Quantum Groups for Restricted SOS Models
We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of π-graded vector spaces for a groupoid π.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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Інститут математики НАН України
2021
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Quantum Groups for Restricted SOS Models. Giovanni Felder and Muze Ren. SIGMA 17 (2021), 005, 26 pages |
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| citation_txt | Quantum Groups for Restricted SOS Models. Giovanni Felder and Muze Ren. SIGMA 17 (2021), 005, 26 pages |
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| description | We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of π-graded vector spaces for a groupoid π.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 005, 26 pages
Quantum Groups for Restricted SOS Models
Giovanni FELDER † and Muze REN ‡
† Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
E-mail: felder@math.ethz.ch
‡ Department of Mathematics, University of Geneva,
2-4 rue du Lièvre, c.p. 64, 1211 Geneva 4, Switzerland
E-mail: muze.ren@unige.ch
Received October 05, 2020, in final form January 05, 2021; Published online January 12, 2021
https://doi.org/10.3842/SIGMA.2021.005
Abstract. We introduce the notion of restricted dynamical quantum groups through their
category of representations, which are monoidal categories with a forgetful functor to the
category of π-graded vector spaces for a groupoid π.
Key words: elliptic quantum groups; dynamical R-matrices; groupoid grading; RSOS models
2020 Mathematics Subject Classification: 17B37; 18M15
Dedicated to Vitaly Tarasov and Alexander Varchenko
on their round birthdays
1 Introduction
The theory of quantum groups was designed in the 1980s to describe the algebraic structure
underlying the theory of exactly solvable models of statistical mechanics and quantum mechan-
ics. Since then the theory of quantum groups has entered diverse fields of mathematics and
mathematical physics and the world of exactly solvable models is entirely explained by quantum
groups, in the guise of Yangians, quantum loop algebras, and elliptic quantum groups. Well,
not entirely . . . One small village of indomitable models, called the RSOS models still holds out
against the invaders.1 These Restricted Solid-On-Solid models, introduced in special cases by
Baxter in his studies of the eight-vertex model and the hard hexagon model, and generalized
by Andrews, Baxter and Forrester, are lattice models of two-dimensional statistical mechanics
for which the technology of exact solutions has provided some of the most spectacular results.
They play a central role also in conformal field theory (CFT) as their critical behaviour is (or
is conjectured to be) given by the universality classes of minimal unitary CFT models. While
the unrestricted SOS models, whose local degrees of freedom take values in an infinite set, are
by now well-described by the representation theory of dynamical elliptic quantum groups, the
RSOS models, with finitely many allowed states at every lattice point, are much less understood.
We propose a theory of dynamical quantum groups with discrete dynamical parameter with
the goal to establish the representation theory underlying RSOS models and their higher rank
generalizations. We introduce a new approach to this problem, based on groupoid-graded vector
spaces, which may be of independent interest and applicability in representation theory.
This paper is a contribution to the Special Issue on Representation Theory and Integrable Systems in honor
of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html
1Actually there are also other exactly solvable models whose quantum group description is unknown, such as
the Inozemtsev spin chain [28].
mailto:felder@math.ethz.ch
mailto:muze.ren@unige.ch
https://doi.org/10.3842/SIGMA.2021.005
https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html
2 G. Felder and M. Ren
1.1 Quantum groups and exactly solvable models
The notion of quantum group [12] emerged in the Leningrad school in the 1980s as the algebraic
structure underlying exactly solvable models of statistical mechanics in 2 dimensions and inte-
grable quantum field theory in 1+1 dimensions, see [18]. While “groups” may be a misnomer for
these Hopf algebras, quantum groups share with groups the fact that they have an interesting
representation theory for which tensor products of representations are defined. Excellent text-
books on quantum groups are [8, 33, 39]. It soon appeared that quantum groups have a much
wider scope of applications, ranging from low dimensional topology to conformal field theory,
algebraic geometry, gauge theory, representation theory of affine Lie algebras etc.
