W . The tensor product W ⊠ U with some other
module U is defined in [41] by the universal property that there exists an intertwining operator
Y(w, z) : W ⊗ U → (W ⊠ U)[logz]{{z}},
where {{z}} denotes power series with arbitrary complex exponents, and their matrix elements
are multivalued analytic functions on C\{0}. If we evaluate Y on our fixed element w ∈ W , we
get a map
Y(w, z) : U → (W ⊠ U)[logz]{{z}}.
Integrating the variable z over the unit circle around z = 0, lifted to a path in the multivalued
covering, we get linear maps into the algebraic closure
Zw : U → W ⊠ U.
These maps are called screening operators. If W = V is the vertex algebra itself, then the
intertwining operator is simply the vertex operator, which is a power series with integer expo-
nents and without logarithms; in this case the integration returns simply the z−1-coefficient or
residue of Y. If we ask in addition that the conformal weight is h(w) = 1, then the commutator
formula shows that the screening operator commutes with the action of the Virasoro algebra.
On the other hand, for non-local screening operators, as defined above, all non-integer z-powers
contribute to the integral in the multivalued covering, which is why we get a result in the alge-
braic closure. Also the consequence of h(w) = 1 is more subtle, one would expect that at least
certain suitable powers of screening operators commute with the Virasoro algebra. Defining and
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 17
studying non-local screening operators in this way is new and might appear strange – however,
non-local screening operators do appear prominently in literature for a long time. For example,
the Felder complex [22] consists of suitable powers of non-local screening operators.
If the singularity of products Y(w1, z1) · · ·Y(wn, zn) at points zi = zj is mild enough so that
certain integrals converge (subpolar, see below), then we expect these screening operators to
fulfill the relations of the Nichols algebra B(W ) associated with the module W and its braiding
in the representation category Rep(V), which expresses the multivaluedness (or non-locality) of
the intertwining operator. If the singularity at z = 0 is more severe, then the screening operators
should generate some algebra extension of the Nichols algebra.
Establishing this result for general VOAs is work in progress, but for Heisenberg and lattice
VOA and their explicit intertwiners, it is proven in [49]:
Definition 3.3 ([49, Definition 5.4]). Let I = {1, . . . , n}. A set of complex parameters (mij)
with 1 ≤ i ≤ j ≤ n is called subpolar, if for every subset J ⊂ I with |J | ≥ 2 the following
inequality holds:∑
i<j, i,j∈J
Re(mij) > −|J |+ 1.
As a weakening, (mij) is called subpolar on intervals with respect to a total order of the index
set, if the inequality holds for all J ⊂ I with |J | ≥ 2 which have the form of an interval
J = [a, b] = {x | a ≤ x ≤ b}.
We later consider monomials xι(1) · · ·xι(n) of total degree n in r variables x1, . . . , xr. Corre-
spondingly, we now consider the following sets of parameters:
Definition 3.4. Suppose (mij) with 1 ≤ i ≤ j ≤ r is given. Then for any map
ι : {1, . . . , n} → {1, . . . , r}
we define a new set of complex parameters (mι
i′j′)
mι
i′j′ := mι(i′),ι(j′), 1 ≤ i′ ≤ j′ ≤ n.
Up to permuting indices, (mι
i′j′) is determined by (mij) and the degrees of ι
dιi :=
∣∣ι−1(i)
∣∣, r∑
i=1
dιi = n.
Remark 3.5. Λ = Zr with basis α1, . . . , αr and inner product (α1, αj) = mij . Then, as
a geometric reformulation, subpolarity of mι
i′j′ is equivalent to the cube
∏r
i=1[0, di] intersecting
the ball defined by
1
2
(α, α)−
(
α, ρ− ρ∨
)
≤ 1
only in α = 0 and α = αi, which is on the boundary, if ρ, ρ∨ are vectors in the complexification
of Λ satisfying (αi, ρ) = mii and
(
αi, ρ
∨) = 1.
Lemma 3.6 ([49, Lemma 5.5]). If Λ is positive definite and (ai, ai) ≤ 1 for a fixed basis ai,
then (mij) and (mι
i′j′) for all ι are subpolar.
18 I. Flandoli and S.D. Lentner
Theorem 3.7 ([49, Theorem 7.1]). For a non-integral lattice Λ of rank r and elements a1, . . . ,
an ∈ Λ, we consider the elements eai in modules of the associated Heisenberg VOA H. The
braiding between two elements is
eai ⊗ eaj 7→ qij e
aj ⊗ eai , where qij := eiπmij , mij := (ai, aj).
Consider the diagonal Nichols algebra B(qij) with braiding matrix (qij) and generators xi with
1 ≤ i ≤ r. Then for any relation in the Nichols algebra, homogeneous in degree (d1, . . . , dr) ∈ Nr,
the same relation holds for the screening operators Zai, assuming that
(
mι
ij
)
is subpolar for
di = ι−1(i).
Example 3.8. In the case Λ = 1√
pΛg, with Λg the root-lattice of a complex finite-dimensional
simple Lie algebra g of rank r, and ℓ = 2p an even integer, we obtain as B(qij), the small
quantum group uq(g)
+, where q is a primitive ℓ-th root of unity and the braiding is
qij = e
iπ
(
1√
p
αi,
1√
p
αj
)
= e
2iπ
ℓ
(αi,αj) = q(αi,αj),
for α1, . . . , αr the basis of simple roots in Λg.
Since we want to improve Theorem 3.7, we first recall the steps of the proof in [49]: Any
iteration of screening operators (in a rather general type of vertex algebra) can by Theorem 4.3
cit. loc. be expanded as follows
Theorem 3.9.(
n∏
i=1
Zai
)
v =
∑
(mi),(mij)
∑
(ki)∈Nn
[(ki), (mi), (mij), (ai)] · F((mi + ki), (mij)),
where we introduce
[(ki), (mi), (mij), (ai)]
:= v(n+1)
1∏
i=n
∂ki
ki!
a
(n+1)
i
∏
1≤i≤n
〈
a
(1)
i , v(i)
〉
mi
∏
1≤i<j≤n
〈
a
(n−j+2)
i , a
(n−i+1)
j
〉
mij
,
F((mi), (mij)) :=
∑
(kij)∈N
(n2)
0
n∏
i=1
res
(
z
(mi+ki)+
∑
i<j(mij−kij)+
∑
j<i kji
i
)∏
i<j
(−1)kij
(
mij
kij
)
.
The notation in these formulae is as follows:
� [(ki), (mi), (mij), (ai)] as well as (ai), v are elements in modules of the vertex algebra, which
is modeled by some (in our case commutative, cocommutative) Hopf algebra, see Section 3
cit. loc. Expressions a(1) ⊗ a(2) denote in Sweedler notation the iterated coproduct and
⟨−,−⟩m are certain pairings, nonzero only for a finite number of values mi,mij ∈ C over
which we sum. The only properties of [(ki), (mi), (mij), (ai)] we require in the following
are
– It is invariant under permutation of the index set, i.e., simultaneously permuting (ki),
(mi), (mij), (ai), if commutativity and cocommutativity is in place. Thus commuta-
tivity of screening operators depends entirely on F((mi), (mij)).
– It is invariant under permuting only the (ki), if the permutation preserves the parti-
tion of the index into subsets of indices with equal ai. In particular, the product of
screenings only depends on the symmetrization F((mi), (mij))
sym over these subsets
of indices.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 19
� F((mi), (mij)) is an infinite
(
n
2
)
-fold series of complex numbers (a generalized hyperge-
ometric series on the boundary of its convergence disc) depending on a set of complex
parameters (mij) and (mi) for 1 ≤ i < j ≤ n. The symbol Res(zm) denotes a formal
residue of the possibly multivalued function
Res(zm) :=
0, m ∈ Z\{−1},
1, m = −1,
(e2πi(m+1) − 1)
2πi(m+ 1)
, m ̸∈ Z.
Convergence of this series is rather subtle and holds for example if (mij) is subpolar, see
Lemmas 5.2 and 5.22 cit. loc.
Note that the quantity (mij) is coupled to the pairing of ai, aj, and the quantity (mi) is coupled
to the pairing of ai with the element v acted upon.
We call F((mi), (mij)) the quantum monodromy numbers, and they play in some analytical
sense the role of structure constants for screening operators. They can be expressed as an integral
of a multivalued function in n complex variables over a suitable lift of (S1)n to the multivalued
covering. The integral converges for subpolar (mij), see Section 5.2 cit. loc.
F((mi), (mij)) =
∫
· · ·
∫
[e0,e2π ]n
dz1 · · · dzn
∏
i
zmi
i
∏
i<j
(zi − zj)
mij .
With the Main Theorem 5.20 cit. loc. we express this function as quantum symmetrizer of
another integral, which converges if (mij) is subpolar on intervals
F((mi), (mij)) =
∑
σ∈Sn
q(σ)F̃
((
mσ−1(i)
)
,
(
mσ−1(i)σ−1(j)
))
,
F̃((mi), (mij)) =
∫
· · ·
∫
△
dz1 · · · dzn
∏
i
zmi
i
∏
i<j
(zi − zj)
mij ,
where △ :=
{(
e2πit1 , . . . , e2πitn
)
| 0 < t1 < · · · < tn < 1
}
and where q(σ) is the braiding factor
in the quantum symmetrizer with respect to the braiding qij = eπimij . The contour integral F̃
can by Lemma 5.15 cit. loc. be deformed to a sum of real integrals, which have additional poles
depending on (mi):
F̃((mi), (mij)) :=
1
(2πi)n
n∑
k=0
(−1)n−k
(
n∏
i=n−k+1
e2πi mi
) ∑
η∈Sn−k,k
∏
i<j, η(i)>η(j)
eπi mij
× Sel
((
mη−1(i)
)
, (0),
(
mη−1(i)η−1(j)
))
,
where Sn−k,k := {η ∈ Sn | ∀i<j≤n−k η(i) < η(j) and ∀n−k<i<j η(i) > η(j)} denotes a version of
(n− k, k)-shuffles and Sel the generalized Selberg integral
Sel((mi), (m̄i), (mij)) :=
∫
· · ·
∫
1>z1>···>zn>0
dz1 · · · dzn
∏
i
zmi
i
∏
i
(1− zi)
m̄i
∏
i<j
(zi − zj)
mij .
Now we restrict ourselves to the Heisenberg vertex algebra Hr of rank r, to a not necessarily
integral lattice Λ = Zr with basis α1, . . . , αr and scalar product (αi, αj), and to screening
operators Z1, . . . ,Zr associated to the elements ai = eαi . An arbitrary monomial of total degree n
in these screening operators is
Zι(1) · · ·Zι(n), ι : {1, . . . , n} → {1, . . . , r}
20 I. Flandoli and S.D. Lentner
and if we expand the action of this monomial on an element v ∈ Hλ by the formula above, then
quantum monodromy numbers F((mi + ki), (mij)) with mi = (αi, λ) and mij = (αi, αj) and
ki ∈ Z appear.
Consider accordingly the braided vector space Cr with basis x1, . . . , xr and braiding qij =
e(αi,αj). Consider the Nichols algebra of diagonal type (be it finite or infinite-dimensional).
A monomial in the tensor algebra is of the form
xι(1) · · ·xι(n), ι : {1, . . . , n} → {1, . . . , r}.
A linear combination of such monomials∑
ι
cι · xι(1) · · ·xι(n), ι : {1, . . . , n} → {1, . . . , r}
is zero in the Nichols algebra iff it is in the kernel of the quantum symmetrizer Xq with respect
to the braiding matrix qι(i),ι(j) and the associated braiding factor qι(σ). This can be reformulated
to a condition on the coefficients cι:
0 =
∑
ι
cι
∑
σ∈Sn
qι(σ) · ·
(
xι(σ−1(1)) · · ·xι(σ−1(n))
)
=
∑
ι
(∑
σ∈Sn
qι◦σ(σ)cι◦σ
)(
xι(1) · · ·xι(n)
)
,
which means that the bracket vanishes for each ι.
Take on the other hand a corresponding product of screenings Zι(1) · · ·Zι(n), expand it by
Theorem 3.9 and then rearrange by substituting ι ◦ σ for ι, which also changes aι(i) to aι(σ−1(i)),
and then by using the simultaneous symmetry in [(ki), (mι(i)), (mι(i)ι(j))] while renaming the
summation indices kσ−1(i) again ki:
∑
ι
cι
(
n∏
i=1
Zι(i)
)
v =
∑
ι
cι
∑
(ki)
[(ki), (mι(i)), (mι(i)ι(j))] · F((mι(i) + ki), (mι(i),ι(j)))
=
∑
ι
cι
∑
(ki)
[(ki), (mι(i)), (mι(i)ι(j))]
∑
σ∈Sn
qι(σ) F̃((mι(σ−1(i)) + kσ−1(i)), (mι(σ−1(i))ι(σ−1(j))))
=
∑
ι
∑
σ∈Sn
qι◦σ(σ)cι◦σ
∑
(ki)
[(ki), (mι(σ(i))), (mι(σ(i))ι(σ(j)))]) · F̃((mι(i) + kσ−1(i)), (mι(i)ι(j)))
=
∑
ι
∑
(ki)
[(ki), (mι(i)), (mι(i)ι(j))] ·
(∑
σ∈Sn
qι◦σ(σ)cι◦σ
)
F̃((mι(i) + ki), (mι(i)ι(j))).
Assume that the integral F̃((mi), (mij)) converges at the given set of parameters, then by the
assumed Nichols algebra relation the bracket vanishes and(∑
σ∈Sn
qι◦σ(σ)cι◦σ
)
F̃((mι(i) + ki), (mι(i)ι(j))) = 0.
As already stated, this is the case for subpolar parameters (mij). Otherwise we are interested
in the nonzero result after analytic continuation. As discussed, only the symmetrizations over
all permutations of the index set enter in this formula. In particular for the Nichols algebra
relations to hold, it suffices that(∑
σ∈Sn
qι◦σ(σ)cι◦σ
)
F̃(ι(1) · · · ι(n))sym = 0,
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 21
where we abbreviate the following functions in (ki),
(
mι(i)
)
,
(
mι(i)ι(j)
)
by
F(ι(1) · · · ι(n)) := F
((
mι(i) + ki
)
,
(
mι(i)ι(j)
))
,
F̃(ι(1) · · · ι(n)) := F̃
((
mι(i) + ki
)
,
(
mι(i)ι(j)
))
.
A different way to put the previous computation would be writing the quantum symmetrizer
formula in the new notation and in a formal basis of the tensor algebra xι(1) · · ·xι(n) as∑
ι
F(ι(1) · · · ι(n)) · (xι(1) · · ·xι(n))sym
=
∑
σ∈Sn
qι(σ)F
(
ι
(
σ−1(1)
)
· · · ι
(
σ−1(n)
))sym · xι(σ−1(1)) · · ·
(
xι(σ−1(n))
)
.
This concludes the overview of the proof of Theorem 3.7.
3.3 Central charge
A realization in the sense of our article provides a set of elements in a Euclidean vector space
a1, . . . ar ∈ Cr, the corresponding screening operators of the Heisenberg algebra Hr, and their
algebra relations. We now also wish to fix an action of the Virasoro algebra on Hr. As discussed
in [23], it is usually desirable to choose the Virasoro structure, where all conformal weights
h(ai) = 1, which gives a unique choice for Q. This implies for local screenings that the screenings
commute with the Virasoro algebra action, but for nonlocal screenings the implication is more
subtle: Their action on the vacuum modules commutes with the Virasoro action, and in general
we expect that suitable powers of nonlocal screenings commute with the Virasoro action, just
like in the Felder complex [22]. Regardless, we now fix the Virasoro action such that h(vi) = 1
holds.
Proposition 3.10. For the Heisenberg algebra, there is a family of Virasoro structures para-
metrized by the choice of an element Q ∈ Cr, called background charge [20]. There is a unique Q
solving
h(ai) =
1
2
(ai, ai)− (ai, Q) = 1, i = 1, . . . , n.
The central charge of the system will be
c = r − 12(Q,Q).
In particular for rank 2, we have as in [57] the explicit formula
c = 2− 3
|a1(m22 − 2)− a2(m11 − 2)|2
m11m22 −m2
12
. (3.1)
4 Analytical continuation of screening relations
A product of screening operators does not necessarily converge beyond subpolar (mij). However,
we can attempt to analytically continue the functions F, F̃, Sel to the subpolar region. Our main
interest is in which regions of parameters Theorem 3.7 holds or which extensions of Nichols
algebras we find.
We consider linear combinations of monomials Za1 · · ·Zan of n screening operators, acting on
the Hr-module Hv generated by ev. We set mi := (ai, v) and mij := (ai, aj) for 1 ≤ i, j ≤ n.
22 I. Flandoli and S.D. Lentner
Problem 4.1. For each ι : {1, . . . , n} → {1, . . . , r} and each (mij) attached to a finite-dimensio-
nal Nichols algebra in this article, find the full analytic continuation and poles of the functions
Sel(ι(1) · · · ι(n)) = Sel((mι(i) + ki), (mι(i)ι(j))).
For a realization of Cartan type mij = (αi, αj)gm with m ∈ Q as in Definition 6.1, a lin-
ear combination of these integrals is the so-called g-Selberg integral and can be expressed as
a product of Gamma functions. This existence of such an formula in certain cases is the Mukhin–
Varchenko conjecture [52] and a general g-Selberg integral formula was proven for A2 in [60] and
for An in [62]. The last source also contains in Theorem 6.1 a version of Kadell’s integral with
Jack polynomials necessary for the case (ki) ̸= (0). It would be tempting to use these results
to get at least in the case An a full analytic continuation of all monomials, and to check other
relations, such as the truncation relation of non-simple roots and the additional relations for
q = −1.
Problem 4.2. Does there exist a Selberg integral formula in the sense of [52, 60, 62] attached
to any Nichols algebra root system for Sel((mι(i) + ki), (mι(i)ι(j))) with the parameters (mij)
obtained in the present article?
We start with a toy case.
4.1 Commutativity relations
For n = 2 the subpolarity condition reads m12 > −1. From the formulas in [49, Example 5.21]
we immediately obtain the following analytic continuations, where B(x, y) = Γ(x)Γ(y)
Γ(x+y) is the Euler
Beta function, and Γ(x) is the Gamma function, which is meromorphic on C with simple poles
at x ∈ −N0:
Sel(m1,m2, 0, 0,m12) =
1
2 +m1 +m2 +m12
B(m2 + 1,m12 + 1).
This function has poles at m2,m12 ∈ −N and at m1 +m2 +m12 = −2:
F̃(m1,m2,m12) =
1
(2πi)2
(
1− e2πim2
)B(m2 + 1,m12 + 1)
m1 +m2 +m12 + 2
− 1
(2πi)2
e2πim2+πim12
(
1− e2πim1
)B(m1 + 1,m12 + 1)
m1 +m2 +m12 + 2
.
This function has poles at most at m12 ∈ −N. The poles at mi ∈ −N are removed by the
exponential prefactor, and the pole at m1 + m2 + m12 = −2 is removed by an equality of the
two summands at these values, visible after applying the Euler reflection formula
Γ(z)Γ(1− z) =
π
sin(πz)
=
1
(2πi)2
(
−2ieπim2
)
sin(πm2)
Γ(m12 + 1)Γ(m2 + 1)Γ(m2 +m12 + 2)−1
m1 +m2 +m12 + 2
− 1
(2πi)2
e2πim2+πim12
(
−2ieπim1
)
sin(π(m1 +m12 + 1))
× Γ(m12 + 1)Γ(−m1)
−1Γ(−m1 −m12 − 1)
m1 +m2 +m12 + 2
.
We now turn finally to
F(m1,m2,m12) =
e2πim2 − 1
2πi
e2πim1+2πim12 − 1
2πi
1
m1 +m2 +m12 + 2
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 23
×
(
B(m2 + 1,m12 + 1) +
sinπm1
sinπ(m1 +m12)
B(m1 + 1,m12 + 1)
)
.
This function has poles at a subset of m12 ∈ −N, depending on m1, m2. We rearrange the sum
to make the quantum symmetrizer formula
F(m1,m2,m12) = F̃(m1,m2,m12) + eπim12F̃(m2,m1,m12)
visible:
=
1
(2πi)2
(
1− e2πim2
)B(m2 + 1,m12 + 1)
m1 +m2 +m12 + 2
− 1
(2πi)2
e2πim2+πim12
(
1− e2πim1
)B(m1 + 1,m12 + 1)
m1 +m2 +m12 + 2
+ eπim12 · 1
(2πi)2
(
1− e2πim1
)B(m1 + 1,m12 + 1)
m1 +m2 +m12 + 2
− eπim12 · 1
(2πi)2
e2πim1+πim12
(
1− e2πim2
)B(m2 + 1,m12 + 1)
m1 +m2 +m12 + 2
.
Corollary 4.3. The product of two screenings Z1Z2 can be analytically continued to parameters
with m12 ̸∈ −N (or further, depending on m1, m2).
In the tensor algebra C⟨x1, x2⟩ we have
Xq[x1, x2]q = Xq(x1x2 − q12 x2x1) = (1− q12q21)x1x2,
which is zero for q12q21 = 1. In the Nichols algebra, this gives in this case the quadratic relation
[x1, x2]q = 0 in Example 2.5, which is the quantum Serre relation for c12 = 0 in Lemma 2.14.
The corresponding expression of screening operators depends by the results in the last section
on the analyticity and zeroes of
F(12)− q12F(21) = (1− q12q21)F̃(12).
Corollary 4.4. The expression [Z1,Z2]q can be analytically continued to all values of m1, m2,
m12 and it vanishes for all m12 except if m12 ∈ Z (where q12q21 = e2πim12 = 1) and m12 < 0
(where F̃ has poles).
Remark 4.5. In the realizations (mij) derived below condition (5.1A) requires mij = 0 when-
ever qijqji = 1. Hence in these cases, commutativity always continues to hold for screening
operators.
Example 4.6. A typical example where the q-commutator [Z1,Z2]q is nonzero in contrast to
the Nichols algebra is the standard case of two local screenings, i.e., m1,m2,m12 ∈ Z. In this
case the standard (anti-)commutator formula
[Z1,Z2]± = Res
(
Y
(
Z1e
α2 , z
))
,
where Z1e
α2 is zero unless Y(eα1 , z)eα2 has a pole at z = 0, which is the case for m12 = (α1, α2)
< 0. For example, in the case m12 = −1 we have2
[Z1,Z2]+ = Res
(
Y
(
eα1+α2 , z
))
=: Z1+2.
2This formula appears in the Frenkel–Kac–Segal construction and shows that the local screenings in the lattice
vertex algebra of the root lattice of a Lie algebra g generates U(g). See [44, Section 5.6] or in our context [49,
Section 3.6].
24 I. Flandoli and S.D. Lentner
4.2 Truncation relations
Let ι : {1, . . . , n} → {1} be the constant function ι(i) = 1, then (mι(i)ι(j)) = m11 and mι(i) = m1
for all i, j. Consider
F̃(11 · · · 1︸ ︷︷ ︸
n
) = F̃(ι(1)ι(2) · · · ι(n)) = F̃((mι(i) + ki), (mι(i)ι(j))).
Lemma 4.7 (n-th power). The function F̃(11 · · · 1) extends analytically to
m11 ̸∈ −N
2
k
, k = 2, . . . , n
with at most (depending on m1) double poles for k = 2, . . . , n− 1 and a simple pole for k = n.
Before we prove this, we state some consequences. In the tensor algebra C[x] we have by
Lemma 2.4
Xqx
n =
(
n∏
k=1
1− qk
1− q
)
xn,
which is zero for qk = 1 for k = 2, . . . , n and thus gives the truncation relation xn = 0 for
n ≥ ord(q).
The corresponding expression of screening operators depends by the results of the previous
section on the analyticity and zeroes of the function
F(11 · · · 1)sym =
(
n∏
k=1
1− qk
1− q
)
F̃(11 · · · 1)sym.
Corollary 4.8. The power of a screening Zn
1 can be analytically continued to parameters with
m11 ̸∈ −N
2
k
, k = 2, . . . , n− 1
and it vanishes for all m11 fulfilling the condition m11 ∈ −Z 2
k for k = 2, . . . , n (where the
associated q-polynomial is zero) and m11 ≥ 0 (where F̃ has no poles).
Note that by this result the truncation relation Zn, n = ord
(
eπim11
)
can always be analytically
continued, but higher powers Zn, n > ord
(
eπim11
)
may not converge. This is related to the
product not converging, see Corollary 4.3.
A typical example where Zn is nonzero in contrast to the Nichols algebra is the casemij = −2
p ,
where one can compute that Zp is proportional to a single screening associated to epα. Since
(pα, α) ∈ 2Z we call this screening local screening since epα is an element in a suitably defined
integral lattice vertex algebra. This computation appears in the Liouville model and higher rank
analogy, where the new screening is a long screening, see [49, Section 6.4].
Proof of Lemma 4.7. In our special situation with equal mij = m11 and mi = m1 we use the
factorization in Lemma 6.3 cit. loc.
F̃((mi + ki), (mij)) :=
n−1∏
s=0
((
eπim11
)s
e2πim1 − 1
)
· Sel((m11 + ki), (0), (m11))
and evaluate Sel for (ki) = (0) with the Selberg integral formula [56]
Sel((a− 1), (b− 1), (2c)) =
n−1∏
k=0
Γ(a+ kc)Γ(b+ kc)Γ(1 + (k + 1)c)
Γ(a+ b+ (n+ k − 1)c)Γ(1 + c)
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 25
for a− 1 = m1, b− 1 = 0, 2c = m11, and for arbitrary (ki) with the following refinement called
Kadell’s integral [45] and [51, Section VI.10]: For any partition in at most n parts l = (l1, . . . , ln),
l1 ≥ · · · ≥ ln ≥ 0 and for P
(1/c)
l (z1, . . . , zn) the associated Jack polynomial, we have∫
· · ·
∫
1>z1>···>zn>0
P
(1/c)
l (z1, . . . , zn)
∏
i
za−1
i
∏
i
(1− zi)
b−1
∏
i<j
(zi − zj)
2c dz1 · · · dzn
=
∏
1≤i<j≤n
Γ((j − i+ 1)c+ li − lj)
Γ((j − i)c+ li − lj)
n−1∏
k=0
Γ(a+ kc+ ln−k)Γ(b+ kc)
Γ(a+ b+ (n+ k − 1)c+ ln−k)
.
We want to study the analyticity of F̃. Since the Jack polynomials form a basis of the symmetric
polynomials, it is sufficient to study analyticity of Kadell’s integral for all partitions l. All
possible poles come from the Gamma-functions in the numerators.
� Consider Γ(a+ kc+ ln−k) with simple poles at most at
a+ kc ∈ −N0, k = 0, . . . , n− 1.
These cancel with the zeroes in the prefactor
((
eπimαα
)s
e2πimαλ − 1
)
, so at these values F̃
is analytic. We remark however, that these exceptional non-zero values of F̃ give rise to
reflection operators [49, Section 6.3].
We remark that the poles in Sel depending on mi must always disappear in F̃, because F̃
is a contour integral avoiding the singularity at zi = 0.
� Consider Γ(b+ kc) for b = 1, which gives simple poles for
kc ∈ −N, k = 2, . . . , n− 1,
while the possible pole for c ∈ −N for k = 1 cancels with the many zeroes coming from
the denominators Γ((j − i)c+ li − lj) for j − i = 1 with li − lj ≤ 0.
� Consider Γ((j − i+ 1)c+ li − lj) for all i, j with fixed k := j − i+ 1, which are n− k + 1
terms that can together produce a pole up to this order for
(li − lj) + kc ∈ −N0, k = 2, . . . , n.
On the other hand consider the n− k− 1 terms in the denominator Γ((j′ − i′)c+ li′ − lj′)
for j′ = j, i′ = i− 1 for i ̸= 1. Since li′ − lj′ ≥ li − lj these zeroes cancel with the possible
poles above, leaving only a possible single pole at l1 − l2 + kc ∈ −N0. A possible pole for
c = 0 cancels again with the many zeroes coming from the denominators Γ((j−i)c+ li− lj)
for j − i = 1 with li − lj ≤ 0 (note that previously we have only used these denominators
for j − i > 1 and for c ∈ −N), leaving possible poles
kc ∈ −N, k = 2, . . . , n.
This proves the assertion. ■
We remark that our calculation can be compared with the special case (ki) = (0) described by
the easier Selberg integral above. Conversely, in the case of general (ki) we integrate additional
positive integer powers zi, so it is reasonable that the case (ki) = (0) already exhibits the
maximal set of poles. However, in presence of Kadell’s integral we chose to be explicit here.
26 I. Flandoli and S.D. Lentner
4.3 Analytical continuation by recursion
We introduce a way of analytically continuing F̃ without explicitly computing it. We will use
this in the Serre relations, but also it seems suitable also for more complicated relations.
A trivial recursion relation is obtained by splitting off a factor
(zk − zl)
mkl = (zk − zl)(zk − zl)
mkl−1
in the integrand, multiplying (zk − zl) out and joining additional zk resp. zl to the powers zmk
k
resp zml
l . Thus
F̃((mi), (mij)) = F̃((mi + δi,k), (mij − δi,kδj,l))− F̃((mi + δi,l), (mij − δi,kδj,l)).
In particular F̃ can be analytically continued from some set of (mij) to positive translates
(mij)+(Nij) for Nij ∈ N0. Note that if (mij) is subpolar, so is (mij)+(Nij). It is more difficult
to continue to smaller mij , where poles appear:
Lemma 4.9. Assume k < l are fixed indices, then we have the following recursion relation
F̃((mi), (mij)) =
∑
k′ ̸=k,l
− mkk′
1 +mkl
sgn(k′ − k)F̃((mi), (mij + δi,kδj,l − δi,kδj,k′))
− mk
1 +mkl
F̃(mi − δi,l), (mij + δi,kδj,l)),
where sgn(x) = ±1 denotes the sign of x and we restrict ourselves for definiteness, e.g., to the
set of (mij) with all mij > 0. For k = 1 the formula gets an additional summand
1
1 +mkl
∫ e2πi
1
dz2 · · ·
∫ e2πi
zn−1
dzn−1
∏
1̸=i
zmi
i (1− zi)
m1i
∏
1̸=i<j
(zi − zj)
mij .
When we write
∫ e2πit
1 we mean the corresponding arc on the unit circle, lifted to the universal
covering of C\{0}.
Proof. We integrate by parts with respect to the variable zk and with respect to the factor
(zk − zl)
mkl :
F̃((mi), (mij)) =
∫ e2πi
1
dz1
∫ e2πi
z1
dz2 · · ·
∫ e2πi
zn−1
dzn
∏
i
zmi
i
∏
i<j
(zi − zj)
mij
=
1
1 +mkl
∫ e2πi
1
dz1 · · ·
∫ e2πi
zk−2
dzk−1
×
[∫ e2πi
zk
dzk+1 · · ·
∫ e2πi
zn−1
dzn
∏
i
zmi
i
∏
i<j
(zi − zj)
mij+δi,kδj,l
]zk=e2πi
zk=zk−1
−
∑
k′ ̸=k,l
mkk′
1 +mkl
sgn(k′ − k)
∫ e2πi
1
dz1 · · ·
∫ e2πi
zn−1
dzn
∏
i
zmi
i
∏
i<j
(zi − zj)
mij+δi,kδj,l−δi,kδj,k′
− mk
1 +mkl
∫ e2πi
1
dz1 · · ·
∫ e2πi
zn−1
dzn
∏
i
z
mi−δi,k
i
∏
i<j
(zi − zj)
mij+δi,kδj,l ,
and the derivative of
∫ e2πi
zk
vanishes due to (zk − zk+1)
mk,k+1 . Also it should be silently implied
that if k′ < k then the roles of i, j are switched where appropriate. Now the boundary term
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 27
in the square bracket vanishes for zk = e2πi, because then the integration domain for dzk+1 is{
e2πi
}
and for k ̸= 1 it vanishes for zk = zk−1, again because of a term (zk−1 − zk)
mk−1,k . For
k = 1 the boundary term for z1 = 1 produces an additional term
1
1 +mkl
∫ e2πi
1
dz2 · · ·
∫ e2πi
zn−1
dzn−1
∏
1̸=i
zmi
i (1− zi)
m1i
∏
1̸=i<j
(zi − zj)
mij . ■
This is still not satisfactory, because mkl is increased at the expense of mkk′ being decreased,
but at least the overall sum of the mij does not decrease. We now demonstrate how this can
be applied in the case n = 3. Inductively applying the previous lemma n times to (k, l) = (2, 3)
gives
F̃(m1,m2,m3,m12,m13,m23)
=
∑
i+j=n
(−1)j
(
m13
i
)(
m3
j
)(
m23+n
n
) F̃(m1,m2 − j,m3,m12,m13 − i,m23 + n).
Now suppose (mij) fulfills m12+m13+m23 > −2 (subpolarity for I = {1, 2, 3}) and m12 > −1
(subpolarity for I = {1, 2}) but not necessarily m23 > −1 (subpolarity for I = {2, 3}). Then for
n sufficient large such that m23 + n > −1 all parameters in the previous formula are subpolar
on intervals.
Corollary 4.10. The previous recursion formula gives an analytic continuation of F̃((mi),(mij))
for n = 3 to all parameters (mij) with m12 + m13 + m23 > −2 and m12 > −1, with at most
simple poles at m23 ∈ −N.
4.4 Serre relations for Cartan matrix entry −1
We want to study the Serre relation [Z1, [Z1,Z2]q]q = 0, which holds by Lemma 2.14 for q12q21 =
q−1
11 or q211 = 1. We start again by setting up an analytical continuation. Let, in more generality
ιN : {0, . . . , n} → {1, 2}, ι(N) = 2, ι(i) = 1, i ̸= N.
Lemma 4.11 (N = 0). There exists an analytic continuation of
Sel(211 · · · 1) = Sel((mι0(i) + ki), (0), (mι0(i)ι0(j)))
as a product of Gamma functions, with at most simple poles at
m1 + k
m11
2
∈ −N, k = 0, . . . , n− 1,
m12 + k
m11
2
∈ −N, k = 0, . . . , n− 1,
k
m11
2
∈ −N, k = 2, . . . , n,
and at n+ nm1 +m2 + nm12 +
(
n
2
)
m11 ∈ −N.
Proof. By the substitution z̃i = zi/z0 we can isolate the first variable in the Selberg integral
and integrate it
Sel((mι(i) + ki), (0), (mι(i)ι(j)))0≤i<j≤n
=
∫
· · ·
∫
1>z0>···>zn>0
zm2+k0
0
n∏
i=1
zm1+ki
i
n∏
j=2
(z1 − zj)
m12
∏
1≤i<j≤n
(zi − zj)
m11dz0dz1 · · · dzn
28 I. Flandoli and S.D. Lentner
=
∫ 1
0
z
n+(m2+k0)+
∑
j m12+
∑
i<j m11+
∑
i(m1+ki)
0 dz0
×
∫
· · ·
∫
1>z1>···>zn>0
n∏
i=1
z̃m1+ki
i
n∏
j=1
(1− z̃j)
m12
∏
2≤i<j≤n
(z̃i − z̃j)
m11dz̃1 · · · dz̃n
=
1
1 + m̃
Sel((mι(i) + ki), (0), (mι(i)ι(j)))0≤i<j≤n,
where m̃ := n + nm1 + m2 + nm12 +
(
n
2
)
m11 +
∑n
i=0 ki. Hence analytic continuation is again
possible using Kadell’s integral, as in the previous section and we thus analyze the poles in
1
1 + m̃
∫
· · ·
∫
1>z1>···>zn>0
P
(1/c)
l (z1, . . . , zn)
∏
i
za−1
i
∏
i
(1− zi)
b−1
∏
i<j
(zi − zj)
2c dz1 · · · dzn
=
1
1 + m̃
∏
1≤i<j≤n
Γ((j − i+ 1)c+ li − lj)
Γ((j − i)c+ li − lj)
n−1∏
k=0
Γ(a+ kc+ ln−k)Γ(b+ kc)
Γ(a+ b+ (n+ k − 1)c+ ln−k)
for a− 1 = m1, b− 1 = m12, 2c = m11 as in the last section
� The fraction produces a simple pole at m̃ = −1.
� Γ(a+ kc+ ln−k) again produces possible poles at
a+ kc ∈ −N0, k = 0, . . . , n− 1.
� Γ(b+ kc) is changed and produces possible simple poles at
b+ kc ∈ −N0, k = 0, . . . , n− 1
� Γ((j− i+1)c+ li− lj) for all i, j with fixed k = j− i+1 again produces a possible simple
pole at
kc ∈ −N, k = 2, . . . , n ■
Example 4.12. For n = 2 the formula in the proof reads
Sel(211) =
1
1 + m̃
Γ(m11 + l1 − l2)
Γ(m11/2 + l1 − l2)
× Γ(1 +m1) + l2)Γ(1 +m12)
Γ(2 +m1 +m12 +m11/2 + l2)
Γ(1 +m1 +m11/2 + l1)Γ(1 +m12 +m11/2)
Γ(2 +m1 +m12 + nm11/2 + l2)
.
This has simple poles at most at m11/2 ∈ −1
2 −N0 and m12 ∈ −N and m12 +m11/2 ∈ −N, and
other poles involving m1. We discuss two examples that are relevant later-on, see condition (5.1)
� For m11 = 2m, m12 = −m with m ̸∈ Z we have simple poles at
m11/2 = m ∈ −1
2
− N0.
� For m11 = 1, m12 = −m with m ̸∈ Z we have simple poles at
m12 + 1/2 = −m+ 1/2 ∈ −N ⇔ m ∈ 1
2
+ N.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 29
Unfortunately we have found no version of the previous lemma if the distinguished index is
not the first index in Sel. For a specific choice of (mij) associated to A2 we can use the Selberg
integral associated to the Lie algebra A2 [60, 62], which gives an expression in terms of Gamma
functions for a specific linear combination:
Lemma 4.13 (n = 2). Let m11 = 2m and m12 = −m. Consider the linear combination of
Selberg integrals
S = −Sel(112)− sin(πm)
sin(2πm)
Sel(121).
Then S has an analytic continuation with at most the following poles
� at 2m ∈ ±N a simple pole,
� simple poles at m1,m2 ∈ −N and m1 +m2 −m ∈ −2− N0.
Proof. The expression S is a special case of [62, Theorem 1.2]: In the notation of cit. ļoc. we
set
k1 = 1, k2 = 1, α = 1, βi = 1 +mi, γ =
m11
2
= m,
then after a slight change in variables zi = 1 − ti, which reverses the order of the integration
variables and causes the factor (−1), then the expression in question reads
S =
Γ(β1)
Γ(1 + β1 − 2γ)
Γ(β1 + β2 − γ)
Γ(1 + β1 + β2)
Γ(β2)
Γ(1 + β2 + γ)
× Γ(1− 2γ)Γ(γ)
Γ(γ)
Γ(1)Γ(γ)
Γ(γ)
Γ(1 + γ)Γ(2γ)
Γ(γ)
=
Γ(1 +m1)
Γ(2 +m1 −m)
Γ(2 +m1 +m2 −m)
Γ(3 +m1 +m2)
Γ(1 +m2)
Γ(2 +m2 +m)
Γ(1− 2m)(m/2)Γ(2m).
We again read off the possible poles from the Gamma-functions: In the first numerator 1+m1 ∈
−N0, in the second numerator 2+m1+m2−m12 ∈ −N0, in the third numerator 1+m2 ∈ −N0,
and in the rest 2m ∈ ±N. ■
The last assertion does not determine F̃(121) and F̃(112) individually, but if we assume
m < 1, then F̃(121) is subpolar on intervals, so it is analytic. Hence in this case we can combine
the information on the poles of Sel(211) and S, Sel(112) to determine all possible poles of F̃(112)
and F̃(211) (and F̃(121), but this is again analytic by subpolarity on intervals). Again, since F̃
is an integral over a contour not on 0, the poles depending on m1, m2 will disappear in the
linear combinations of Sel.
Corollary 4.14 (later-on the case A2, m < 1). Suppose m11 = 2m, m12 = −m and suppose in
addition m < 1, then the previous two lemmas and the analyticity of Sel(121) give an analytic
continuation of F̃(211), F̃(112) to all values of m < 1 with at most simple poles at 2m ∈ −N.