Returning to the origin in quantum mechanics and statistical mechanics, the basic equation
is the Yang–Baxter equation, which appeared in the 1960s in the works of J.B. McGuire and
C.N. Yang on one-dimensional many-body quantum systems [40, 47] and later in R. Baxter’s
work on statistical mechanics [4]. In its basic form it is an equation for a meromorphic function
z 7→ R(z) ∈ EndC(V ⊗V ) of one complex variable (called the spectral parameter) with values in
the linear endomorphisms of the tensor square of a finite dimensional complex vector space V .
The Yang–Baxter equation is
R(z − w)(12)R(z)(13)R(w)(23) = R(w)(23)R(z)(13)R(z − w)(12)
in End(V ⊗ V ⊗ V ). Here the superscripts in the notation indicate the factors on which the
endomorphisms act: for example R(w)(12) means R(w) ⊗ id. One also requires that R(z) is
invertible for generic z. One of the simplest non-trivial solutions is the R-matrix R(z) = Id +
z−1PV V of McGuire and Yang where PV V is the flip v ⊗ w 7→ w ⊗ v. Here V is any finite-
dimensional vector space.
As noted by Baxter, solutions of the Yang–Baxter equation give rise to families of commuting
operators on n-fold tensor powers W = V ⊗n: fix complex numbers z1, . . . , zn and consider the
operator valued function L(z) ∈ End(V ⊗ W ) given by the product R(z − zn)(0n) · · ·R(z −
z2)(02)R(z−z1)(01) (we number the factors from 0 to n where 0 refers to the “auxiliary space” V
in V ⊗W ). Then the (row-to-row) transfer matrices defined by the partial traces
T (z) = trV L(z)
are commuting endomorphisms of W = V ⊗n:
T (z)T (w) = T (w)T (z),
as a consequence of the Yang–Baxter equation. The relation to statistical mechanics is that
the trace of T (z)m, say with all zi = 0, when written out using matrix multiplication, is a sum
of products of matrix entries of R(z) over all the ways to assigning a basis vector to pairs of
nearest neighbours of an n ×m lattice with periodic boundary conditions. These assignments
are configurations of local states of a system and the sum is the partition function of a statistical
mechanics model.
The Bethe ansatz, invented by H. Bethe in 1931 [7] in the case of the Heisenberg spin chain,
and further developed by E. Lieb, R. Baxter, B. Sutherland, C.N. Yang and others in the 1960s,
is a technique to find simultaneous eigenvectors and eigenvalues of the T (z). The Leningrad
school, see [18] for a review of the results in the early phase, reformulated this technique under
the name “algebraic Bethe ansatz” in the framework of the “quantum inverse scattering method”
in terms of representation theory of an algebra with quadratic relations (called RLL or RTT
relations) whose coefficients are matrix entries of a solution of the Yang–Baxter equation. For
example the Yangian Y (glN ) corresponds to the McGuire–Yang R-matrix with V = CN . It can
be defined as the algebra with generators Lij;n, i, j = 1, . . . , N , n = 1, 2, . . . with relations
R(z − w)L(z)⊗ L(w) = L(z)⊗ L(z)R(z − w), R(z) = Id +
1
z
PV V ,
Quantum Groups for Restricted SOS Models 3
where L(z) is the N×N matrix with entries δij+
∑
n≥1 Lij,nz
−n. It is a Hopf algebra deformation
of the universal enveloping algebra of the current Lie algebra glN [t] and has a universal R-
matrixR in a completion of Y (glN )⊗Y (glN ) relating the opposite coproduct ∆′ to the coproduct
via ∆′(x) = R∆(x)R−1. Evaluating R in pairs Vi ⊗ Vj of finite dimensional representations of
the Yangian yields solutions of the Yang–Baxter equation, in the generalized form
R
(12)
V1V2
R
(13)
V1V3
R
(23)
V2V3
= R
(23)
V2V3
R
(13)
V1V3
R
(12)
V1V2
. (1.1)
The spectral parameter may be viewed as a parameter of the representations Vi, and in fact there
is an issue of convergence of the action of R on finite dimensional representations, resulting in
the fact that RVi,Vj is a meromorphic function of the spectral parameters.