Observe that Sel(211) only contributes poles for negative half-integer m, which apparently
are related to the power Z2
1. On the other hand Sel(112) also contributes poles for positive
half-integer m, which apparently are related to the product Z1Z2. The possible pole at m = 1
2
must in fact be analytic, since subpolarity holds – it appears that all possible poles m ∈ 1
2 +N0
are artifacts from the sine-fraction appearing in the definition of S.
As a second method (with different range of applicability), we now consider directly for F̃(112)
the analytic continuation via recursion in Corollary 4.10. Note that this could also be applied
to the previous case, but then the additional boundary term in Lemma 4.9 would come into
consideration.
30 I. Flandoli and S.D. Lentner
Corollary 4.15 (later-on the cases A2, m > −1
2 and A(1|0), m < 3
2). We have the following
analytic continuations:
� For m11 = 2m, m12 = −m with m > −1
2 we have
m11 = 2m > −1, m11 +m12 +m12 = 2m−m−m = 0 > −2,
so Corollary 4.10 applies and we have an analytic continuation with at most simple poles
at m ∈ N.
� For m11 = 1, m12 = −m with m < 3
2 we have
m11 = 1 > −1, m11 +m12 +m12 = 1−m−m > −2,
so Corollary 4.10 applies and we have an analytic continuation with at most a simple pole
at m = 1. Note that for m < 1 the expression is subpolar.
We are now ready to check the Serre relations with cij = −1. We calculate in the tensor
algebra explicitly:
[x1, [x1, x2]q]q = x1x1x2 − q12(q11 + 1)x1x2x1 + q11q
2
12x2x1x1,
X[x1, [x1, x2]q]q = x1x1x2 − q12(q11 + 1)x1x2x1 + q11q
2
12x2x1x1
+ q11x1x1x2 − q12(q11 + 1)q12x2x1x1 + q11q
2
12q21x1x2x1 + q12x1x2x1
− q12(q11 + 1)q21x1x1x2 + q11q
2
12q11x2x1x1 + q12q12x2x1x1
− q12(q11 + 1)q21q11x1x1x2 + q11q
2
12q11q21x1x2x1 + q11q12x1x2x1
− q12(q11 + 1)q12q11x2x1x1 + q11q
2
12q21q21x1x1x2 + q11q12q12x2x1x1
− q12(q11 + 1)q12q11q21x1x2x1 + q11q
2
12q21q21q11x1x1x2
= (q11 + 1)(q12q21 − 1)(q11q12q21 − 1)x1x1x2.
In the factorization we see the three possibilities for this relation to hold, namely: q-truncation,
q-Cartan with cij = 0 and q-Cartan with cij = −1.
Now by the results in Section 3.2 this translates to an equation
F(112)sym − q12(q11 + 1)F(121)sym + q11q
2
12F(211)
sym
= (q11 + 1)(q12q21 − 1)(q11 q12q21 − 1)F̃(112)sym.
Combining the analytic continuations in Corollaries 4.14 and 4.15 we thus find:
Corollary 4.16 (later-on the case A2). The function F̃(112) can be analytically continued to
parameters m11 = 2m, m12 = −m with at most simple poles for m ∈ −1
2N and m ∈ N. For
m ∈ −1
2 −N0 we have mii ∈ −1− 2N0 and qii = −1, for which the factor above (q11 + 1)(q12q21
−1)(q11q12q21−1) has a double zero, and thus its product with F̃(112) is zero. As a consequence,
for m ̸∈ Z two screening operators Z1, Z2 with such (mij) fulfill the quantum Serre relation
[Z1, [Z1,Z2]q]q = 0.
Corollary 4.17 (later-on the case A(1, 0)). For m < 3
2 the function F̃(112) can be analytically
continued to parameters m11 = 1, m12 = −m with at most a simple pole at m = 1. As
a consequence, for m ̸∈ Z two screening operators Z1, Z2 with such (mij) fulfill the quantum
Serre relation
[Z1, [Z1,Z2]q]q = 0.
Here we make no assertion if the Serre relation holds for m > 3
2 .
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 31
5 Formulation of the classification problem
Definition 5.1. Let Λ be a lattice of rank r, basis {a1, . . . , ar}, bilinear form ( , ) and Cartan
matrix (cij) and let mij := (ai, aj). Given a braiding matrix (qij), we say that the lattice Λ
described by (mij) realizes (qij) iff
� the matrix elements mij exponentiate to the matrix elements qij :
eiπmij = qij ,
� for each pair (i, j) one of the following conditions hold:
A: 2mij = cijmii or B: (1− cij)mii = 2 (5.1)
(this condition originates in [57], but not the next condition),
� a base change by precomposing with a Weyl reflection σk(αi) = αi − ckiαk returns a new
matrix rk(mij) with
rk(mij)ij = (σk(αi), σk(αj))
and as suggested by notation rk(mij) trivially fulfills condition (1) with the reflected braid-
ing matrix rk(qij). We now demand in addition that all iterated reflections rk1 · · · rkn(mij)
again satisfy condition (5.1).
In complete analogy to Definition 2.19 we define
Definition 5.2. Let (mij) be a realization.
1. A pair (i, j) is calledm-Cartan ifmij satisfies (5.1A), andm-truncation if it satisfies (5.1B).
2. A root α is called m-Cartan resp. m-truncation if in any Weyl chamber containing α = αi
as a simple root and any neighbour j ∼ i the pair (i, j) is m-Cartan resp. m-truncation.
3. A root α is called only m-Cartan if in any Weyl chamber containing α = αi as a simple
root and any neighbour j ∼ i the pair (i, j) is only m-Cartan.
We visualize a realization by a diagram decorated on the top as a q-diagram by qii, qijqji and
on the bottom by the realization mii, mij +mji = 2mij , e.g.,
q11
m11
q22
m22
q33
m33
q12q21
2m12
q23q32
2m23
and no line connecting the vertices 1, 3 if m13 = 0. Note that if qijqji = 1 and thus cij = 0, then
in any realization the pair (i, j) cannot be m-trunctation, must be m-Cartan, and thus mij = 0.
We now summarize the strategy by which these notions allow to construct and classify real-
izations:
� Any pair (i, j) that is m-Cartan resp. m-truncation is surely q-Cartan resp. q-truncation.
Conversely, if a pair (i, j) is only q-Cartan resp. only q-truncation, than in any realization
(mij) the pair has to be (only) m-Cartan resp. (only) m-Cartan.
� If a simple root αi is not q-Cartan, then mii =
2
ord(qii)
. If a simple root α is q-Cartan,
then (5.1A) determines mij for all j ̸= i, hence one may proceed inductively. Moreover,
for a subsystem generated by only q-Cartan simple roots αi, αi′ , . . . this determines (mij)
for this subsystem to be a rescaled root lattice (see Lemma 6.5).
32 I. Flandoli and S.D. Lentner
� If a root α is q-Cartan and q-truncation, then there might exist different realizations,
depending on the assumption that it is m-Cartan or m-truncation. In more difficult cases
one may argue with individual pairs (i, j). See as example Remark 5.6.
� Conversely, suppose we are given a possible realization (mij) and want to proof that this is
indeed a realization. If mii =
2
ord(qii)
and we already know αi is q-truncation (in particular
if it is fermionic), then (5.1B) holds. Otherwise we check condition (5.1A). Then we have
to go through all reflections and check the same conditions, possibly fixing additionally
open parameters. As for the q-diagrams, the following fact reduces greatly the amount of
computation:
Proposition 5.3. If the pairs (k, i) and (k, j) are m-Cartan (for example if the root αk is
m-Cartan) then we have rk(mij) = (mij).
Proof.
rk(mij)ij = (σkαi, σkαj) = (αi − ckiαk, αj − ckjαk)
= mij − ckjmik − ckimkj + ckickjmkk
= mij − ckj ·
1
2
ckimkk − cki ·
1
2
ckjmkk + ckickjmkk = mij . ■
� For a reflection on a m-truncation root the result is less predictable. We derive in Propo-
sitions 8.23 and 8.24 a sufficient criterion, essentially from performing one reflection and
give conditions when this is m-truncation of m-Cartan. In practice, these conditions are
sufficient to fix (mij) uniquely.
� In particular for the Nichols algebras that do not follow into families, we proceed by
induction: The realizations (mij) restrict on a subset of simple roots to a realization of
the respective q-subdiagram.
Example 5.4. We now show an example of this procedure. We consider row 3 of Table 1 in [33],
described by the braiding matrices:
(
qIij
)
=
[
q2 q−1
q−1 −1
]
,
(
qIIij
)
=
[
−1 −q
−q −1
]
,
and corresponding diagrams:
q2 −1q−2 −1 −1q2
I II
with q ∈ C×, q2 ̸= ±1, simple roots {α1, α2} and {α12, α2} respectively, and a unique associated
Cartan matrix(
cIij
)
=
(
cIIij
)
=
[
2 −1
−1 2
]
.
This describes the Lie superalgebra sl(2|1). The set of positive roots is {α1, α2, α12} where α1 is
a q-Cartan root and α2, α12 are fermionic, thus q-truncation. All pairs (i, j) are only q-Cartan
or only q-truncation, since we assumed q2 ̸= −1.
Proposition 5.5. For this braiding and its reflections, the following are all realizations (mij):(
mI
ij
)
=
[
2m −m
−m 1
]
,
(
mII
ij
)
=
[
1 −1 +m
−1 +m 1
]
for all m = p′
p ∈ Q with (p′, p) = 1 such that eiπm = q.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 33
Proof. We check that condition (5.1B) is satisfied for the pair (2, 1) in I and both pairs (1, 2)
and (2, 1) in II:
mI
22 =
2
1− cI21
= 1, mII
11 =
2
1− cII12
= 1, mII
22 =
2
1− cII21
= 1,
while condition (5.1A) is satisfied for the pair (1, 2) in I:
mI
11 =
2mI
12
cI12
= 2m.
The reflection r1 preserves
(
qIij
)
as well as
(
mI
ij
)
by Proposition 5.3, because α1 is m-Cartan.
We check that the other reflection
r2(α1) = α1 + α2, r2(α2) = −α2
maps
(
mI
ij
)
to our choice of
(
mII
ij
)
:
r2
(
mI
ij
)
=
(α1 + α2, α1 + α2) (α1 + α2,−α2)
(−α2, α1 + α2) (−α2,−α2)
=
[
2m− 2m+ 1 m− 1
m− 1 1
]
=
(
mII
ij
)
.
We now prove conversely, that this is the only realization. Thereby we will find the typical
dichotomy of arguments that we will also find in later cases:
Since (1, 2) in I is only q-Cartan, the realization has to be m-Cartan, which fixes
mI
12 =
cI12
2
mI
11 = −m
once we have defined m via
mI
11 =: 2m.
Since (2, 1) in II is only q-truncation, the realization has to be m-truncation, so
mI
22 =
2
1− cI21
= 1.
Fixing mI
ij already fixes the bilinear form, so for proving uniqueness of the realization, this is
sufficient. ■
Remark 5.6. We now discuss the case q2 = −1 in the previous q-diagram, which is excluded
in row 3. It appears in row 2, which corresponds to sl3:
−1 −1−1
But in some sense, this diagram can be viewed as special case of both the q-diagrams appear-
ing in rows 2 and 3, and there is an exceptional isomorphism uq(sl3)
+ ∼= uq(sl(2|1)) for q2 = −1.
We also find that in this case all pairs i ∼ j are both q-truncation and q-Cartan, which opens the
possibility for different realizations (mij) in which different pairs are m-truncation or m-Cartan.
Indeed we find two solutions, and they are special cases of the two different realizations we can
construct for the diagrams in rows 2 and 3 respectively:
For this diagram, we find precisely the following two families of realizations (mij), each
parametrized by odd p′, p′′ ∈ Z,
34 I. Flandoli and S.D. Lentner
� if we assume (1, 2) and (2, 1) in I to be m-truncation, we find the unique realizations
(
mI
ij
)
=
[
1 −p′′
2
−p′′
2 1
]
,
� if we assume (1, 2) in I to be m-truncation and (2, 1) in I to be m-Cartan (or vice versa),
we find the realizations
(
mII
ij
)
=
[
1 −p′
2
−p′
2 p′
]
.
Reflection r1 maps this realizations
(
mII
ij
)
to the previous
(
mI
ij
)
and back, for p′′ = 2− p′.
Thus we have essentially one solution where (qij) is invariant under reflection, but (mij)
is different in different Weyl chambers. This solution should be viewed as an instance (or
limiting case) of the solution in Proposition 5.5, which corresponds to sl(2|1) at q2 = −1.
� if we assume both pairs (1, 2) and (2, 1) in I to m-Cartan, the unique family of realizations
is given by
(mij) =
[
p′ −p′
2
−p′
2 p′
]
,
which is the rescaled root lattice of sl3. By Proposition 5.3 all reflections leave (mij)
invariant. This solution should be viewed as an instance of the generic solution for Cartan
type in the next section, here sl3, q
2 = −1.
6 Cartan type
6.1 q diagram
Let g be a simple Lie algebra with simple roots α1, . . . , αn and Killing form in the standard
normalization (αi, αj)g ∈ {−3,−2,−1, 0, 2, 4, 6}. Let q ∈ C× be a primitive ℓ-th root of unity
with ℓ ∈ Z and let ord
(
q2
)
> d with d half length of the long roots. Define a braiding matrix (qij)
by
qij = q(αi,αj)g .
The finite-dimensional Nichols algebra B(qij) [8, 50, 55] is called of Cartan type We have that:
� (qij) is invariant under reflections rk,
� the Weyl groupoid is the Weyl group associated to g,
� the set of positive roots is the set of roots associated to g,
� the Cartan matrix cij is the Cartan matrix for g.
6.2 Construction of (mij)
Definition 6.1. Given m ∈ Q we define
mij := (αi, αj)gm.
Hence, the lattice Λ the root lattice of g rescaled by m.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 35
Remark 6.2. If we choose relatively prime integers k, ℓ, such that m
2 = k
ℓ , then q = eπim
is a primitive ℓ-th root of unity. In literature on the logarithmic Kazhdan Lusztig conjecture,
e.g., [21, 28], one usually sets m = p′
p , so q2 is a primitive p-th root of unity and ℓ = 2p, p
depending on the parity of p′.
Lemma 6.3. The matrix (mij) realizes the braiding (qij) for all reflections, and every simple
root is m-Cartan.
Proof. Condition (5.1) asks
2mij = cijmii or (1− cij)mii = 2,
2mji = cjimjj or (1− cji)mjj = 2.
But from the last point of enumeration in Section 6.1 we have cij =
2(αi,αj)g
(αi,αi)g
. Hence
cijmii =
2(αi, αj)g
(αi, αi)
(αi, αi)gm = 2mij ,
which is (5.1A), saying that the roots are m-Cartan. Since by Proposition 5.3, any reflection on
such a root leaves the (mij) invariant, condition (5.1) holds also after reflections. ■
Proposition 6.4. If ℓi > 1 − cij with ℓi := ord
(
q2di
)
, then the pair (i, j) is not q-truncation.
Hence the only Nichols algebras of Cartan type, where roots α are both q-Cartan and q-truncation
are
g q both q-Cartan and q-truncation
An q2 = −1 all roots
Bn, Cn, F4 q4 = −1 long roots
Bn, Cn, F4 q2 ∈ G3 short roots
G2 q6 = −1 long roots
G2 q2 ∈ G4 short roots
Proof. Assume that (i, j) ism-truncation (1−cij)mii = 2, this implies: q
(1−cij)
ii = eiπmii(1−cij) =
eiπ·2 = 1. But ord(qii) = ord
(
q2di
)
> 1 − cij and we find a contradiction. The second claim
follows by writing out these equations for long and short roots, and discarding the cases excluded
by the conditions on the q-diagram (q2 ̸∈ G2 for Bn, Cn, F4 and q2 ̸∈ G2,G3 for G2). ■
Lemma 6.5. If (mij) is a realization, such that all pairs (i, j) are m-Cartan, then (mij) is the
realization in Definition 6.1 for some m.
Proof. If we fix mii =: 2m for some short root αi, then mij for all j is fixed by condition (5.1A)
and so ismjj by the same condition with reversed indices. Hence up to rescaling there is a unique
solution (mij) and Definition 6.1 is such a solution. ■
Corollary 6.6. The realization in Definition 6.1 is unique for all Nichols algebras of Cartan
type except for the cases listed in Proposition 6.4.
As a counterexample, consider the case sl3 and ℓ = 2p = 4, where all pairs are both m-Cartan
and m-truncation. Indeed, we have in this case two realizations, as discussed in Remark 5.6,
corresponding to sl3 and to sl(2|1). In the latter realization, not all pairs are m-Cartan, despite
being q-Cartan.
36 I. Flandoli and S.D. Lentner
6.3 Central charge
Recall {a1, . . . , ar} as basis of Λ with mij = (ai, aj) and r = rank(g).
Proposition 6.7. The central charge of the system is
c = rank(g)− 12
(
1
r
∣∣ρ∨∣∣2
g
− 2
〈
ρ, ρ∨
〉
g
+ r|ρ|2g
)
,
where ρ is half the sum of all positive roots.
Proof. The central charge is
c = r − 12(Q,Q),
where Q =
∑
j cjaj is the unique combination such that for every i
1
2
(ai, ai)− (ai, Q) = 1,
1
2
(ai, ai)−
∑
j
cj(ai, aj)Λ = 1.
Rewriting ai = −
√
mαi, with αi root of g, this set of equations bring us to
Q =
√
1
m
ρ∨ −
√
mρ
that on turn gives the central charge as in the statement. ■
Remark 6.8. The central charge matches with the one of the affine Lie algebra ĝk at level
k+ h∨ = 1
r as in [11]. Conjecturally, the kernel of screening contains the Hamiltonian reduction
of ĝk.
Remark 6.9. For rank 2 and m = p′/p, the central charge is
c = 1− 3
(2p′ − 2p)2
2pp′
= 13− 6
p
p′
− 6
p′
p
,
which is the central charge of the (p, p′) Virasoro model, see, e.g., [61].
6.4 Algebra relations
According to the results in Section 4.4 we restrict ourselves here to the case of simple-laced
g = An, Dn, E6, E7, E8. By [3, Sections 4.1, and 4.4 and 4.5] a set of defining relations is
� The commutation relations resp. Serre relations for Cartan matrix entry cij = 0 are [xi, xj ]q
for i ̸∼ j.
� The Serre relations for Cartan matrix entry cij = −1 are [xi, [xi, xj ]q]q = 0 for i ∼ j for
q2 ̸= −1 (for q2 = −1 the Serre relations are implied by x2i = 0).
� The truncation relations of root vector are xℓαα = 0 for any root α ∈ Φ+ and ℓα = ord
(
q2
)
,
where the root vector xα is defined by repeated reflections using Lusztig’s isomorphism.
� For q2 = −1 additional relations [xj , [xi, [xj , xk]q]q]q = 0 for any subsystem αi, αj , αk of
type A3 (for q2 ̸= −1 these relations are a consequence of the Serre relations).
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 37
We now consider the realization of Cartan type mij = (αi, αj)gm. By Corollary 4.16, the
quantum Serre relations can be analytically continued and hold for all values of m. By Corol-
lary 4.8 the truncation relations of simple root vectors can be analytically continued for all
values of m, but they only hold for m ≥ 0. We make no assertion about the additional relation
for q2 = −1. The Nichols algebra without truncations relations is the Borel part of the Kac–
DeConcini–Procesi quantum group UK
q (g) resp. the distinguished pre-Nichols algebra [10]. In
particular we have proven:
Corollary 6.10. In the realization mij = (αi, αj)m of the braiding qij = eiπ(αi,αj)gm associated
to a simply-laced Lie algebra g at q = eiπm, m ̸∈ 2Z, the Nichols algebra relations hold for the
corresponding screenings as follows:
� For 0 < m < 1 the parameters (mij) are subpolar and all relations hold. Differently spoken,
the algebra of screenings is a surjective image of the Borel part of the small quantum
group uq(g).
� For m < 0 the Serre relations hold. The truncation relations of simple root vectors fail.
Differently spoken, for q2 ̸= 1 the algebra of screenings is a surjective image of the Borel
part of the Kac–DeConcini–Procesi quantum group UK
q (g).
� for m > 1 the Serre relations and the truncation relations of simple root vectors hold.