To such a system of R-matrices we can associate corresponding transfer matrices Ti =
trVi RViV3 , i = 1, 2, acting on V3 and the Yang–Baxter equation with an invertible RV1V2 implies
that T1T2 = T2T1. Baxter’s transfer matrices are the special case V1,2 = V and V3 a tensor
product of vector representations with equal spectral parameters.
This story extends to arbitrary semisimple (or reductive) Lie algebras g, and the Yan-
gians Y (g) provide the algebraic structure underlying several integrable systems based on ratio-
nal solutions of the Yang–Baxter equation such as the Heisenberg spin chain.
The theory admits a trigonometric version, leading to solutions of the Yang–Baxter equation
with trigonometric coefficients. The corresponding quantum group is a Hopf algebra deformation
of the loop Lie algebra g
[
t, t−1
]
. It is (a subquotient of) the Drinfeld–Jimbo quantum enveloping
algebra Uqĝ of the affine Kac–Moody Lie algebra ĝ. The corresponding solvable models are the
six-vertex model, a special case of which is the two-dimensional ice model, and the XXZ spin
chain.
1.2 Elliptic quantum groups and dynamical Yang–Baxter equation
The next level, after the rational and trigonometric functions, are the elliptic functions, which
are meromorphic functions which are periodic with respect to two independent periods. Several
solvable models have an elliptic version and the trigonometric and rational versions are obtained
as degenerate limits as the periods tend to infinity. The relation to quantum groups is more tricky
in the elliptic case. On one hand there is a solution of the Yang–Baxter equation with elliptic
coefficients due to Baxter, corresponding to the XYZ spin chain and the eight-vertex model,
whose underlying algebraic structure is the Sklyanin algebra (which is not a Hopf algebra). On
the other hand there are the SOS (solid-on-solid) models also known as IRF (interaction-round-
a-face) models. They are based on a variant of the Yang–Baxter equation, called the star-triangle
relation. While the existence of the Baxter solution is special for g = slN , we now know that
SOS models exist for all semisimple Lie algebras. Also the Baxter solution can be related to
a solution of the star-triangle relation by the so-called vertex-IRF transformation, also due to
Baxter.
The elliptic quantum groups introduced in [19, 20] provide a generalization of the theory of
quantum groups that applies to elliptic SOS models. They are based on a modification of the
Yang–Baxter equation, nowadays called the dynamical Yang–Baxter equation, see (1.2) below,
which had previously been found by Gervais and Neveu in their study of the exchange relations
of vertex operators in the Liouville conformal field theory [26]. The dynamical Yang–Baxter
equation reappeared in various contexts since, e.g., [1, 9, 10, 14, 15, 29, 32, 42, 43]. A recent
textbook on elliptic quantum groups is [36].
The unknown in the dynamical Yang–Baxter equation is a function R(z, a) ∈ Endh(V ⊗ V )
of a second “dynamical” variable a ∈ h∗ with values in the dual vector space to an abelian
Lie algebra h and V = ⊕µ∈h∗Vµ is a finite dimensional semisimple h-module. The (quantum)
4 G. Felder and M. Ren
dynamical Yang–Baxter equation is
R
(
z − w, a+ h(3)
)(12)
R(z, a)(13)R
(
w, a+ h(1)
)(23)
= R(w, a)(23)R
(
z, a+ h(2)
)(13)
R(z − w, a)(12). (1.2)
The “dynamical shift” notation is adopted here: for example R(w, a+ h(3))(12) acts as R(w, a+
µ3) ⊗ Id on the product of weight subspaces Vµ1 ⊗ Vµ2 ⊗ Vµ3 . More generally, we can consider
dynamical Yang–Baxter equations in Endh(V1⊗V2⊗V3) for R-matrices RViVj (z, a) ∈ Endh(Vi⊗
Vj) as in (1.1).