We would conjecture that for m < 0 also the additional relation holds and for q2 = −1 holds,
so that also in this case we get the Borel part of UK
q (g), and that for m > 1 also the truncation
relations of non-simple root vectors hold, so that also in this case for q2 ̸= −1 we get the Borel
part of uq(g). We make no assertion about the additional relation for q2 = −1 for m > 1. We
would conjecture that all surjections above are in fact isomorphism.
6.5 Examples: Cartan type realizations in rank 2
Heckenberger row 2 (Cartan type A2)
This case of the list is described by the braiding diagram:
q2 q2q−2
with q ∈ C q2 ̸= 1 and simple roots {α1, α2}. The set of positive roots is given by {α1, α2, α12}
with unique associate Cartan matrix:
(cij) =
[
2 −1
−1 2
]
.
Definition 6.1 gives the following realization
(mij) =
[
2m −m
−m 2m
]
,
q2
2m
q2
2m
q−2
−2m
For q2 ̸= −1, these are all solutions by Lemma 6.5. For q2 = −1 there is a second family of
solutions associated to A(1, 0), as discussed in Remark 5.6:
(
mI
ij
)
=
[
2m −m
−m 1
]
,
(
mII
ij
)
=
[
1 −1 +m
−1 +m 1
]
,
38 I. Flandoli and S.D. Lentner
−1
2m
−1
1
−1
−2m
−1
1
−1
1
−1
−2 + 2m
for m = p′
2 for p′ odd. The algebra relations for this realization are discussed in Section 7.4.
Heckenberger row 4 (Cartan type B2)
This case of the list is described by the braiding diagram:
q2 q4q−4
with q ∈ C, q2 ̸= ±1 and simple roots {α1, α2}. The set of positive roots is given by {α1, α2, α12,
α112} with unique associate Cartan matrix:
(cij) =
[
2 −2
−1 2
]
.
Definition 6.1 gives the following realization
(mij) =
[
2m −2m
−2m 4m
]
,
q2
2m
q4
4m
q−4
−4m
If all pairs are m-Cartan, in particular if ord
(
q2
)
> 3, ord
(
q4
)
> 2, then this is the unique
realization by Lemma 6.5. We now discuss the other cases:
1. When q4 = −1, the pair (2, 1) is q-Cartan and q-truncation. We find an additional family
of solutions where (2, 1) is m-truncation:(
mI
ij
)
=
[
2m −2m
−2m 1
]
,
(
mII
ij
)
=
[
−2m+ 1 2m− 1
2m− 1 1
]
,
±i
2m
−1
1
−1
−4m
±i
−2m+ 1
−1
1
−1
4m− 2
for m = p′
4 and p′ odd, with simple roots I: {α1, α2} and II: {α12,−α2}.
This can be interpreted as a limiting case of the Lie superalgebra B(1, 1) described in case
Heckenberger row 5, which for this choice of q2 has the same q-diagram.
2. When q2 = ζ ∈ G3, the pair (1, 2) is q-Cartan and q-truncation. We find an additional
family of solutions where (1, 2) is m-truncation:(
mI
ij
)
=
[
2
3 −2m
−2m 4m
]
,
(
mII
ij
)
=
[
2
3 −4
3 + 2m
−4
3 + 2m 8
3 − 4m
]
,
ζ
2
3
ζ2
4m
ζ
−4m
ζ
2
3
ζ2
8
3 − 4m
ζ
−8
3 + 4m
for m = 2+3p′
6 , p′ ∈ Z, with simple roots I: {α1, α2} and II: {−α1, α112}.
It can be interpreted as a limiting case of Heckenberger row 6 (a color Lie algebra), which
for this choice of q2 has the same q-diagram.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 39
Heckenberger row 11 (G2)
This case of the list is described by the braiding diagram:
q2 q6q−6
with q2 ̸= ±1, q2 ̸∈ G3 and simple roots {α1, α2}. The set of positive roots is given by
{α1, α2, α12, α112, α1112, α11122} with unique associate Cartan matrix:
(cij) =
[
2 −3
−1 2
]
.
Definition 6.1 gives the following realization
(mij) =
[
2m −3m
−3m 6m
]
,
q2
2m
q6
6m
q−6
−6m
If all pairs are m-Cartan, in particular if ord
(
q2
)
> 4, ord
(
q6
)
> 2, then this is the unique
realization by Lemma 6.5. We now discuss the other cases, where we ultimately find no additional
realizations:
1. When q2 ∈ G4, the pair (1, 2) is q-Cartan and q-truncation. We now look for a possible
additional realization, where (1, 2) is m-truncation: This assumption fixes m11 = 1
4 , and
since (2, 1) is stillm-Cartan, m12 is still fixed as in the previous case. Hence the assumption
uniquely determines the following possible realization (mij) =:
(
mI
ij
)
, and we also compute
the reflection r1(m
I
ij) =:
(
mII
ij
)
:
(
mI
ij
)
=
[
1
2 −3m
−3m 6m
]
,
(
mII
ij
)
=
[
1
2 −3
2 + 3m
−3
2 + 3m 9
2 − 12m
]
with simple roots I: {α1, α2} and II: {−α1, α1112}. The pair (2, 1) in II is only q-Cartan, so
it must be m-Cartan, which requires m = 1
4 . But for this value of m the realization agrees
with the previous realization. It appears here a second time, because for this realization
(1, 2) is both m-Cartan and m-truncation
2. When q2 ∈ G6, the pair (2, 1) is m-Cartan and m-truncation, and we now look for a pos-
sible additional realization where it is m-truncation. Again, this uniquely determines
the following possible realization (mij) =:
(
mI
ij
)
, and we also compute the reflection
r1
(
mI
ij
)
=:
(
mII
ij
)
:
(
mI
ij
)
=
[
2m −3m
−3m 1
]
,
(
mII
ij
)
=
[
1− 4m −1 + 3m
−3
2 + 3m 1
]
with simple roots I: {α1, α2} and II: {α12,−α2}. The pair (1, 2) in II is only q-Cartan, so
it must be m-Cartan, which requires m = 1
6 . But for this value of m again the realization
agrees with the first realization.
40 I. Flandoli and S.D. Lentner
7 Super Lie type
In the classification of finite-dimensional Nichols algebras of diagonal type in [35] several infinite
series occur, which are not of Cartan type, but which are linked to the root system of certain
Lie superalgebras. Corresponding quantum super groups had been defined, e.g., in [46]. In [39,
Example 9 and Theorem 24] it is shown that an earlier definition of a generalized root system for
Lie superalgebras [59] is a special case of the generalized root system in the sense of Section 2.3.
In [6] the contragradient Lie superalgebras and their quantum supergroups are related to the
corresponding Nichols algebras by the process of bozonization.
7.1 q diagram
Let g = g0 ⊕ g1 be a simple Lie superalgebra of classical, basic type [24], i.e., of type A(m,n),
B(m,n), C(n+1),D(m,n), F (4), G(3),D(2, 1;α). For these Lie superalgebras a (non degenerate
or zero) Killing form ( , )g is defined.
We now choose a Weyl chamber α1, . . . , αf−1, αf , αf+1, . . . , αn with just one simple fermionic
root αf . We call it the standard chamber according to [43]. Given α positive root in the standard
chamber, we define f(α) the multiplicity of αf in α.
We can then split g as the direct sum of vector spaces
g = g′ ⊕ g′′ ⊕m,
where g′ and g′′ are two bosonic connected component generated by the simple roots α1, . . . , αf−1
and αf+1, . . . , αn respectively, while m is the g′ ⊕ g′′-module spanned by all other roots.
We have that m contains g1 and thus in particular contains the g′ ⊕ g′′-submodule generated
by the fermion αf , i.e., the vector space of fermions γ, with f(γ) = 1. Moreover m may contain
bosonic roots δ, with f(δ) ∈ 2N.
Definition 7.1. We can write the inner product ( , )g of two arbitrary simple roots as
(αi, αj)g = (αi, αj)g′ + (αi, αj)g′′ =
(αi, αj)g′ if i ≤ f, j < f,
0 if i ≤ f ≤ j,
(αi, αj)g′′ if i ≥ f, j > f.
In particular we assume (αf , αf )g = (αf , αf )g′ = (αf , αf )g′′ = 0.
Definition 7.2. Let q′, q′′ be primitive roots of unity. Then to the above data in the standard
chamber we associate the braiding matrix (qij) with
qij =
(q′)(αi,αj)g′ if i ≤ f, j < f,
(q′′)(αi,αj)g′′ if i ≥ f, j > f,
1 if i > f > j,
−1 if i = f = j.
Under certain conditions relating q′, q′′, these braiding gives a finite-dimensional Nichols alge-
bra B(qij), which we call of super Lie type. We will continue our general considerations without
having to specify these conditions on q′, q′′. In the process of establishing a realization (mij)
depending on m′, m′′ we will encounter additional conditions relating m′, m′′. These additional
conditions will in each case imply the conditions relating q′, q′′ in Heckenberger’s list for this
specific Nichols algebra of super Lie type. For an explicit example, see Example 7.15, in general
these conditions will arise for the exceptional cases in Corollary 7.12 and will be spelled out
when we go through all cases in Sections 7.5 and 7.6.
The reflections will act on the braiding as follow:
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 41
� Reflections rk around bosonic roots αk leave (qij) invariant.
� Reflections rk around fermionic roots αk interchange fermionic and bosonic roots.
Remark 7.3. In the classification of Nichols algebras in [33] and [34] we find that the fermion
(as in the Lie superalgebra sense of the term) in the standard chamber αf has qff = −1, i.e.,
it is q-truncation. This is not true in general for every fermion as we can see in the following
example.
Example 7.4. The case Heckenberger row 5 of Table 1 in [33] is described by two diagrams:
−1 q2q−4 −1 −q−2q4
I II
corresponding to the simple roots:
I : {α1, α2}, II : {−α1, α12}.
This is the Lie superalgebra B(1, 1) and α12 is a fermion with q(α12, α12) ̸= −1.
We will describe this example in detail later in this section.
7.2 Construction of (mij)
Definition 7.5. Given p′, p′′ ∈ Z such that (p′, p) = (p′′, p) = 1, we define m′ := p′
p , m
′′ := p′′
p
and in the standard chamber:
mS
ij =
(αi, αj)g′m
′ if i ≤ f, j < f,
(αi, αj)g′′m
′′ if i ≥ f, j > f,
0 if i > f > j,
1 if i = f = j.
We notice that if we restrict to g′ or g′′, we get exactly the same result as in the Cartan type
section for p′, p respectively p′′, p.
Lemma 7.6. If we call q′ = eiπm
′
and q′′ = eiπm
′′
, then qij = eiπmij is the braiding defined in
Definition 7.2.
Proof. We have mij = 0 if αi and αj are disconnected, so that 1 = eiπ·0 and mff = 1 for the
fermionic root which gives −1 = eiπ·1 as demanded. ■
Lemma 7.7. In an arbitrary chamber C with simple roots γ1, . . . , γr we have
mij
C = (γi, γj)g′m
′ + (γi, γj)g′′m
′′ + f(γi)f(γj).
Proof. We write γi =
∑
k xikαk and γj =
∑
l xjlαl and we extend for linearity:
mij
C =
∑
k,l
xikxjlmkl
S
=
∑
k,l∈g′∪{f}
xikxjl(αk, αl)g′m
′ +
∑
k,l∈g′′∪{f}
xikxjl(αk, αl)g′′m
′′ + xifxjf
= (γi, γj)g′m
′ + (γi, γj)g′′m
′′ + f(γi)f(γj),
where the last equality follows from the definition of f(γ) as the multiplicity of αf in γ and the
fact that on each component g′ and g′′ the roots are spanned as in a Lie algebra. ■
42 I. Flandoli and S.D. Lentner
Corollary 7.8. A root γ in an arbitrary chamber is
– m-truncation if (γ, γ)g′m
′ + (γ, γ)g′′m
′′ + f(γ)f(γ) = 1,
– m-Cartan if, for every simple root αi in the standard chamber,
2(γ, αi)g′m
′ + 2(γ, αi)g′′m
′′ + 2f(γ)f(αi)
= cγ,βi
((γ, γ)g′m
′ + (γ, γ)g′′m
′′ + f(γ)f(γ)).
Example 7.9. We consider as an example the Lie superalgebra A(1, 1) of rank 3. The simple
roots in the standard chamber are {α1, α2 = αf , α3} with inner product:
(αi, αj) =
2 −1 0
−1 0 −1
0 −1 2
.
Hence,
(
mS
ij
)
=
2m′ −m′ 0
−m′ 1 −m′′
0 −m′′ 2m′′
, (qij) =
(q′)2 (q′)−1 1
(q′)−1 −1 (q′′)−1
1 (q′′)−1 (q′′)2
.
We check, under which conditions
(
mS
ij
)
above is indeed a realization in all Weyl chambers.
Lemma 7.10. If γ =
∑f−1
i=1 aiαi ∈ g′ (resp. for γ ∈ g′′) the root γ is m-Cartan.
Proof. Suppose γ ∈ g′; then
(γ, γ) = (γ, γ)g′ , (γ, γ)g′′ = 0,
(γ, αi) = (γ, αi)g′ , (γ, αi)g′′ = 0
for an arbitrary simple root αi. Moreover f(γ) = 0. We then write out (5.1A)
2(γ, αi)m
′ = cγ,αi((γ, γ)m
′).
This is true because of definition of cγαi in the Lie algebra setting. By linearity in the simple
roots αi it is possible to extend this result to an arbitrary root α =
∑
biαi. ■
Lemma 7.11. If γ ̸= αf is isotropic, i.e., (γ, γ) = (γ, γ)g′ = (γ, γ)g′′ = 0, and f(γ) = ±1 then γ
is m-truncation.
Proof. Condition (5.1B) for a root to be m-truncation reads:
(γ, γ)g′ + (γ, γ)g′′ + f(γ)f(γ) = 1,
which is clearly true under these hypothesis. ■
We summarize these results in the following:
Corollary 7.12. The matrix (mij) defined in Definition 7.5 corresponds to the given braid-
ing (qij) and the realization condition (5.1) holds for every root α in every Weyl chamber con-
taining α as simpe root, with the following possible exceptions:
1. α is a boson in g′ ∪ g′′, i.e., f(α) is a strictly positive even integer.
2. α is an isotropic fermion with f(α) ̸= ±1.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 43
3. α is a non-isotropic fermion.
4. α is a fermion strong orthogonal to another fermion γ, i.e., in their real span ⟨α, γ⟩R there
are no roots.
Proof. If a boson α is contained in g′ or g′′, then Lemma 7.10 asserts that it must be m-Cartan.
Otherwise, α must have the exceptional case (1) with f(α) > 0 even.
Let now α be a fermion which is not strong orthogonal to any other fermions. If it is isotropic
and f(α) = ±1, thanks to Lemma 7.11, it satisfies condition (5.1B). If f(α) ̸= ±1 or it is non-
isotropic, we have the exceptional case (2) and (3).
If α and γ are two strong orthogonal fermions, then cαβ = 0. ■
In the cases (1)–(4) we have to check explicitly condition (5.1A) or (5.1B) by Corollary 7.8.
Note that it is possible that in a realization a fermion is m-Cartan but not m-trunctation. In
case (4) the condition simplifies to In this case we have to check for which m′ and m′′
mα,β = (α, β)g′m
′ + (α, β)g′′m
′′ + f(α)f(β) = 0.
Remark 7.13. Going through all cases, we do not find any boson with f(α) > 2 and any
fermion with f(α) > 1. Thus, point (1) concerns then just bosons with f(α) = 2, for example
in type D(m,n) and F (4), and point (2) never occurs.
As last general result we now state conversely a classification lemma:
Lemma 7.14. If all the bosonic roots are m-Cartan, then the unique possible realizing solution
for the given braiding is the matrix (mij) of Definition 7.5. In particular this is the case if
ℓi > 1− cij for ∀i ̸= f.
Proof. Condition (5.1) gives a unique solution (mij) in the standard chamber: the fermionic
root is m-truncation and thus fixed to mff = 1, while, since all the other roots are m-Cartan,
restricting our study to the two bosonic sectors separately we end up in the same situation of
Lemma 6.5. Moreover the bilinear form is uniquely fixed by its values on one basis. ■
Example 7.15. We apply Lemma 7.12 to Example 7.9: after reflecting the standard chamber
set of roots around the fermion α2, we find for new simple roots: {α12,−α2, α23} the matrix:
(
m
r2(S)
ij
)
=
1 −1 +m′ −1 +m′ +m′′
−1 +m′ 1 −1 +m′′
−1 +m′ +m′′ −1 +m′′ 1
.
Exception (4) of Lemma 7.12 appears. We then have to ask m23 = 0, i.e., m′ + m′′ = 1 and
thus q′q′′ = −1. In that case (mij) is a realizing solution. This construction realizes the Nichols
algebra B(qij) described by case row 8 of Table 2 in [33] when q ̸= ±1.
7.3 Central charge
We will compute the central charge of systems associated to a Lie superalgebra g of rank r, with
non degenerate Killing form ( , ).
Proposition 7.16. The central charge is c = r − 12(Q,Q) with
Q =
ρ∨g′√
m′
− ρg′
√
m′ +
ρ∨g′′√
m′′
− ρg′′
√
m′′ − ρ∨rest,
where we denoted by ρg′ the half sum of positive roots in g′, ρg′′ the half sum of positive roots
in g′′ and ρrest the half sum of the remaining positive roots of g.
44 I. Flandoli and S.D. Lentner
Proof. By Proposition 3.10 the vector Q is characterized uniquely by the following condition
for all simple roots αi of g
1
2
(
−
√
miαi,−
√
miαi
)
−
(
−
√
miαi, Q
)
= 1, where mi =
p′
p
if i < f,
1 if i = f,
p′′
p
if i > f.
Let λ∨
j =
λj
dj
be such that (αi, λ
∨
j ) = δij . Since ρg =
n∑
i=1
λi, we have that ρg′ =
∑
i<f
λi, ρg′′ =
∑
i>f
λi
and then ρrest = λf . We can thus rewrite Q as
Q =
ρ∨g′√
m′
− ρg′
√
m′ +
ρ∨g′′√
m′′
− ρg′′
√
m′′ − ρ∨rest =
∑
i
(
1
√
mi
−
√
midi
)
λ∨
i .
Hence the previous equation becomes
1
2
(
−αi
√
mi,−αi
√
mi
)
−
(
−αi
√
mi, Q
)
=
1
2
2dimi +
∑
j
√
mi
(
1
√
mj
−√
mjdj
)(
αi, λ
∨
j
)
= 1. ■
7.4 Algebra relations
According to the results in Section 4.4 we restrict ourselves here to the case of simple-laced
g = A(n,m), D(m,n), D(2, 1;α). By [3, Sections 4.1, 4.4 and 4.5] a set of defining relations is
� Commutation relation [xi, xj ]q for i ̸∼ j.
� Serre relations [xi, [xi, xj ]q]q = 0 for i ∼ j for i bosonic (for αi fermionic the Serre relation
is implied by x2i = 0).
� Root vector truncation relations xℓαα = 0 for any root α ∈ Φ+ and ℓα = ord
(
q2
)
for α
bosonic or ℓα = 2 for α fermionic, where the root vector xα is defined by repeated reflections
using Lusztig’s isomorphism.
� For j fermionic the additional relations [xj , [xi, [xj , xk]q]q]q = 0 for any subsystem αi, αj ,
αk of type A3.
We now consider the realization in Definition 7.5. We first clarify, which parameters m′, m′′
give subpolar (mij).
Lemma 7.17. Subpolarity for (mij) in Definition 7.5 holds under the condition
1
2d′
≥ m′ > 0,
1
2d′′
≥ m′′ > 0, det(mij) > 0.