The elliptic quantum group associated with a solution of the dynamical Yang–Baxter equation
and its tensor category of representations can be again defined by quadratic relations similar to
those of the Yangian but with dynamical shifts at the appropriate places, see [19, 23, 25]. The
main new feature is that the representations are vector spaces over the field of meromorphic
functions of the dynamical variables and the elements of the elliptic quantum group act as
difference operators in these variables. The underlying generalization of the notion of Hopf
algebra was formalized by Etingof and Varchenko [16] who called it h-Hopf algebroid.
The transfer matrix construction generalizes to the dynamical setting [22]: suppose that we
have invertible operators RViVj (z, λ) ∈ Endh(Vi ⊗ Vj), i < j ∈ {1, 2, 3} obeying the dynamical
Yang–Baxter equation (1.2) on V1 ⊗ V2 ⊗ V3, depending meromorphically on z ∈ C, λ ∈ h∗.
Then the transfer matrix is defined as an operator acting on meromorphic functions of λ with
values in the zero-weight subspace of V3:
Ti(z) =
∑
µ
trViµ RViV3(z, λ)tµ. (1.3)
Here the partial trace is over the weight-µ subspace of Vi, the R-matrix acts as a multiplication
operator and (tµf)(λ) = f(λ+µ). In the special case where V1 = V2, RV1,V3 has the interpretation
of an L-operator obeying a dynamical version of the RLL-relations.
1.3 Restricted SOS models
The restricted solid-on-solid (RSOS) models introduced by Andrews, Baxter and Forrester [3],
generalizing models previously considered by Baxter [4, 5] in his study of the eight-vertex model
and of the hard hexagon model, form a general class of models of statistical mechanics in two
dimensions. A configuration of a solid-on-solid model on a subset M of a square lattice in the
plane or 2-dimensional torus is described by assigning an integer li (height) to each lattice site
i ∈ M , with the restriction that |li − lj | = 1 for neighbouring sites i, j. We can think of the
graph of i 7→ li as a discrete random surface modeling the interface between a solid and a gas.
The probability of a configuration (li) is proportional to a product over the faces (unit
squares with vertices in M) of Boltzmann weights W(li, lj , lk, lm) depending on the heights
on the corners i, j, k, l of the face. In the “solvable” SOS models the Boltzmann weights are
part of a one-parameter family W(z; a, b, c, d) obeying the star-triangle relation
∑
g
W(z − w; f, g, d, e)W(z; a, b, g, f)W(w; b, c, d, g)
=
∑
g
W(w; a, g, e, f)W(z; c, d, e, g)W(z − w; a, b, c, g),
Quantum Groups for Restricted SOS Models 5
which is best understood graphically:
∑
g
a
f
b
g
e
c
d
=
∑
g
a g
c
f
b
e
c
d
The star-triangle equation admits interesting families of solutions in terms of elliptic theta
functions. Andrews, Baxter and Forrester considered a special limit of parameters so that the
equation holds for the heights in a finite interval
li ∈ {1, 2, . . . , r − 1}.
For these families of models (depending essentially on an elliptic curve and a point of order r
on it) they were able to compute several quantities in the thermodynamic limit M → Z2 (under
some physically motivated assumptions on the asymptotic behaviour), including the probability
distribution of the height at the origin as a function of the boundary conditions in the ordered
phase. One interesting mathematical outcome of this calculation is that it involves for r = 5
(Baxter’s hard hexagon model) the celebrated Rogers–Ramanujan identities, which get general-
ized to arbitrary r. From the point of view of statistical mechanics and conformal field theory,
these models are interesting since their scaling limit at the critical point are conjectured [27]
to be the unitary A-series of minimal models of Belavin–Polyakov–Zamolodchikov and Friedan–
Qiu–Shenker.
We will be mostly concerned with a generalization of the RSOS models in which the heights
take values in the weight lattice of a simple Lie algebra, see [11, 30, 31]. The main difference
is that in general the Boltzmann weights W(a, b, c, d) are no longer scalar-valued, but must be
understood as linear operators.