Proof. Subpolarity in Section 3.2 for all monomials holds under the assumptions |αi| ≤ 1,
which means 2d′m′ ≤ 1, 2d′′m′′ ≤ 1, and (mij) positive definite. By Sylvester’s criterion, this
is equivalent to det(mij) > 0 and to the principal minor being positive definite. The principal
minor is a rescaling of the root lattices g′, g′′, so it is positive definite for m′,m′′ > 0. ■
Example 7.18. For type A(n′, n′′) these conditions read
1
2
≥ m′ > 0,
1
2
≥ m′′ > 0,
n′
n′ + 1
m′ +
n′′
n′′ + 1
m′′ < 1,
where m′ +m′′ = 1.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 45
By Corollary 4.16, for αi bosonic the quantum Serre relations hold for all m and by Corol-
lary 4.17, for αi fermionic, the quantum Serre relations can be analytically continued and hold
for m < 3
2 . By Corollary 4.8, the truncation relations of simple root vectors xℓαα = 0 can be
analytically continued for all values of m, but they only hold for m ≥ 0 for α bosonic and for
all m for α fermionic.
The additional relation in degree αi + 2αj + αk with mij = −m′, mjj = 1, mjk = −m′′,
mik = 0 is subpolar for m′,m′′ < 1, because going through all subsets with multiplicities J ,
subpolarity amounts to the following inequalities
−m′ > −1,
−m′′ > −1,
−2m′ + 1 > −2,
−2m′′ + 1 > −2,
−m′ −m′′ > −2,
−2m′ + 1− 2m′′ > −3.
The Nichols algebra without truncations relations of bosonic roots is the Borel part of the
Kac–DeConcini–Procesi quantum super group UK
q (g) resp. the distinguished pre-Nichols alge-
bra [10]. We hence find:
Corollary 7.19. In (mij) in Definition 7.5, which is the realization of the braiding (qij) asso-
ciated to a simply-laced Lie superalgebra g at q = eiπm, m ̸∈ 1
2Z or the alternate realization of
the braiding associated to a simply-laced Lie algebra g at q = eiπm = i, m ∈ 1
2 + Z, the Nichols
algebra relations hold for the corresponding screenings as follows:
� For 1
2 ≥ m′,m′′ > 0 and det(mij) < 0 subpolarity holds, so that all relations hold. Differ-
ently spoken, the algebra of screenings is a surjective image of the Borel part of the small
quantum super group uq(g).
� For m′ < 0 the Serre relations and the additional relation of a root α in g′ with the
fermionic simple root αf holds. The truncation relations of simple fermionic root vectors
hold, the truncation relations of simple bosonic root vectors fail.
� The analogous statment holds for m′′ and simple roots in g′′.
� For 3
2 > m′,m′′ > 1
2 the Serre relation holds and for 1 > m′,m′′ > 1
2 the additional relation
holds, and we can make no assertion for larger values of m′, m′′.
We would conjecture that for m′,m′′ < 0 also the truncation relations of non-simple root
fermionic vectors hold, so that this case we get the Borel part Kac–DeConcini–Procesi quan-
tum super group UK
q (g) where the truncation relations for one or both subsets of bosonic roots
hold. We would conjecture that all surjections above are in fact isomorphism.
In the example A(n′, n′′) we have the additional condition m′+m′′ = 1, so we expect (proven
except for truncation of non-simple root vectors) for 1 > m > 0 the small quantum supergroup
and for m′ < 0, m′′ > 1 (or vice versa) the Borel part of a version of the Kac–DeConcini–Procesi
quantum super group UK
q (g) where the bosonic root vectors of A(n′) fail and those of A(n′′)
hold and for the latter the additinal relation is in question.
7.5 Examples in rank 2
We now present the cases of Table 1 in [33] that come from Lie superalgebras of rank 2. We
will check in every case whether the exceptions of Corollary 7.12 appear. In rank 2, there is
obviously always just one bosonic sector g′.
We also remark in each case how the simple roots in the standard chamber can be expressed
using the standard basis ϵi and δi in [43].
46 I. Flandoli and S.D. Lentner
Heckenberger row 3
The case row 3 of Table 1 in [33], studied in Example 5.4, is realized by the Lie superalgebra
lattice A(1, 0). This case is described by the diagrams
q2 −1q−2 −1 −1q2
I II
with q2 ̸= ±1 and simple roots I : {α1, α2}, II : {α12,−α2}. The set of positive roots is given by
{α1, α2, α12} with unique associate Cartan matrix and inner products
(cij) =
[
2 −1
−1 2
]
, (αi, αj) =
[
2 −1
−1 0
]
.
Therefore the matrix (mij) in the standard basis and after reflecting around α2 are given by
(
mI
ij
)
=
[
2m −m
−m 1
]
,
(
mII
ij
)
=
[
1 −1 +m
−1 +m 1
]
.
None of the exceptions of Lemma 7.12 appears; therefore (mij) is a realizing solution for all m.
This result matches with Example 5.4.
Remark 7.20. As observed in Example 5.4, if we allow the value q2 = −1 we obtain row 2 of
Table 1 in [33].
Remark 7.21. The simple roots in the standard chamber of A(1, 0) can be expressed by
α1 = ϵ1 − ϵ2, α2 = αf = ϵ2 − δ1.
Heckenberger row 5
Row 5 of Table 1 in [33] is realized by the Lie superalgebra lattice B(1, 1). This case is described
by the diagrams
q2 −1q−4 −q−2 −1q4
I II
with q2 ̸= ±1, q2 ̸∈ G4 and simple roots I : {α1, α2}, II : {α12,−α2}. The set of positive roots is
given by {α1, α2, α12, α112} with unique associate Cartan matrix
(cij) =
[
2 −2
−1 2
]
and inner product
(αi, αj) =
[
2 −2
−2 0
]
.
Therefore the matrix (mij) and its reflecting around α1 are given by
(
mI
ij
)
=
[
2m −2m
−2m 1
]
,
(
mII
ij
)
=
[
−2m+ 1 2m− 1
2m− 1 1
]
.
None of the exceptions of Lemma 7.12 appears; therefore (mij) is a realizing solution for all m.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 47
Remark 7.22. When q2 ∈ G3, the root α1 is q-Cartan and q-truncation. When it is m-
truncation we get
(
mI
ij
)
=
[
2
3 −2m
−2m 1
]
,
(
mII
ij
)
=
[
5
3 − 4m 2m− 1
2m− 1 1
]
,
(mIII
ij ) =
[
2
3 2m− 4
3
2m− 4
3
11
3 − 8m
]
,
where III: {−α1, α112} comes after reflecting around α1. The root α112 is never m-Cartan and
it is m-truncation iff m = 1
3 . But for this value of m, α1 is also m-Cartan and thus this is not
a new solution.
Remark 7.23. The roots can be expressed by
α1 = ϵ1, α2 = αf = δ1 − ϵ1.
7.6 Arbitrary rank
We generalize our study to arbitrary rank cases. In every case we will see under which additional
assumptions on m′, m′′ the matrices (mij) in Definition 7.5 are indeed realizing solutions.
A(m,n)
q2 q2q−2
· · · −1 · · ·
q−2 q2 q−2
The simple roots in the standard chamber are
α1, . . . , αf = αm+1, . . . , αm+n+1
with inner product matrix
(αi, αj) =
2 −1
−1
. . .
. . .
. . .
. . .
. . .
. . . 0
. . .
. . .
. . . −1
−1 2
We list all the positive roots. We denote by ∆0 the set of bosons and by ∆1 the set of fermions
according to the literature [43].
∆0 = {αl + · · ·+ αk, with l, k < f or l, k > f} ,
∆1 = {αl + · · ·+ αk, with l ≤ f ≤ k} .
We now apply the lemmas of the previous section to determine possible conditions on m′
and m′′ such that the matrix (mij) defined as in Definition 7.5 is a realizing solution.
� All the bosons are either in g′ or g′′. By Lemma 7.10, we know they are always m-Cartan.
� All the fermions are isotropic and have f(α) = ±1. By Lemma 7.11 we know that if they
are not strong orthogonal to any other root they are m-truncation.
48 I. Flandoli and S.D. Lentner
� We now focus on the case of strong orthogonal fermions. Let us consider two fermions:
γ1 = αl1 + · · ·+ αk1 with l1 ≤ f ≤ k1,
γ2 = αl2 + · · ·+ αk2 with l2 ≤ f ≤ k2.
They are strong orthogonal if l1 ̸= l2, k1 ̸= k2. In this case we have to check that
m12 = (γ1, γ2)g′m
′ + (γ1, γ2)g′′m
′′ + f(γ1)f(γ2) = 0.
We thus compute the inner products in the two bosonic sides. We assume l1 < l2 and
k1 < k2, because every other combination works analogously and gives the same result.
Without loss of generality we can assume l2 = l1 + 1 and k2 = k1 + 1 and thus
(γ1, γ2) = (αl1 , γ2) + (αl1+1, γ2) + · · ·+ (αf , γ2) + · · ·+ (γk1 , γ2)
= (αl1 , αl1+1)g′ + (αl1+1, αl1+1)g′ + (αl1+1, αl1+2)g′ + · · ·
+ (αf , αf−1)g′ + (αf , αf ) + (αf , αf+1)g′′ + · · ·
+ (αk1 , αk1 − 1)g′′ + (αk1 , αk1)g′′ + (αk1 , αk1+1)g′′ .
The only term that contributes is (αf , αf−1)g′ + (αf , αf ) + (αf , αf+1)g′′ since the previous
terms sum up to zero in g′, and the following terms sum up to zero in g′′. Hence we have
(γ1, γ2)g′ = (γ1, γ2)g′′ = −1. Asking m12 to be zero, means to ask
−1 ·m′ − 1 ·m′′ + 1 = 0 ⇒ m′ +m′′ = 1.
To conclude, the only condition needed for the matrix (mij) to be a realizing solution is m′ +
m′′ = 1. This condition matches with the formulation of A(m,n) in terms of Nichols algebra
diagram [34, Table C, row 2], where q′ = q and q′′ = −q−1. Indeed, if m′ + m′′ = 1 then
q′q′′ = eiπm
′
eiπm
′′
= −1.
Remark 7.24. We can write the simple roots in the standard chamber using as in [43] the
standard basis ϵ1, . . . , ϵm+1, δ1, . . . , δn+1:
{α1 = ϵ1 − ϵ2, α2 = ϵ2 − ϵ3, . . . , αm+1 = ϵm+1 − δ1,
αm+2 = δ1 − δ2, . . . , αm+n+1 = δn − δn+1},
B(m,n)
q−4 q−4q4
· · · −1 · · ·
q4 q−4 q2
The simple roots in the standard chamber are
α1, . . . , αf = αn, . . . , αm+n
with inner product matrix
(αi, αj) =
4 −2
−2
. . .
. . .
. . .
. . .
. . .
. . . 0
. . .
. . .
. . . −2
−2 2
.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 49
All the positive roots are
∆0 =
{
αl + · · ·+ αk with l, k < f,
αl + · · ·+ αk with l, k > f, k ̸= m+ n,
αl + · · ·+ αm+n with l > f,
αl + · · ·+ 2αk + · · ·+ 2αm+n with l < f, k ≤ f,
αl + · · ·+ 2αk + · · ·+ 2αm+n with l, k > f
}
,
∆1 =
{
αl + · · ·+ αm+n with l ≤ f,
αl + · · ·+ 2αk + · · ·+ 2αm+n with l < f < k,
αl + · · ·+ αk with l < f < k, k ̸= m+ n
}
.
We now apply the lemmas of the previous section to determine possible conditions on m′
and m′′ such that the matrix (mij) matrix defined as in Definition 7.5 is a realizing solution.
� All the bosons which are not of the type γlk := αl + · · ·+ 2αk + · · ·+ 2αm+n, with l < f,
k ≤ f, are either in g′ or g′′. By Lemma 7.10, we know they are always m-Cartan.
� For γlk, we need to explicitly ask condition (5.1).
The inner product is (γlk, γlk)g′ = −2, (γlk, γlk)g′′ = −4.
– γlk is m-truncation if 2m′ + 4m′′ = 3.
– γlk is m-Cartan if m′ +m′′ = 1.
� All the fermions which are not of the type γl := αl + · · · + αm+n, are isotropic and have
f(α) = ±1. By Lemma 7.11 we then know that if they are not strong orthogonal to any
other root they are m-truncation.
� For γl, we need to explicitly ask condition (5.1).
The inner product is (γl, γl)g′ = 0, (γl, γl)g′′ = −1.
– γl is m-truncation if m′′ = 0.
– γl is m-Cartan if m′ +m′′ = 1.
� We now focus on the case of strong orthogonal fermions. Let us consider the fermions:{
γ1 := αl1 + · · ·+ αm+n, γ2 := αl2 + · · ·+ 2αk2 + · · ·+ 2αm+n,
γ3 := αl3 + · · ·+ αk3
}
.
The fermions γ1 and γ2 are strong orthogonal iff l1 ̸= l2;
The fermions γ2 and γ3 are strong orthogonal iff l2 ̸= l3 or k2 ̸= k3 + 1;
The fermions γ1 and γ3 are strong orthogonal iff l1 ̸= l3;
Two fermions of type γ2 are strong orthogonal for different l2 and k2;
Two fermions of type γ3 are strong orthogonal for different l3 and k3; Asking the condition
mij = 0 for those cases, we find again the condition m′ +m′′ = 1.
In conclusion, the only condition needed for the matrix (mij) to be a realizing solution is
m′ +m′′ = 1. If this condition is satisfied the bosons with f(α) = 2 as well as the non isotropic
fermions are m-Cartan. If moreover m′ = m′′ = 1
2 then the bosons with f(α) = 2 are also
m-truncation.
Remark 7.25. We can write the simple roots in the standard chamber using as in [43] the
standard basis ϵ1, . . . , ϵm, δ1, . . . , δn:
{α1 = δ1 − δ2, α2 = δ2 − δ3, . . . , αn = δn − ϵ1, αn+1 = ϵ1 − ϵ2, . . . , αm+n = ϵm}.
The bosons with f(α) = 2 will be of the form δi+ δj , while the non isotropic fermions will be δi.
50 I. Flandoli and S.D. Lentner
C(n)
−1 q2q−2
· · ·
q2 q2q−2 q−4 q4
The simple roots in the standard chamber are
αf = α1, . . . , αn
with inner product matrix
(αi, αj) =
0 −1
−1 2
. . .
. . .
. . .
. . .
. . .
. . . −1
−1 2 −2
−2 4
.
All the positive roots are
∆0 =
{
αl + · · ·+ αk with l ̸= 1, k ̸= n,
αl + · · ·+ 2αk + · · ·+ 2αn−1 + αn with l ̸= 1, k ̸= n,
αl + · · ·+ αn with l ̸= 1,
2αl + · · ·+ 2αn−1 + αn with l ̸= 1
}
,
∆1 =
{
α1 + · · ·+ αn,
α1 + · · ·+ αk with k ̸= 1,
α1 + · · ·+ 2αk + · · ·+ 2αn−1 + αn with k ̸= n
}
.
We now apply the lemmas of the previous section to determine possible conditions on m′
such that the matrix (mij) defined as in Definition 7.5 is a realizing solution.
� Since there is just one bosonic side and there are no bosons with f(α) > 0 it is obvious
that all the bosons are m-Cartan.
� All the fermions are isotropic, not strong orthogonal to each other, and have f(α) = ±1.
By Lemma 7.11 we then know that they are m-truncation.
To conclude, the matrix (mij) is always a realizing solution.
Remark 7.26. We can write the simple roots in the standard chamber using as in [43] the
standard basis ϵ1, δ1, . . . , δn−1:
{α1 = ϵ1 − δ1, α2 = δ1 − δ2, . . . , αn−1 = δn−2 − δn−1, αn = 2δn−1}.
D(m,n)
q−2 q−2q2
· · · −1 · · ·
q2 q−2 q2
q2
q−2
q2
q−2
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 51
The simple roots in the standard chamber are
α1, . . . , αn = αf , . . . , αn+m
with inner product matrix
(αi, αj) =
2 −1
−1
. . .
. . .
. . . 0
. . .
. . . 2 −1 −1
−1 2 0
−1 0 2
.
All the positive roots are
∆0 =
{
αl + · · ·+ αk with l, k < f,
αl + · · ·+ αk with l, k > f,
αl + · · ·+ αm+n−2 + αm+n with l > f,
αl + · · ·+ 2αk + · · ·+ 2αm+n−2 + αm+n−1 + αm+n with l < f, k ≤ f,
αl + · · ·+ 2αk + · · ·+ 2αm+n−2 + αm+n−1 + αm+n with l, k > f,
2αl + · · ·+ 2αk + · · ·+ 2αm+n−2 + αm+n−1 + αm+n with l < f, k ≤ f
}
,
∆1 =
{
αl + · · ·+ αk with l ≤ f ≤ k,
αl + · · ·+ αn+m−2 + αn+m with l ≤ f,
αl + · · ·+ 2αk + · · ·+ 2αm+n−2 + αn+m−1 + αn+m with l < f < k
}
.
We now apply the lemmas of the previous section to determine possible conditions on m′ and m′′
such that the matrix (mij) defined as in Definition 7.5 is a realizing solution.
� All bosons except the IV or VI type in the list, are either in g′ or g′′. Then, thanks to
Lemma 7.10, we know they are always m-Cartan.
� The bosons of type IV have inner product −2 in g′ and −4 in g′′:
– it is m-truncation if 2m′ + 4m′′ = 3,
– it is m-Cartan if m′ +m′′ = 1.
The bosons of type VI have inner product −4 in g′′:
– it is m-truncation if 4m′′ = 3,
– it is m-Cartan if m′ +m′′ = 1.
� All fermions are isotropic and have f(α) = ±1. By Lemma 7.11 we then know that if they
are not strong orthogonal to any other root they are m-truncation.
� There are many possibility for two fermions to be strong orthogonal. Asking the condition
mij = 0 for those cases, we find again the condition m′ +m′′ = 1.
In conclusion, the only condition needed for the matrix (mij) to be a realizing solution is
m′ +m′′ = 1. If this condition is satisfied the bosons with f(α) = 2 are m-Cartan. If moreover
m′ = m′′ = 1
2 then the boson of type IV are also m-truncation. Instead if m′ = 1
4 , m
′′ = 3
4 then
the boson of type VI are also m-truncation.
52 I. Flandoli and S.D. Lentner
Remark 7.27. As in the previous cases the condition m′+m′′ = 1 matches with the formulation
of D(m,n) in terms of Nichols algebra diagram [34, Table C, row 10], where q′ = q and q′ = q−1.
Remark 7.28. We can write the simple roots in the standard chamber using as in [43] the
standard basis ϵ1, . . . , ϵm, δ1, . . . , δn:
{α1 = δ1 − δ2, . . . , αn = δn − ϵ1, αn+1 = ϵ1 − ϵ2, . . . ,
αm+n−1 = ϵm−1 − ϵm, αm+n = ϵm−1 + ϵm}.
The bosons of type IV will be of the form δi+δj , while the one of type VI will be of the form 2δi.
7.7 Sporadic cases
7.7.1 G(3)
−1 q2 q6q−2 q−6
The simple roots in the standard chamber are {αf = α1, α2, α3} with inner product
(αi, αj) =
0 −1 0
−1 2 −3
0 −3 6
.
There is only one bosonic part g′ and the positive roots are
{α1, α2, α3, α12, α23, α223, α123, α1223, α12223, α2223, α22233, α1222233, α122233}.
The matrix (mij) is given by
(
mI
ij
)
=
1 −m 0
−m 2m −3m
0 −3m 6m
.
� Since there is just one bosonic side and there are no bosons with f(α) > 0, it is obvious
that all the bosons satisfy are m-Cartan.
� All the fermions, except for α1223, are isotropic and have f(α) = ±1. By Lemma 7.11 we
then know that they are m-truncation.
� The fermion α1223 is m-Cartan without further assumptions on m by Corollary 7.8 and ex-
plicit computation (hence this is a rare example that a fermionic root in a Lie superalgebra
can be m-Cartan and not m-truncation).
� There are no pairs of strong orthogonal fermions.
To conclude, the matrix (mij) is a realizing solution for all m. This construction realize the
Nichols algebra B(qij) described row 7 of Table 2 in [33] when q ̸= ±1, q ̸∈ G3.
For convenience we show explicitly all the reflections of the matrix (mij): Reflecting
(
mI
ij
)
with r1 around α1 we find the following
(
mII
ij
)
=
1 −1 +m 0
−1 +m 1 −3m
0 −3m 6m
in basis {−α1, α12, α3}.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 53
Reflecting it with r2 around α12 we find the following
(
mIII
ij
)
=
2m −m −2m
−m 1 −1 + 3m
−2m −1 + 3m 1
in basis {α2,−α12, α123}.