The relation with the dynamical quantum groups comes from the simple observation that the
above star-triangle equation is essentially a rewriting of the dynamical Yang–Baxter equation.
The row-to-row transfer matrix of the RSOS model is the transfer matrix (1.3) for suitable
representations of the elliptic quantum group associated with gl2, acting on functions with
support on a finite set.
This restriction of a difference operator such as (1.3) with meromorphic coefficients to a finite
set is rather subtle as one needs to avoid the poles and check that the support condition is
preserved. This was done in the case of the RSOS model in [24], where it was also shown that
the gl2-elliptic weight functions of [21] obey “resonance conditions”, guaranteeing that their
restrictions to suitable discrete or finite subsets of the values of the dynamical variable provide,
via the Bethe ansatz, well-defined eigenvectors of the row-to-row transfer matrix of the RSOS
model.
1.4 Categories of representations
Instead of talking of quantum groups it is more convenient to talk about their tensor category of
representations, and we take this approach in this paper. In the representation theory of elliptic
quantum groups [13, 23], the representation space of a representation is defined as a graded
vector space over the field of meromorphic functions of the dynamical variables, where the
grading is by weights of the underlying Lie algebra. The representation structure is defined
by C-linear endomorphisms obeying quadratic relations and commutations relations with scalar
multiplication by meromorphic functions.
6 G. Felder and M. Ren
For the application to RSOS models, where the dynamical variables take values in a discrete
set, the approach with meromorphic functions is not suitable. In this paper we propose that the
vector spaces underlying representations of quantum groups with discrete dynamical variables
should be groupoid-graded vector spaces. More precisely we propose that representations of
such quantum groups are monoidal categories equipped with a faithful monoidal functor to
the category of π-graded vector spaces of finite type for a certain groupoid π (see Section 2
for the definitions). For applications to generalized RSOS models the groupoids are certain
subgroupoids of the transformation groupoid for the translation action of the weight lattice of
a semisimple Lie algebra. It turns out that in this approach the various shifts of dynamical
variables appearing in the dynamical context appear naturally and one can immediately apply
the standard technology of the quantum inverse scattering method (R-matrices, RLL relations,
transfer matrices, Bethe ansatz). An instance of this is the fusion procedure, which consists in
constructing solutions of the Yang–Baxter or star-triangle relations from known ones by taking
subquotients of tensor products.
An interesting new feature in the groupoid-graded case is that the Grothendieck ring of the
category of π-graded vector spaces is non-commutative in general, even in the case of action
groupoids of abelian groups. Thus characters of representations of dynamical quantum groups
live in a non-commutative ring. However if a collection of representations (Vi) admit R-matrices,
which are isomorphisms Vi⊗Vj ∼= Vj⊗Vi, then their characters generate a commutative subring
of the Grothendieck ring of π-graded vector spaces. In the case of transformation groupoids
these rings are realized as rings of commuting difference operators.
1.5 Outline of the paper
We introduce the category Vectk(π) of π-graded vector spaces of finite type over a field k in
Section 2. It is a variant of a special case of the category of π-graded modules considered in [38].
It is an abelian monoidal category with duality. We discuss the notion of character of a π-graded
vector space taking values in the convolution ring of π. In Section 3 we adapt the machinery of
Yang–Baxter equations and transfer matrices to the case of π-graded vector spaces and explain
the relation with the star-triangle relation. We introduce the notion of partial traces in this
context and prove that solutions of the Yang–Baxter equation give rise to commuting transfer
matrices. In the case of transformation groupoids and their subgroupoids, we show that the
Yang–Baxter equation can be written as a dynamical Yang–Baxter equation, and that transfer
matrices produce commuting difference operators. In Section 4 we consider in more detail
the example of the elliptic quantum group of type An−1, which admits a dynamical R-matrix
with restricted dynamical variables and thus a monoidal category with a forgetful functor to
π-graded vector spaces for a finite groupoid π. We compute a few characters, in particular the
characters of (analogues of the) exterior powers of the vector representation, obtained by the
fusion procedure. Finally in Section 5 we consider the case of dynamical R-matrices arising from
quantum groups at root of unity, which may be viewed as a toy model for restricted models, with
R-matrices that are independent of the spectral parameters. The construction uses a semisimple
rigid braided category Cq(g) of representations of quantum groups for each simple Lie algebra g
and root of unity q. Technically it is a semisimple quotient of the category of tilting modules
of the Lusztig quantum groups. It has finitely many isomorphism classes of simple objects. We
construct a faithful monoidal functor from Cq(g) to the categories of π-graded vector spaces of
finite type for a suitable finite groupoid π. The braiding in Cq(g) is then mapped to a system
of dynamical R matrices with dynamical variable restricted to a finite set. This construction is
a formalization of the “passage to the shadow world” of [34, 44] and is a version with discrete
dynamical variable of [17]. The characters of simple modules define a representation of the
Verlinde algebra by difference operators.
Quantum Groups for Restricted SOS Models 7
2 Grading by groupoids
2.1 Groupoids
A groupoid π on a set A is a small category with set of objects A whose morphisms, called arrows,
are invertible. The set of morphisms from an object a to an object b is denoted by π(a, b). The
composition of arrows γ ∈ π(a, b) and η ∈ π(b, c) is denoted by η ◦ γ ∈ π(a, c) or by ηγ in case
of typographical constraints. The inverse of γ ∈ π(a, b) is γ−1 ∈ π(b, a). We identify A with the
subset of identity arrows and denote a groupoid by its set of arrows π when no confusion arises.
The maps s, t : π → A sending γ ∈ π(a, b) to a and b, respectively, are called source and target
map, respectively.
A subgroupoid of a groupoid π is a subset of (the set of arrows of) π that is closed under
composition and inversion. It is a groupoid on the set of its identity arrows. The full subgroupoid
of π on a subset B ⊂ A is the subgroupoid s−1B ∩ t−1B of arrows between objects of B.
2.2 The convolution ring of a groupoid
To a groupoid π we associate the convolution ring of π, which is a unital associative ring Z(π)
with an involutive anti-automorphism.
As an abelian group Z(π) consists of the maps n : π → Z such that for all a ∈ A, the set of
arrows α ∈ s−1(a) ∪ t−1(a) with n(α) 6= 0 is finite. The product is the convolution product
n ∗m : γ 7→
∑
β◦α=γ
n(α)m(β).
The sum has finitely many non-zero terms because of the finiteness assumption. The unit is the
characteristic function on identity arrows and the involutive anti-automorphism σ sends n to
σ(n) : γ → n
(
γ−1
)
. The assignment π 7→ Z(π) is a contravariant functor from the category of
groupoids to the category of unital involutive associative rings.
Remark 2.1. For any commutative ring R we have an R-algebra R(π) = R ⊗Z Z(π) obtained
by extension of scalars. For transfer matrices we will need a more general construction where R
is also π-graded, see Section 2.10 below.
2.3 Convolution rings of subgroupoids
It will be convenient to view the convolution ring of a subgroupoid π′ ⊂ π as a subring of Z(π).
Lemma 2.2. The characteristic functions χA′ of subsets A′ ⊂ A of the set of identity arrows
are idempotents in Z(π).
Proof. By definition χA′(γ) = 0 unless γ is an identity arrow a ∈ π(a, a) for a ∈ A′. In this
case χA′(a) = 1. Thus χA′ ∗ χA′(γ) vanishes unless γ is an identity arrow a ∈ A′, in which case
χA′ ∗ χA′(a) = χA′(a)χA′(a) = 1. �
Lemma 2.3. Let π be a groupoid on A and π′ the full subgroupoid on A′ ⊂ A. Then the induced
morphism Z(π)→ Z(π′) restricts to a unital ring isomorphism
χA′ ∗ Z(π) ∗ χA′ → Z(π′),
where χA′ is the unit element of the subring χA′ ∗Z(π) ∗ χA′. Moreover the left-hand side is the
subring of functions vanishing on the complement of π′.