Reflecting it around with r3 around α123 and permuting the indices we find
(
mIV
ij
)
=
6m −3m 0
−3m 1 −1 + 2m
0 −1 + 2m 1− 2m.
in basis {α3,−α123, α1223}.
Remark 7.29. If q2 ∈ G6, α3 is both q-Cartan and q-truncation. When it is m-truncation we
find
−1
1
ζ
2m
−1
1
ζ−2
−2m
−1
−6m
with ζ ∈ G6. This is a solution iff m = 1
6 . But for this value of m, α3 is also m-Cartan and thus
this is not a new solution.
Remark 7.30. The roots can be expressed by
α1 = αf = δ + ϵ1, α2 = ϵ2, α3 = ϵ3 − ϵ2.
F (4)
q4 q4 q2 −1q−4 q−4 q−2
The simple roots in the standard chamber are {α1, α2, α3, α4 = αf} with inner product
(αi, αj) =
4 −2
−2 4 −2
−2 2 −1
−1 0
.
There are 18 roots: 9 in the one bosonic part g′ of type C3, furthermore 8 fermionic roots
{α4, α34, α234, α1234, α2334, α12334, α122334, α1223334}
and one bosonic root α12233344 with f(α) = 2.
� All bosons in g′ are automatically m-Cartan.
� The boson α12233344 is m-Cartan without further assumptions by Corollary 7.8 and explicit
computation.
� All fermions are isotropic and have f(α) = ±1. By Lemma 7.11 we then know they are
m-truncation.
� We have two couples of strong orthogonal fermions:
{α34, α122334}, {α234, α12334}
which give the condition m = 1
3 .
To conclude, the condition for the matrix (mij) to be a realizing solution is m = 1
3 .
54 I. Flandoli and S.D. Lentner
D(2, 1;α)
−1
−1 −1
q′ q′′
q′′′
1
1 1
m′ − 2 m′′ − 2
m′′′ − 2
The simple roots in the standard chamber are {α1, α2 = αf , α3} with inner product:
(αi, αj) =
2 −2 0
−2 0 −2
0 −2 2
.
The positive roots are
{α1, α2, α3, α12, α23, α123, α1223}.
Reflecting the diagram around, say, α2 (the system is completely symmetric in the three roots),
we obtain
q′
m′
−1
1
q′′′
m′′′
(q′)−1
−m′
(q′′′)−1
−m′′′
Exception (4) of Lemma 7.12 appears. Imposing that the first and the third roots are not con-
nected we find the condition m′+m′′+m′′′ = 2. This corresponds to the condition q′ · q′′ · q′′′ = 1
of rows 9, 10, 11) in Table 2 of [33]. Hence these matrices (mij) are realizing solution.
8 Other cases in rank 2
8.1 Construction of (mij)
In this section we are going to present the examples of rank 2 Nichols algebras that do not
belong to the Cartan and super Lie study of the previous two sections. We use abbreviations
such as q112,112 := q(α112, α112) and similarly m112,112 := (α112, α112).
Heckenberger row 6
This corresponds to a Z3-(color-)Lie algebra [4, 64]. In Table 1 in [33] it is described by two
diagrams:
ζ q2q−2 ζ ζq−2ζ−1q2
I II
where ζ ∈ G3 and q2 ̸= 1, ζ, ζ2 and with respectively simple roots:
I : {α1, α2}, II : {−α1, α112}.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 55
There is just one associate Cartan matrix:
(cij) =
[
2 −2
−1 2
]
.
The set of positive roots is {α1, α2, α12, α112} where α2 and α112 are only q-Cartan while the
others are only q-truncation.
Proposition 8.1. The following matrices (mij) are realizing solutions of the given braiding and
its reflections:(
mI
ij
)
=
[
2
3 −m
−m 2m
]
,
(
mII
ij
)
=
[
2
3 −4
3 +m
−4
3 +m 8
3 − 2m
]
.
Proof. First we check that condition (5.1)B is satisfied for α1:
m11 =
2
1− c12
=
2
3
and condition (5.1A) is satisfied for α22 and α112:
m22,22 =
2m12
c21
= 2m,
m112,112 =
2m112,−1
c112,1
=
8
3
− 2m.
We then check that the reflection around α1 sends one matrix (mij) to the other as follows
(
mII
ij
)
=
[
2
3 −4
3 +m
−4
3 +m 8
3 − 2m
]
=
[
−1 2
0 1
]T [ 2
3 −m
−m 2m
] [
−1 2
0 1
]
= r1
(
mI
ij
)
. ■
Remark 8.2. When q2 ∈ G2, the root α2 is q-Cartan and q-truncation. When we assume it is
m-truncation, we get the additional solution:(
mI
ij
)
=
[
2
3 −m
−m 1
]
,
(
mII
ij
)
=
[
2
3 −4
3 +m
−4
3 +m 11
3 − 4m
]
,
(
mIII
ij
)
=
[
5
3 − 2m m− 1
m− 1 1
]
.
with III: {α12,−α2}. The root α112 is never m-truncation and it is m-Cartan iff m = 1
2 . But
for this value of m, α2 is also m-Cartan and thus this is not a new solution.
Since this is the only other series in rank 2 and the only other example with nontrivial
behaviour of the Nichols algebra relations, we briefly discuss them: The relations are given
in [40] and [3, Section 7.2]. For q ̸= −1 the truncation and Serre relations are the only defining
relations for this Nichols algebra, for q = −1 there is an additional relation [x112, x12]q = 0
Proposition 8.3.
1. For 0 < m < 1
2 the condition |mii| < 1 holds and (mij) is positive definite, so in this we
are in the subpolar region and all Nichols algebra relations hold.
2. The truncation relation for the root α1 always holds. The truncation relation for α2 holds
iff m > 0. In chamber II the truncation relation for α112 holds if m < 4
3 . We make no
assertion about the truncation relation for the non-simple root vectors.
3. The quantum Serre relation always holds.
4. We make no assertion about the additional relation.
Note that the classification Lemma 7.14 from Lie superalgebras extends to this case mutatis
mutandis.
56 I. Flandoli and S.D. Lentner
Heckenberger row 9
This case of Table 1 in [33] is described by three diagrams:
−ζ2 −ζ2ζ −ζ2 −1ζ3 −ζ−1 −1−ζ3
I II III
where ζ ∈ G12 and with respectively simple roots:
I : {α1, α2}, II : {−α1, α112}, III : {α12,−α122}.
The associate Cartan matrices are
(
cIij
)
=
[
2 −2
−2 2
]
,
(
cIIij
)
=
[
2 −2
−1 2
]
,
(
cIIIij
)
=
[
2 −3
−1 2
]
.
The set of positive roots is {α1, α2, α12, α112, α122} where α12 is only q-Cartan while the others
are only q-truncation.
Proposition 8.4. The following mij matrices are realizing solutions of the given braiding and
its reflections:
(
mI
ij
)
=
[
2
3 − 7
12
− 7
12
2
3
]
,
(
mII
ij
)
=
[
2
3 −3
4
−3
4 1
]
,
(
mIII
ij
)
=
[
1
6 −1
4
−1
4 1
]
.
Proof. First we check that condition (5.1B) is satisfied for all the roots:
m11 =
2
1− c12
=
2
3
, m22 =
2
1− c21
=
2
3
,
m112,112 =
2
1− c112,1
= 1, m122,122 =
2
1− c122,12
= 1
and condition (5.1A) is satisfied for the root α12:
m12,12 =
2m−122,12
c12,112
=
1
6
.
We then check that the reflections send the matrices (mij) above to each other:
(
mII
ij
)
=
[
2
3 −3
4
−3
4 1
]
=
[
−1 2
0 1
]T [ 2
3 − 7
12
− 7
12
2
3
] [
−1 2
0 1
]
= r1
(
mI
ij
)
,
(
mIII
ij
)
=
[
1
6 −1
4
−1
4 1
]
=
[
1 0
1 −1
]T [ 2
3 −3
4
−3
4 1
][
1 0
1 −1
]
= r122r2
(
mI
ij
)
. ■
Corollary 8.5. By formula (3.1) for rank 2, we have that the central charge of the system is
c = −126.
Corollary 8.6. Since
(
mI
ij
)
is positive definite and has diagonal entries |mii| ≤ 1, by Lemma 3.6
the screening algebra is the Nichols algebra.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 57
We conclude this case with a picture illustrating how the set of simple roots behave under
reflections. We write I, II, III, to indicate to which diagram do the simple roots in each case
belong:
{α1,α2}I
{−α1,α112}II {α122,−α2}II
{α12,−α112}III {−α122,α12}III
{α112,−α12}III
r1 r2
r112 r122
sign swap r12
Heckenberger row 10
This case of Table 1 in [33] is described by three diagrams:
−ζ ζ3ζ−2 ζ3 −1ζ−1 −ζ2 −1ζ
I II III
where ζ ∈ G9 and with respectively simple roots:
I : {α1, α2}, II : {−α2, α122}, III : {α12,−α122}.
The associate Cartan matrices are:
(
cIij
)
=
[
2 −2
−2 2
]
,
(
cIIij
)
=
[
2 −2
−1 2
]
,
(
cIIIij
)
=
[
2 −4
−1 2
]
.
The set of positive roots is {α1, α2, α12, α112, α122, α11122} where α1 and α12 are only q-Cartan
while the others are only q-truncation.
Proposition 8.7. The following mij matrices are realizing solutions of the given braiding and
its reflections:
(
mI
ij
)
=
[
5
9 −5
9
−5
9
2
3
]
,
(
mII
ij
)
=
[
2
3 −7
9
−7
9 1
]
,
(
mIII
ij
)
=
[
1
9 −2
9
−2
9 1
]
.
Proof. We check that the roots {α2, α112, α122, α11122} satisfy condition (5.1B), while the root
α1 and α12 satisfy condition (5.1A). We check that the reflections send one (mij) to the other. ■
Corollary 8.8. By formula (3.1) for rank 2, we have that the central charge of the system
is −1088
5 .
Corollary 8.9. Since
(
mI
ij
)
is positive definite and has diagonal entries |mii| ≤ 1, by Lemma 3.6
the screening algebra is the Nichols algebra.
58 I. Flandoli and S.D. Lentner
Heckenberger row 12
This case of Table 1 in [33] is described by three diagrams:
ζ2 ζ−1ζ ζ2 −1−ζ−1 ζ −1−ζ
I II III
where ζ ∈ G8 and with respectively simple roots:
I : {α1, α2}, II : {−α1, α1112}, III : {α112,−α1112}.
There is just one associate Cartan matrix:
(cij) =
[
2 −3
−1 2
]
.
The set of positive roots is {α1, α2, α12, α112, α1112, α11122} where α2 and α112 are only q-Cartan
while the others are only q-truncation.
Proposition 8.10. The following mij matrices are realizing solutions of the given braiding and
its reflections:
(
mI
ij
)
=
[
1
2 −7
8
−7
8
7
4
]
,
(
mII
ij
)
=
[
1
2 −5
8
−5
8 1
]
,
(
mIII
ij
)
=
[
1
4 −3
8
−3
8 1
]
.
Proof. We check that the roots {α1, α12, α1112, α11122} satisfy condition (5.1B), while the
root α2 and α112 satisfy condition (5.1A). We check that the reflections send one (mij) to
the other. ■
Corollary 8.11. By formula (3.1) for rank 2, we have that the central charge of the system
is −874
7 .
Corollary 8.12. Since
(
mII
ij
)
is positive definite and has diagonal entries |mii| ≤ 1, by Lem-
ma 3.6 the screening algebra is the Nichols algebra.
Heckenberger row 13
This case of Table 1 in [33] is described by four diagrams:
ζ6 −ζ−4−ζ−1 ζ6 ζ−1ζ −ζ−4 −1ζ5 ζ −1ζ−5
I II III IV
where ζ ∈ G24 and with respectively simple roots:
I : {α1, α2}, II : {−α1, α1112}, III : {−α2, α122}, IV: {α12,−α122}.
The associate Cartan matrices are(
cIij
)
=
[
2 −3
−2 2
]
,
(
cIIij
)
=
[
2 −3
−1 2
]
,
(
cIIIij
)
=
[
2 −2
−1 2
]
,
(
cIVij
)
=
[
2 −5
−1 2
]
.
The set of positive roots is {α1, α2, α12, α112, α122, α1112, α11122, α1111222} where α12 and α1112
are the only q-Cartan roots while the others are only q-truncation.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 59
Proposition 8.13. The following (mij) matrices are realizing solutions of the given braiding
and its reflections:
(
mI
ij
)
=
[
1
2 −13
24
−13
24
2
3
]
,
(
mII
ij
)
=
[
1
2 −23
24
−23
24
23
12
]
,
(
mIII
ij
)
=
[
1 −19
24
−19
24
2
3
]
,
(
mIV
ij
)
=
[
1 − 5
24
− 5
24
1
12
]
.
Proof. We check that the roots α12 and α1112 satisfy condition (5.1A), while the others satisfy
condition (5.1B). We check that the reflections send one (mij) to the other. ■
Corollary 8.14. By formula (3.1) for rank 2, we have that the central charge of the system
is −7826
23 .
Corollary 8.15. Since
(
mI
ij
)
is positive definite and has diagonal entries |mii| ≤ 1, by Lem-
ma 3.6 the screening algebra is the Nichols algebra.
Heckenberger row 14
This case of Table 1 in [33] is described by two diagrams:
ζ −1ζ2 −ζ−2 −1ζ−2
I II
where ζ ∈ G5 and with respectively simple roots:
I : {α1, α2}, II : {α12,−α2}.
The associate Cartan matrices are
(
cIij
)
=
[
2 −3
−1 2
]
,
(
cIIij
)
=
[
2 −4
−1 2
]
.
The set of positive roots is {α1, α2, α12, α112, α1112, α1111222, α11122, α11111222} where α1, α12, α112
and α11122 are only q-Cartan while the others are only q-truncation.
Proposition 8.16. The following mij matrices are realizing solutions of the given braiding and
its reflections:
(
mI
ij
)
=
[
2
5 −3
5
−3
5 1
]
,
(
mII
ij
)
=
[
1
5 −2
5
−2
5 1
]
.
Proof. We check that the roots α1, α12, α112 and α11122 satisfy condition (5.1A), while the
others satisfy condition (5.1B). We check that the reflections send one (mij) to the other. ■
Corollary 8.17. By formula (3.1) for rank 2, we have that the central charge of the system
is −364.
Corollary 8.18. Since
(
mI
ij
)
is positive definite and has diagonal entries |mii| ≤ 1, by Lem-
ma 3.6 the screening algebra is the Nichols algebra.
60 I. Flandoli and S.D. Lentner
Heckenberger row 17
This case of Table 1 in [33] is described by two diagrams:
−ζ −1−ζ−3 −ζ−2 −1−ζ3
I II
where ζ ∈ G7 and with respectively simple roots:
I : {α1, α2}, II : {α12,−α2}.
The associate Cartan matrices are(
cIij
)
=
[
2 −3
−1 2
]
,
(
cIIij
)
=
[
2 −5
−1 2
]
.
The set of positive roots is
{α1, α2, α12, α112, α1112, α11122, α1111222, α111112222, α111111122222,
α11111222, α1111111122222, α11111112222},
where {α1, α12, α112, α11122, α1111222, α11111222} are only q-Cartan while the others are only q-
truncation.
Proposition 8.19. The following (mij) matrices are realizing solutions of the given braiding
and its reflections:
(
mI
ij
)
=
[
6
14 − 9
14
− 9
14 1
]
,
(
mII
ij
)
=
[
2
14 − 5
14
− 5
14 1
]
.
Proof. We check that the roots {α1,α12,α112,α11122,α1111222,α11111222} satisfy condition (5.1A),
while the others satisfy condition (5.1B). We check that the reflections send one mij-matrix to
the other. ■
Corollary 8.20. By formula (3.1) for rank 2, we have that the central charge of the system
is −962.
Corollary 8.21. Since
(
mI
ij
)
is positive definite and has diagonal entries |mii| ≤ 1, by Lem-
ma 3.6 the screening algebra is the Nichols algebra.
8.2 Classification
In this section we are going to prove the following
Theorem 8.22. For all finite-dimensional diagonal Nichols algebras of rank 2, the realiza-
tions (mij) constructed in Sections 6, 7 or 8.1 are all realizations.
In order to prove it, we are going to go through Table 1 in [33], see which roots are q-
truncation, q-Cartan and compute for every diagram the corresponding (mij) from some neces-
sary conditions in the following Lemma. We will see that for every case this already fixes (mij)
uniquely, and of course we recover what we computed in the previous section.
To prove this result we will need the following two Propositions giving necessary conditions
for a realization. In essence, it lists the conditions on a general (mij) of rank 2, such that after
one reflection condition (5.1) holds.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 61
Proposition 8.23. We consider a diagram
qii qjjqijqji
where we assume that both {αi, αj} are q-truncation, and apply a reflection ri around the root αi
σi : αi 7−→ −αi,
αj 7−→ α
arriving to a new diagram with simple roots {−αi, α := αj − cijαi}. We have
1) if α is m-truncation then
mij =
cij
1− cij
− 1
cij(1− aβ,−αi
)
+
1
cij(1− cji)
, (8.1)
2) if α is m-Cartan then
mij =
cij
1− cij
+
(
1
1−cji
− cij
(1−cij)cβαi
)
(− 1
cβαi
+ cij)
. (8.2)
Proof. Since {αi, αj} are only q-truncation, thus m-truncation, we have the relations
mii =
2
1− cij
, mjj =
2
1− cji
.
1. If β is m-truncation then mββ = 2
1−cβ,−αi
. But for definition of β we have
mββ = mjj − 2cijmij + c2ijmii.
Gathering all the information together we get
2
1− cβ,−βi
=
2
1− cji
− 2cijmij + c2ij
2
1− cij
and from this the final result.
2. This case is completely analogous, with the only difference that β is m-Cartan and thus
mββ =
2mβ,−i
cβαi
we will then have
mββ =
2mβ,−i
cβαi
= −2
mij
cβαi
+
2cij(
2
1−cij
)
cβαi
,
mββ =
2
1− cji
− 2cijmij + c2ij
2
1− cij
.
The two equations together give the thesis. ■
Proposition 8.24. We consider a diagram
qii qjjqijqji
where we assume that {αi, αj} are the first q-Cartan and the latter q-truncation. We apply
a reflection around the q-truncation root αj,
σj : αj 7−→ −αj ,
αi 7−→ β
arriving to a new diagram rj(qij) associated to the roots: {β := αi−cjiαj ,−αj}. In this diagram
we have the necessary conditions
62 I. Flandoli and S.D. Lentner
1) if β is m-truncation then
mij =
cij
1− cijcji
(
1
1− cβ,−αj
−
c2ji
1− cji
)
, (8.3)
2) if β is m-Cartan then
mij =
aijcji
1− cji
·
cjicβ,−αj
− 2
cjicijcβ,−j − cβ,−j − cij
. (8.4)
Heckenberger row 2
We have d = 1 and then ℓ1 = ℓ2 = ℓ
gdc(ℓ,2) . Therefore ℓ ̸= 2 and since cij = −1 we have the
following: If ℓ > 4 or ℓ = 3 then by classification Lemma 6.5 we get a unique solution, presented
in Section 6 Heckenberger row 2. If ℓ = 4 then qii = q2 = −1 and the roots are both q-Cartan
and q-truncation:
� If both are m-Cartan, we find a unique solution, by Lemma 6.5 presented in Section 6
Heckenberger row 2, in the limit case q2 = −1.
� If one of the two is m-truncation, we find a unique solution, presented in Section 7,
Heckenberger row 3, in the limit case q2 = −1. This result is a consequence of Lemma 7.14.
� If both are only m-truncation we recognize the matrix
[
1 − p′
2
− p′
2
1
]
, which is the other Weyl
chamber in Example 5.4.
Heckenberger row 3
We have d = 1 and then ℓ1 = ℓ2 = ℓ
gdc(ℓ,2) . Therefore ℓ ̸= 2 and since c12 = −1 we have the
following: If ℓ > 4 or ℓ = 3 then by classification Lemma 7.14 we get a unique solution, presented
in Section 7 case Heckenberger row 3. If ℓ = 4, α1 is both q-Cartan and q-truncation.
� If it is m-Cartan, we find again the unique solution presented in Section 7 Heckenberger
row 3, in the limit case q2 = −1. This result is a consequence of Lemma 7.14.
� If it is m-truncation we recognize again the matrix
[
1 − p′
2
− p′
2
1
]
which is the other Weyl
chamber in Example 5.4.
Heckenberger row 4
We have d = d2 = 2 and then ℓ1 = ℓ
gdc(ℓ,2) , ℓ2 = ℓ
gdc(ℓ,4) . Moreover ℓ ̸= 2, 4, because q2 ̸= ±1,
and since c12 = −2, c21 = −1 we have the following: If ℓ > 8 or ℓ = 5, 7 then by classification
Lemma 6.5 we get a unique solution, presented in Section 6 Heckenberger row 4. If ℓ = 8 then
the long root α2 is both q-Cartan and q-truncation, while α1 is only q-Cartan.
� If α2 is m-Cartan, we find again the unique solution presented in Section 6, Heckenberger
row 4, by Lemma 6.5.
� If α2 is m-truncation, we find the unique solution presented in Section 7, Heckenberger
row 5, in the limit case q2 = i, by Lemma 7.14.
If ℓ = 3, 6 then the short root α1 is both q-Cartan and q-truncation, while α2 is only q-Cartan.
� If α1 is m-Cartan, we find a unique solution, presented in Section 6 Heckenberger row 4,
again thanks to Lemma 6.5.
� If α1 is m-truncation, we find a family of solution, presented in Section 8.1, Heckenberger
row 6, up to rescaling. The uniqueness follows from Lemma 7.14.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 63
Heckenberger row 5
We have d = 1 and then ℓ1 = ℓ
gdc(ℓ,2) . Moreover ℓ ̸= 2, 4, because q2 ̸= ±1, and since c12 = −2
we have the following: If ℓ > 6 or ℓ = 5 then by classification Lemma 7.14 we get a unique
solution, presented in Section 7 Heckenberger row 5. If ℓ = 3, 6 then the bosonic root α1 is both
q-Cartan and q-truncation.
� If α1 is m-Cartan, we find again the unique solution presented in Section 7 Heckenberger
row 5, by Lemma 7.14.
� If α1 is m-truncation, we recognize the matrix
[
2
3
−2m
−2m 1
]
of Remark 7.22 which is a so-
lution only for m = 1
3 .
Heckenberger row 6
We have d = 1 and then ℓ2 = ℓ
gdc(ℓ,2) . Moreover ℓ ̸= 2, 3, 6, because q2 ̸= 1, ζ, ζ2, with ζ ∈ G3.
Since c12 = −1 we have the following: If ℓ > 6 or ℓ = 5 then by classification Lemma 7.14 we
get a unique solution, presented in Section 8.1 Heckenberger row 6. If ℓ = 4 then the root α2 is
both q-Cartan and q-truncation.
� If α2 is m-Cartan, we find again the unique solution presented in Section 8.1 Heckenberger
row 6, by Lemma 7.14.
� If α2 is m-truncation, we recognize the matrix
[
2
3
−m
−m 1
]
of Remark 8.2 which is a solution
only for m = 1
2 .
Heckenberger row 7
We apply formula (8.1) to the reflection r1 and r2, since the simple roots α1 and α2 as well as the
ones after reflections are only q-truncation and thus m-truncation. From the first reflection we
obtain m12 = −2
3 , while from the latter m12 = −1
2 . Since these results do not match, it means
that there is no possible formulation of the Nichols algebra braiding in terms of the matrix (mij).
Remark 8.25. We have q-truncation roots αi, αj , with qii = ζ, qjj = ζ−1, both third roots of
unity and it is not possible to realize both of them with mii = mjj =
2
3 . This is another way to
see that this case is not realizable.
Heckenberger row 8
We apply formula (8.1) to the reflections r1 and r2, since the simple roots α1 and α2 as well as
the ones after reflections are only q-truncation and thus m-truncation. From the first reflection
we obtain m12 = −3
4 , while from the latter m12 = − 7
12 . Since these results do not match,
it means that there is no possible formulation of the Nichols algebra braiding in terms of the
matrix (mij).
Heckenberger row 9
We apply formula (8.1) to the reflection r1 or r2, since the simple roots α1 and α2 as well as
the ones after reflections are only q-truncation and thus m-truncation. The resulting m12 shows
that this is the matrix (mij) appearing in Section 8.1. This is thus the only possible solution.
64 I. Flandoli and S.D. Lentner
Heckenberger row 10
We apply formula (8.3) to the reflection r2, since the simple root α1 is only q-Cartan and thus
m-Cartan, while α2 as well as the ones after reflections are only q-truncation and thus m-
truncation. The resulting m12 shows that this is the mij appearing in Section 8.1. This is thus
the only possible solution.
Heckenberger row 11
We have d = d2 = 3 and then ℓ1 =
ℓ
gdc(ℓ,2) , ℓ2 =
ℓ
gdc(ℓ,6) . Moreover ℓ ̸= 2, 3, 4, 6 because q2 ̸= ±1,
q2 ̸∈ G3. Since c12 = −3 and c21 = −1 we have the following: If ℓ > 12 or ℓ = 5, 7, 9, 10, 11
then by classification Lemma 6.5 we get a unique solution, presented in Section 6 Heckenberger
row 11. If ℓ = 12 then the root α2 is both q-Cartan and q-truncation, while the root α1 is only
q-Cartan.
� If α2 is m-Cartan, we find again the unique solution presented in Section 6 Heckenberger
row 11, by Lemma 6.5.
� If α2 is m-truncation, we recognize the matrix
[
2m −3m
−3m 1
]
, which is a solution only for
m = 1
6 .
If ℓ = 8 then the root α1 is both q-Cartan and q-truncation, while the root α2 is only q-Cartan.
� If α1 is m-Cartan, we find again the unique solution presented in Section 6 Heckenberger
row 11, by Lemma 6.5.
� If α1 is m-truncation, we recognize the matrix
[
1
2
−3m
−3m 6m
]
which is a solution only for
m = 1
4 .
Heckenberger row 12
We apply formula (8.3) to the reflections r1, since the simple roots α1 as well as the ones after
reflections are only q-truncation and thus m-truncation, while α2 is only q-Cartan, and thus
m-Cartan. The result is m12 = −7
8 , which matches with the one of Section 8.1.
Heckenberger row 13
We apply formula (8.1) to the reflection r1 or r2, since the simple roots α1 and α2 as well as
the ones after reflections are only q-truncation and thus m-truncation. The resulting m12 shows
that this is the mij appearing in Section 8.1. This is thus the only possible solution.
Heckenberger row 14
We apply formula (8.4) to the reflections r2, since the simple roots α1 as well as the ones
after reflections are only q-Cartan and thus m-Cartan, while α2 is only q-truncation, and thus
m-truncation. The result is m12 = −3
5 , which matches with the one of Section 8.1.
Heckenberger row 15
We apply formula (8.1) to the reflections r1 and (8.2) to r2 since the simple roots α1 and α2 as
well as the ones after r1 are only q-truncation and thus m-truncation, while the ones after r2
are only q-Cartan, and thus m-Cartan. From the first reflection we obtain m12 = −4
5 , while
from the latter m12 = −11
20 . Since these results do not match, it means that there is no possible
formulation of the Nichols algebra braiding in terms of the matrix (mij).
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 65
Heckenberger row 16
The root α1 is q-Cartan so we can not start with the system of simple roots α1, α2 if we want
to compare the results of the reflections around them. We then start with the simple roots α122
and −α2 which are only q-truncation and thus m-truncation. After reflection r122 we obtain
a only q-Cartan, and thus m-Cartan, simple root. While after reflection r2 we obtain a only
q-truncation, and thus m-truncation, simple root. We then apply (8.2) to r122 and (8.1) to r2
obtaining two different results. Hence there is no possible formulation of the Nichols algebra
braiding in terms of the matrix (mij).
Heckenberger row 17
We apply formula (8.3) to the reflections r2, since the simple roots α2 as well as the ones after
reflections are only q-truncation and thus m-truncation, while α1 is only q-Cartan, and thus
m-Cartan. The result is m12 = − 5
14 , which matches with the one of Section 8.1.
9 Rank 3
We now rise the rank by one and construct all matrices (mij) which realize finite-dimensional
diagonal Nichols algebras of rank 3, listed in Table 2 of [33].
For Cartan type we will refer to the study of Section 6. For super Lie type we will explicitly
compute the realizing solutions.
For the other cases, we will see that the matrices (mij) matrices are completely fixed by the
lower rank: this will imply uniqueness of the solution and make it not just a construction result
but also a classification one.
In particular for these latter cases we will proceed as follows:
� Given a q-diagram in rank 3, we will consider it as two rank 2 q-diagrams joined in the
middle node. We will then associate to both sides the matrices (mij) realizing them, found
in the rank 2 study. For these matrices (mij) to be compatible, some restriction on the
parameter of which they depend will possibly appear.
� We will then reflect the q-diagram on its q-truncation roots and proceed again as in the
first point for the new diagram. We reflect until we arrive not just to an already found q-
diagram, but also when the realization (mij) is repeated (the matrix (mij) can be different
also if associated to the same q-diagram).
� We will then have to make sure that all the conditions found on the parameters are
compatible and acceptable, in order for the rank 3 matrices (mij) to be realizing solutions.
The q-diagrams and the associated realizing solutions are listed in Table 2 of the appendix.
Heckenberger row 1
This case belongs to the Cartan section. In particular it corresponds to the Lie algebras A3 and
it is described by the following q-diagram with corresponding realization (mij):
q2
2m
q2
2m
q2
2m
q−2
−2m
q−2
−2m
Remark 9.1. When q2 ∈ G2 the roots are both q-Cartan and q-truncation and the q-diagram
reads
−1 −1 −1−1 −1
66 I. Flandoli and S.D. Lentner
We have the following extra solutions:
– When α1 is m-truncation and α2, α3 are m-Cartan we find
−1
1
−1
2m
−1
2m
−1
−2m
−1
−2m
which is one chamber of the Lie superalgebra A(2, 0) described in Heckenberger row 4.
– When α1, α2 are m-truncation and α3 is m-Cartan we find
−1
1
−1
1
−1
2m
−1
m′
−1
−2m
which is a m-solution just for m = 1
2 and m′ = −1. But for these values of m, m′ the roots
α1, α2 are also m-Cartan and thus this is not a new solution.
– When α2 is m-truncation and α1, α3 are m-Cartan we find
−1
2m′
−1
1
−1
2m′′
−1
−2m′
−1
−2m′′
This is a solution either for m′ = 1
2 for which we end up again in the previous point, or
for m′ = 1−m′′, which gives us one chamber of the Lie superalgebra A(1, 1) described in
Heckenberger row 8.
– When α1, α3 are m-truncation and α2 is m-Cartan we find
−1
1
−1
2m
−1
1
−1
−2m
−1
−2m
which is another chamber of the Lie superalgebra A(1, 1) described in Heckenberger row 8.
– When the roots are all m-truncation we find
−1
1
−1
1
−1
1
−1
m′
−1
m′′
This is a solution either for m′ = −m′′−2 which is again a chamber of the Lie superalgebra
A(1, 1), or for m′ = m′′ = −1 for which the roots are also m-Cartan and thus does not
give a new solution.
Heckenberger row 2
This case belongs to the Cartan section. In particular it corresponds to the Lie algebras B3 and
it is described by the following q-diagram with corresponding realization (mij):
q4
4r
q4
4m
q2
2m
q−4
−4m
q−4
−4m
Remark 9.2. When q2 ∈ G4 the roots α1, α2 are both q-Cartan and q-truncation and the
q-diagram reads
−1 −1 i−1 −1
For all the possible combinations of m-truncation and m-Cartan roots, no new solution is found.
In some cases we find the Lie superalgebra B(2, 1) described in Heckenberger row 5.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 67
Remark 9.3. When q2 ∈ G3 the root α3 is both q-Cartan and q-truncation and the q-diagram
reads
ζ2 ζ2 ζζ−2 ζ−2
with ζ ∈ G3. The case when it is m-truncation is a solution only for m = 1
3 for which the root
is also m-Cartan and thus does not give a new solution.
Heckenberger row 3
This case belongs to the Cartan section. In particular it corresponds to the Lie algebras C3 and
it is described by the following q-diagram with corresponding realization (mij):
q2
2m
q2
2m
q4
4m
q−2
−2m
q−4
−4m
Remark 9.4. If q2 ∈ G4, α3 is both q-Cartan and q-truncation and the q-diagram reads
i
2m
i
2m
−1
1
−i
−2m
−1
−4m
The case when it is m-truncation is a solution iff m = 1
4 for which it is actually also m-Cartan.
So this is not a new solution.
Heckenberger row 4
Row 4 of Table 2 in [33] corresponds to the Lie superalgebra A(2, 0).
The simple roots in the standard chamber are {α1 = αf , α2, α3}. We then have just a bosonic
part g′. The inner products is given by
(αi, αj) =
0 −1 0
−1 2 −1
0 −1 2
and therefore
−1
1
q2
2m
q2
2m
q−2
−2m
q−2
−2m
Reflecting around α1 we find the following
−1
1
−1
1
q2
2m
q2
−2 + 2m
q−2
−2m
Reflecting around the second root we find a symmetric result. The roots satisfy condition (5.1)
for all m and therefore this (mij) is a realizing solution.
68 I. Flandoli and S.D. Lentner
Heckenberger row 5
Row 5 of Table 2 in [33] corresponds to the Lie superalgebra B(2, 1).
The simple roots in the standard chamber are {α1 = αf , α2, α3}. We then have just a bosonic
part g′. The inner products is given by
(αi, αj) =
0 −2 0
−2 4 −2
0 −2 2
and therefore
−1
1
q4
4m
q2
2m
q−4
−4m
q−4
−4m
Reflecting around α1 we find the following
−1
1
−1
1
q2
2m
q4
−2 + 4m
q−4
−4m
and after another reflection around the second root we find the following
q4
4m
−1
1
−q−2q−4
−4m
q4
−2 + 4m1− 2m
The roots satisfy condition (5.1) for all m and therefore this (mij) is a realizing solution.
Remark 9.5. If q2 ∈ G4 then the root α2 is both q-Cartan and q-truncation. This case has
been already studied in details in Heckenberger row 2 Remark 9.2.
Remark 9.6. If q2 ∈ G3 then the root α3 is both q-Cartan and q-truncation. When it is
m-truncation we get
−1
1
ζ2
4m
ζ
2
3
ζ−2
−4m
ζ−2
−4m
This is a solution iff m = 1
3 . But for this value of m, α3 is also m-Cartan and thus this is not
a new solution.
Heckenberger row 6
Row 6 of Table 2 in [33] corresponds to the Lie superalgebra C(3).
The simple roots in the standard chamber are {α1 = αf , α2, α3}. We then have just a bosonic
part g′. The inner products is given by
(αi, αj) = −
0 −1 0
−1 2 −2
0 −2 4
and therefore
−1
1
q2
2m
q4
4m
q−2
−2m
q−4
−4m
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 69
Reflecting around α1 we find the following
−1
1
−1
1
q4
4m
q2
−2 + 2m
q−4
−4m
Reflecting around α12 we find the following
−1
q2 −1
q−2 q4
q−2
1
2m 1
−2m −2 + 4m
−2m
The roots satisfy condition (5.1) for all m and therefore this (mij) is a realizing solution.
Remark 9.7. If q2 ∈ G4, α3 is both q-Cartan and q-truncation. When it is m-truncation we
find
−1
1
i
2r
−1
1
−i
−2m
−1
−4m
This is a solution iff m = 1
4 . But for this value of m, α3 is also m-Cartan and thus this is not a
new solution.
Remark 9.8. The simple roots in the standard chamber can be expressed according to [43] by
α1 = αf = ϵ1 − δ1, α2 = δ1 − δ2, α3 = 2δ2.
Heckenberger row 7
Row 7 of Table 2 in [33] corresponds to the Lie superalgebra G(3) and it has been already
explicitly treated as sporadic case of super Lie type in Section 7.7.1.
Heckenberger row 8
Row 8 of Table 2 in [33] corresponds to the Lie superalgebra A(1, 1).
The simple roots in the standard chamber are {α1, α2 = αf , α3}. We then have two bosonic
parts g′ and g′′. The inner products is given by
(αi, αj) =
2 −1 0
−1 0 −1
0 −1 2
and therefore
q2
2m′
−1
1
q−2
2m′′
q−2
−2m′
q2
−2m′′
70 I. Flandoli and S.D. Lentner
Reflecting around α2 we find the following
−1
1
−1
1
−1
1
q2
−2 + 2m′
q−2
−2 + 2m′′
Other reflections give different matrices (mij) as shown in Table 2. However, exception (4)
of Lemma 7.12, already appears. Indeed to the latter diagram is associated the following:
mC
ij =
1 −1 +m′ −1 +m′ +m′′
−1 +m′ 1 −1 +m′′
−1 +m′ +m′′ −1 +m′′ 1
.
We then have to ask mC
13 = 0, i.e., m′ +m′′ = 1. In this case these matrices (mij) are realizing
solution.
Remark 9.9. The simple roots in the standard chamber can be expressed according to [43] by
α1 = ϵ1 − ϵ2, α2 = αf = ϵ2 − δ1, α3 = δ1 − δ2,
with vectors ϵi generating g′ and δi generating g′′.
Heckenberger row 9–10–11
Rows 9, 10, 11 of Table 2 in [33] correspond to the Lie superalgebra D(2, 1;α) and it has been
already explicitly treated as sporadic case of super Lie type in Section 7.7.1.
Heckenberger row 12
The first diagram is a composition of the diagrams of rank 2: #2 with q = −ζ−1 and #6 with
q = −ζ−1, with ζ ∈ G3.
−ζ−1
2m′
−ζ−1
2m′
ζ−ζ
−2m′
−ζ
2m′′−2m′′ 2
3
For them to be joint in the middle circle we find m′ = m′′ =: m.
The only q-truncation root is the third. Reflecting on it we find the same diagram and as
matching condition 2m = 8
3 −2m, i.e., m = 2
3 . But q = eiπm ∈ G6. So this case is not realizable.
Heckenberger row 13
This case has two sub cases: ζ ∈ G3 and ζ ∈ G6 and diagram:
ζ
2m′
ζ
2m′
−1ζ−1
−2m′
ζ−2
2m′′−4m′′ 1
1. Suppose ζ ∈ G3. The first diagram is a composition of the diagrams of rank 2: #2 with
q = ζ and #5 with q = ζ. For them to be joint in the middle circle we find m′ = m′′ =: m.
The only q-truncation root is the third. Reflecting on it we find a diagram composition of
#4 with q = −ζ−1 and #5 with q = ζ. As matching condition we find m = −2m+ 1, i.e.,
m = 1
3 . This case is thus realizable by the unique solution with parameter m = 1
3 .
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 71
2. Suppose ζ ∈ G6. We proceed analogously, but after reflecting around the third root we
find a diagram which is composition of #6 with q = ζ and #5 with q = ζ. The condition
now is m = 1
6 which is an acceptable condition. This case is thus realizable by the unique
solution with parameter m = 1
6 .
Heckenberger row 14
This case is not realizable, since one of the diagrams contains diagram #7 of rank 2 which is on
turn not realizable.
Heckenberger row 15
The first diagram is a composition of the diagrams of rank 2: #3 with q = ζ and #5 with q = ζ,
where ζ ∈ G3.
−1
1
ζ
2m′
−1ζ−1
−2m′
ζ
2m′′−4m′′ 1
For them to be joint in the middle circle we find m′ = m′′ =: m. After the reflections around
r12 ◦ r1 we find the condition m = 1
3 which is acceptable and gives a unique realizable solution.
Heckenberger row 16
The first diagram is a composition of the diagrams of rank 2: #3 with q = ζ and #6 with
q = −ζ, where ζ ∈ G3.
−1
1
ζ
2m′
−ζζ−1
−2m′
−ζ−1
2
3
−2m′′ 2m′′
For them to be joint in the middle circle we find m′ = 1
3 . After reflecting on the second root we
find the condition m′′ = 5
6 . This case is thus realizable by the unique solution with parameters
m′ = 1
3 and m′′ = 5
6 .
Heckenberger row 17
This case is not realizable, since one of the diagrams contains diagram #7 of rank 2 which is on
turn not realizable.
Heckenberger row 18
The first diagram is a composition of the diagrams of rank 2: #2 with q = ζ and #6 with q = ζ,
with ζ ∈ G9:
ζ
2m′
ζ
2m′
ζ−3ζ−1
−2m′
ζ−1
2m′′−2m′′ 2
3
For them to be joint in the middle circle we find m′ = m′′ =: m. The only q-truncation root is
the third. Reflecting on it we find the same diagram and as matching condition m = −8
3 + 2r,
i.e., m = 8
9 . This case is thus realizable by the unique solution with parameter m = 8
9 .
72 I. Flandoli and S.D. Lentner
10 Rank ≥ 4
In rank ≥ 4 we do not list all diagram, but we give an effective way to determine all possible
realizations from the list of rank 3 realizations:
Determining all possible realizations is a simple matter of covering a q-diagram with smaller q-
diagrams, looking up their realizations (which are typically unique or depend on one parameter)
and choosing the parameters such that (mij) agrees on the overlap of the subdiagrams. Typically
the result is a unique possible realization.
Verifying on the other hand that a possible realization is indeed a realization can in rank ≥ 4
be in principle done as follows: The entry m
rk(C)
ij after a reflection on αk from mC
ij is entirely
determine in the rank 3 Nichols subalgebra and root system generated by αi, αj , αk. Hence in
principle we go through all simple roots αk in all chamber, which are not m-Cartan (otherwise
the diagram and its realization remains unchanged), and compare our choices of
(
mC
ij
)
,
(
m
rk(C)
ij
)
as follows:
� If αk is connected to αi and αj determine its reflection of this rank 3 subdiagram from
the list and verify that it coincides with the choice of the realization
(
m
rk(C)
ij
)
. If αk is
a branch point, this has to be verified for all combinations of αi and αj . If αk is only
connected to one vertex αi, this has to be verified only for the rank 2 subdiagram.
� For each αi, αj not connected to αk, verify that mC
ij coincides with the choice of realiza-
tion m
rk(C)
ij .
In practical examples, we have chosen large subdiagrams that correspond in many Weyl cham-
bers, and we have tried to mostly have an overlap of rank 2 between these subdiagrams, so that
most verifications above are true by construction.
Example 10.1. We consider rank 4 row 18 with q3 = 1, q ̸= 1:
q−1 q−1 q −1q q q−1
In the first diagram we consider the subdiagram on the nodes 1, 2, 3 of Cartan type B3 (we
slightly rewrite the q’s to make this visible) and the subdiagram on 2, 3, 4 of super Lie type C(3).
Each has a unique family of realizations depending on a parameter m1 resp. m2. The overlap
between the diagrams (Cartan type B2) has decorations 2m1, −2m1, m1 resp. 2m2, −2m2, m2.
Hence the only possible realization is for m1 = m2 =: m
q2
2m
q2
2m
q
m
−1
1
q−2
−2m
q−2
−2m
q−1
−m
The only relevant reflection is r4, and the entire neighborhood is contained in C(3). So we look
up the reflection of the realization of C(3) and leave the remaining realization unchanged:
q2
2m
q2
2m
−1
1
−1
1
q−2
−2m
q−2
−2m
q
−2 +m
We now have to verify that the subdiagram on 1, 2, 3 with this decoration turns into a listed
realization. Indeed this is the realization of A(2|0) at q2 with parameter m3 = 2m.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 73
The only relevant new reflection is r3 and the neighborhood is contained in C(3). So we look
up the reflection there:
q2
2m
−1
1
−1
1
q
m
q−2
−2m
q2
2m− 2
q−1
m
q−1
−m
Since 3 is also in the subdiagram on 1, 2, 3, this is automatically still the realization of A(2|0).
The only relevant new reflection is r2 at the branch point.We introduce a new subdiagram
on 1, 2, 4. This is the q-diagram of A(1|1) at q with realizations parametrized by m4. But
matching the decorations on the right side of 2 requires m4 = m and matching it on the left side
requires 2−m4 = 2m. This is only possible for m = 2
3 .
The only relevant new reflection is r2, a branch point, and part of the neighborhood is
contained in A(2|0) and parts in A(1|1). So we look up the reflection there:
−1
1
−1
1
q2
2m
−1
1
q2
−2 + 2m
q−2
−2m
q
−2 +m
Now there are two relevant new reflections: r1 has its neighborhood in A(2|0) and A(1|1) and
accordingly gives
−1
1
q2
2m
q2
2m
−1
1
q−2
−2m
q−2
−2m
q
−2 +m
We have to verify that the reflections in both subdiagrams agree (which is clear because they
are reflections in a common A(1|0)).
The second reflection r2 has its neighborhood in A(1|1):
−1
1
q
m
q2
2m
−1
1
q−1
−m
q−2
−2m
q−1
−m
We have to verify that the new diagram on 1, 2, 3 appears in the list of realizations, namely super
Lie type C(3) at q with m5 = 5, which again requires on the edge 12 the identity −2+2m = −m
for m = 2
3 .
Applying both reflections in either order gives the following (again this is not problematic
because the neighborhood is in both diagrams and the reflection on A(1|0) gives the same result:
−1
1
−1
1
q2
2m
−1
1
q
−2 +m
q−2
−2m
q−1
−m
74 I. Flandoli and S.D. Lentner
To summarize: The following is the unique realization
q2
4
3
q2
4
3
q
2
3
−1
1
q−2
−4
3
q−2
−4
3
q−1
−2
3
11 Tables: realizing lattices of Nichols algebras in rank 2 and 3
We now list from [33] all finite-dimensional diagonal Nichols algebras in rank 2 and 3 in terms
of their q-diagrams, and below each of them we display the corresponding realizing lattice in
terms of m-diagrams, such that qij = eiπmij and the reflection compatibility (5.1) holds.
The numbers of the rows are Heckenberger’s numbering, but sometimes we subdivide the
cases, e.g., row 2 into 2′ for q = −1 and 2′′ for q ̸= ±1, if they have different number of
realizations. Note that we display the Nichols algebras associated to quantum (super-)groups as
in Heckenberger list (with qii = 1 for a short root) in contrast to the notation used for quantum
(super-)groups (with qii = q(αi,αi) = q2 for a short root), due to the usual normalization of the
Killing form, which we used in Sections 6 and 7.
Table 1. Realization of finite-dimensional diagonal Nichols algebras of rank 2.
row braiding conditions comments
2′
−1 −1−1
−mm m
one solution according to A2 (see 2′′)
−1 −1−1
−mm 1
−1 −1−1
−2 +m1 1
one solution according to A(1, 0) (see 3)
2′′
q qq−1
−mm m
q ̸= ±1 Cartan, A2
3
q −1q−1
−mm 1
−1 −1q
−2 +m1 1
q ̸= ±1 super Lie, A(1, 0)
4′
i −1−1
−2mm 2m
i ∈ G4
one solution according to B2 (see 4′′′)
i −1−1
−2mm 1
i −1−1
−2 + 2m−m+ 1 1
one solution according to B(1, 1) (see 5)
4′′
ζ ζ−1ζ
−2mm 2m
ζ ∈ G3
one solution according to B2 (see 4′′′)
ζ ζ−1ζ
−2m2
3
2m
ζ ζ−1ζ
− 8
3
+ 2m2
3
8
3
− 2m
one solution according to 6
4′′′
q q2
2m
q−2
−2mm
q ̸= ±1,
q ̸∈ G3,G4
Cartan, B2
5
q −1q−2
−2mm 1
−q−1
1−m
−1q2
−2 + 2m 1
q ̸= ±1,
q ̸∈ G4
super Lie, B(1, 1)
6
ζ
2
3
qq−1
−m m
ζ
2
3
ζq−1ζ−1q
− 8
3
+ r 8
3
−m
ζ ∈ G3,
q ̸= 1, ζ, ζ2
7
ζ −1−ζ ζ−1 −1−ζ−1
ζ ∈ G3 no solution
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 75
8
−ζ−2 −ζ2−ζ3 −ζ−2 −1ζ−1 −ζ2 −1−ζ
−ζ3 −1ζ −ζ3 −1−ζ−1 ζ ∈ G12 no solution
9
−ζ2
2
3
−ζ2
2
3
ζ
− 7
6
−ζ2
2
3
−1ζ3
− 3
2 1
−ζ−1
1
6
−1−ζ3
− 1
2 1
ζ ∈ G12
10
−ζ
5
9
ζ3
2
3
ζ−2
− 10
9
ζ3
2
3
−1ζ−1
− 14
9 1
−ζ2
1
9
−1ζ
− 4
9 1
ζ ∈ G9
11
q q3
3r
q−3
−3rr
q ̸∈ G3,
q ̸= ±1
Cartan, G2
12
ζ2
1
2
ζ−1
7
4
ζ
− 7
4
ζ2
1
2
−1−ζ−1
− 5
4
ζ
1
4
−1−ζ
− 3
4
1
1
ζ ∈ G8
13
ζ6
1
2
−ζ−4
2
3
−ζ−1
− 13
12
ζ6
1
2
ζ−1
23
12
ζ
− 23
12
−ζ−4
2
3
−1ζ5
− 19
12
ζ
1
12
−1ζ−5
− 5
12 11
ζ ∈ G24
14
ζ
2
5
−1ζ2
− 6
5
−ζ−2
1
5
−1ζ−2
− 4
5 11 ζ ∈ G5
15
ζ −1ζ−3 −ζ −1−ζ−3
−ζ−2 −1ζ3 −ζ−2 −1−ζ−3 ζ ∈ G20 no solution
16
−ζ ζ5−ζ−3 ζ3 −ζ−4−ζ4
ζ5 −1−ζ−2 ζ3 −1−ζ2
ζ ∈ G15 no solution
17
−ζ
6
14
−1−ζ−3
− 9
7
−ζ−2
2
14
−1−ζ3
− 5
7 11
ζ ∈ G7
Table 2. Realization of finite-dimensional diagonal Nichols algebras of rank 3.
row braiding conditions comments
1′
−1
m
−1
m
−1
m
−1
−m
−1
−m
one solution according to A3 (see 1′′)
−1
1
−1
m
−1
m
−1
−m
−1
−m
−1 −1 −1−1 −1
1 1 m−2 +m −m
one solution according to A(2, 0) (see 4)
76 I. Flandoli and S.D. Lentner
−1
m
−1
1
−1
−1
−m
−1
−2 +m 2−m
−1 −1 −1−1 −1
−1 −1 −1−1 −1
−1 −1 −1−1 −1
1 1
1m− 2 −m
1 m 1−m −m
1 2−m 1−2 +m −2 +m
one solution according to A(1, 1) (see 8)
1′′
q
m
q
m
q
m
q−1
−m
q−1
−m
q ̸= ±1 Cartan, A3
2′
−1
2m
−1
2m
i
m
−1
−2m
−1
−2m
i ∈ G4 one solution according to B3 (see 2′′)
−1
1
−1
2m
i
m
−1
−2m
−1
−2m
−1 −1 i−1 −1
−1 −1 i−1 −1
1 1 m−2 + 2m −2m
2m 1 −m+ 1−2m −2 + 2m
one solution according to B(2, 1) (see 5)
2′′
q2
2m
q2
2m
q
m
q−2
−2m
q−2
−2m
q ̸= ±1,
q ̸∈ G4
Cartan, B3
3
q
m
q
m
q2
2m
q−1
−m
q−2
−2m
q ̸= ±1 Cartan, C3
4
−1 q qq−1 q−1
−1 −1 qq q−1
1 m m−m −m
1 1 m−2 +m −m
q ̸= ±1 super Lie, A(2, 0)
5
−1 q2 qq−2 q−2
−1 −1 qq2 q−2
q2 −1 −q−1q−2 q2
1 2m m−2m −2m
1 1 m−2 + 2m −2m
2m 1 −m+ 1−2m −2 + 2m
q ̸= ±1,
q ̸∈ G4
super Lie, B(2, 1)
6
−1 q q2q−1 q−2
−1
q −1
q−1 q2
q−1
1
m 1
−m 2m− 2
−m
−1 −1 q2q q−2
1 m 2m−m −2m
1 1 2m−2 +m −2m
q ̸= ±1 super Lie, C(3)
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 77
7
−1 q q3q−1 q−3
−1
q −1
q−1 q3
q−2
1
m 1
−m 3m− 2
−2m
−1 −1 q3q q−3
q3 −1 −q−1q−3 q2
1 m 3m−m −3m
1 1 3m−2 +m −3m
3m 1 1−m−3m −2 + 2m q ̸= ±1,
q ̸∈ G3
super Lie, G(3)
8
q −1 q−1q−1 q
−1 −1 −1q q−1
−1 q −1q−1 q−1
−1 q−1 −1q q
m 1 2−m−m −2 +m
1 1 1m− 2 −m
1 m 1−m −m
1 2−m 1−2 +m −2 +m
q ̸= ±1 super Lie, A(1, 1)
9
10
11
q′ −1 q′′(q′)−1 (q′′)−1
−1
−1 −1
q′ q′′
q′′′
1
1 1
m′ − 2 m′′ − 2
m′′′ − 2
q′ −1 q′′′(q′)−1 (q′′′)−1
q′′ −1 q′′′(q′′)−1 (q′′′)−1
m′ 1 m′′−m′ −m′′
m′ 1 m′′′−m′ −m′′′
m′′ 1 m′′′−m′′ −m′′′
q′, q′′, q′′′ ̸= 1,
q′q′′q′′′ = 1
super Lie, D(2, 1;α), m′ +m′′ +m′′′ = 2
12 −ζ−1 −ζ−1 ζ−ζ −ζ ζ ∈ G3 no solution
13′
ζ
2
3
ζ
2
3
−1
1
ζ−1
− 2
3
ζ−2
− 4
3
ζ
2
3
−ζ−1
1
3
−1
1
ζ−1
− 2
3
ζ2
− 2
3
ζ ∈ G3 m = 1
3
13′′
ζ
2
6
ζ
2
6
−1
1
ζ−1
− 2
6
ζ−2
− 4
6
ζ
2
6
−ζ−1
2
3
−1
1
ζ−1
− 2
6
ζ2
− 4
3
ζ
7
3
−ζ−1
2
3
−1
1
ζ−1
− 7
3
ζ2
− 4
3
ζ ∈ G6 m = 1
6
78 I. Flandoli and S.D. Lentner
14
−1 −ζ−1 ζ−ζ −ζ −1 −1 ζ−ζ−1 −ζ
−ζ−1 −1 ζ−1−ζ −ζ−1
ζ ∈ G3 no solution
15
−1
1
ζ
2
3
−1
1
ζ−1
− 2
3
ζ
− 4
3 −1
ζ ζ
ζ−1 ζ−1
ζ−1
1
2
3
2
3
− 2
3
− 2
3
− 2
3
−1
1
−1
1
−1
1
ζ
− 4
3
ζ
− 4
3
−1
1
−ζ−1
1
3
−1
1
ζ−1
− 2
3
ζ−1
− 2
3
ζ ∈ G3 r = 1
3
16
−1
1
ζ
2
3
−ζ
5
3
ζ−1
− 2
3
−ζ−1
− 5
3 −1
ζ −1
−1 −ζ
ζ−1
1
2
3 1
−1 − 1
3
− 2
3
−1
1
−1
1
−ζ
5
3
ζ
− 4
3
−ζ−1
− 5
3
ζ
2
3
−1
1
−ζ
5
3
−1
−1
−ζ−1
− 5
3
ζ
2
3
−ζ
5
3
−ζ
5
3
−ζ−1
− 5
3
−ζ−1
− 5
3
ζ ∈ G3 r = 5
6
17
−1 −1 −1−1 ζ
ζ
−ζ −1
−ζ−1 ζ−1
−ζ−1
−1 ζ −1−1 ζ−1
−1 −1 −1ζ −ζ −1 −ζ −1ζ −ζ−1
−1 ζ−1 −1ζ−1 −ζ−1
−1
−1 ζ
−1 ζ−1
−ζ
−1 ζ −1ζ−1 −ζ
−1 −1 ζ−1−1 −ζ−1
ζ ∈ G3 no solution
18
ζ
16
9
ζ
16
9
ζ−3
2
3
ζ−1
− 16
9
ζ−1
− 16
9
ζ
16
9
ζ−4
8
9
ζ−3
2
3
ζ−1
− 16
9
ζ4
− 8
9
ζ ∈ G9 r = 8
9
Acknowledgements
IF and SL are partially supported by the RTG 1670 “Mathematics inspired by String theory
and Quantum Field Theory”. Many thanks to Christian Reiher for suggesting the analytic
continuation by partial integration in Section 4.3, to Ivan Angiono for explaining to us Proposi-
tion 2.20, to Sven Ole Warnaar for answering questions on the g-Selberg integral formula, and
to the anonymous referees for many improving comments on the manuscript.
Algebras of Non-Local Screenings and Diagonal Nichols Algebras 79
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1 Introduction
2 Preliminaries on Nichols algebras
2.1 Definition and properties
2.2 Examples
2.3 Generalized root system and Weyl groupoid
3 Preliminaries on screening operators
3.1 Vertex algebras and their representation theory
3.2 Screening operators and Nichols algebra relations
3.3 Central charge
4 Analytical continuation of screening relations
4.1 Commutativity relations
4.2 Truncation relations
4.3 Analytical continuation by recursion
4.4 Serre relations for Cartan matrix entry -1
5 Formulation of the classification problem
6 Cartan type
6.1 q diagram
6.2 Construction of (mij)
6.3 Central charge
6.4 Algebra relations
6.5 Examples: Cartan type realizations in rank 2
7 Super Lie type
7.1 q diagram
7.2 Construction of (mij)
7.3 Central charge
7.4 Algebra relations
7.5 Examples in rank 2
7.6 Arbitrary rank
7.7 Sporadic cases
7.7.1 G(3)
8 Other cases in rank 2
8.1 Construction of (mij)
8.2 Classification
9 Rank 3
10 Rank 4
11 Tables: realizing lattices of Nichols algebras in rank 2 and 3
References
|
| id | nasplib_isofts_kiev_ua-123456789-211527 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T08:48:00Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Flandoli, Ilaria Lentner, Simon D. 2026-01-05T12:26:00Z 2022 Algebras of Non-Local Screenings and Diagonal Nichols Algebras. Ilaria Flandoli and Simon D. Lentner. SIGMA 18 (2022), 018, 81 pages 1815-0659 2020 Mathematics Subject Classification: 16T05;17B69 arXiv:1911.11040 https://nasplib.isofts.kiev.ua/handle/123456789/211527 https://doi.org/10.3842/SIGMA.2022.018 In a vertex algebra setting, we consider non-local screening operators associated with the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a quantum shuffle algebra or Nichols algebra associated with a diagonal braiding, which encodes the non-locality and non-integrality. In the present article, we take all finite-dimensional diagonal Nichols algebras, as classified by Heckenberger, and find all lattice realizations of the braiding that are compatible with reflections. Usually, the realizations are unique or come as one- or two-parameter families. Examples include realizations of Lie superalgebras. We then study the associated algebra of screenings with improved methods. Typically, for positive definite lattices we obtain the Nichols algebra, such as the positive part of the quantum group, and for negative definite lattices we obtain a certain extension of the Nichols algebra generalizing the infinite quantum group with a large center. IF and SL are partially supported by the RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory”. Many thanks to Christian Reiher for suggesting the analytic continuation by partial integration in Section 4.3, to Ivan Angiono for explaining to us Proposition 2.20, to Sven Ole Warnaar for answering questions on the -Selberg integral formula, and to the anonymous referees for many helpful comments on the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Algebras of Non-Local Screenings and Diagonal Nichols Algebras Article published earlier |
| spellingShingle | Algebras of Non-Local Screenings and Diagonal Nichols Algebras Flandoli, Ilaria Lentner, Simon D. |
| title | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_full | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_fullStr | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_full_unstemmed | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_short | Algebras of Non-Local Screenings and Diagonal Nichols Algebras |
| title_sort | algebras of non-local screenings and diagonal nichols algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211527 |
| work_keys_str_mv | AT flandoliilaria algebrasofnonlocalscreeningsanddiagonalnicholsalgebras AT lentnersimond algebrasofnonlocalscreeningsanddiagonalnicholsalgebras |