8 G. Felder and M. Ren
Proof. The map Z(π) → Z(π′) is the restriction map r : n 7→ n|π′ . The extension by zero
Z(π′)→ Z(π) is a right inverse. Its image consists of the functions vanishing outside π′. Thus r
restricts to an isomorphism from the functions vanishing outside π′ and Z(π). Now a function
n ∈ Z(π) vanishes outside the full subgroupoid π′ if and only it vanishes everywhere except on
arrows between elements of A′. But this is equivalent to n = χA′ ∗ n ∗ χA′ . Since χA′ is an
idempotent, χA′ ∗ Z(π) ∗ χA′ is a subring with unit element χA′ , which is sent to 1 ∈ Z(π′). �
2.4 Action groupoids
The main examples of groupoids for our purpose are action groupoids and their subgroupoids.
Let G be a group with identity element e and A be a set with a right action A×G → A. The
action groupoid AoG has set of objects A and an arrow a→ a′ for each g ∈ G such that a′ = ag.
Thus an arrow is described by a pair (a, g) ∈ A × G. The source and target are s(a, g) = a,
t(a, g) = ag and the composition is
(a′, g′) ◦ (a, g) = (a, gg′), whenever a′ = ag.
The identity arrows are (a, e), a ∈ A and the inverse of (a, g) is
(
ag, g−1
)
.
The convolution ring Z(A o G) contains the subring ZA of functions with support on the
identity arrows as in the general case and a subring ZG isomorphic to the group ring of G via
the injective ring homomorphism t : ZG→ Z(AoG) sending g ∈ G to
tg : (a, h) 7→ δg,h.
The right action of G defines a group homomorphism r : G→ Aut(ZA): for g ∈ G and f ∈ ZA,
rgf(a) = f(ag).
Proposition 2.4. The convolution ring Z(A oG) is the crossed product ZA or ZG of its sub-
rings ZA and ZG. The involution acts trivially on ZA and as tg 7→ tg−1 on ZG.
This means that Z(AoG) is isomorphic to the algebra generated by ZA and elements tg for
g ∈ G with relations
tgth = tgh, tgf = rg(f)tg, g, h ∈ G, f ∈ ZA.
Explicitly, a function n ∈ Z(AoG) corresponds to the element
∑
g∈G ngtg where ng(a) = n(a, g).
Remark 2.5. In particular the convolution ring acts on the space of functions on A by difference
operators (i.e., operators acting on functions by translations of the argument and multiplication
by functions). This is the scalar case of the more general case of a convolution algebra acting
on vector-valued functions on A by difference operators, which we construct in Section 3.7.
2.5 The category of π-graded vector spaces of finite type
Definition 2.6. Let π be a groupoid with set of objects A. A π-graded vector space of finite
type over a field k is a collection (Vα)α∈π of finite-dimensional vector spaces indexed by the
arrows of π such that for each a ∈ A there are finitely many arrows α with source or target a
and nonzero Vα.
The π-graded vector spaces over k form an abelian category Vectk(π): the k-vector space
Hom(V,W ) of morphisms between objects V,W consists of families (fα)α∈π of linear maps
fα : Vα →Wα and the composition is defined componentwise.
Quantum Groups for Restricted SOS Models 9
2.6 Tensor product
The finite type condition allows us to define a monoidal structure (tensor product) on Vectk(π).
The tensor product of objects is
(V ⊗W )γ = ⊕β◦α=γVα ⊗Wβ. (2.1)
The direct sum is over all pairs of arrows whose composition is γ and has finitely many nonzero
summands. Similarly the tensor product f ⊗ g of morphisms has components ⊕β◦α=γfα ⊗ gβ.
For any three objects U , V , W of Vectk(π) and δ ∈ π,
((U ⊗ V )⊗W )δ = ⊕α◦β◦γ=δ(Uγ ⊗ Vβ)⊗Wα,
(U ⊗ (V ⊗W ))δ = ⊕α◦β◦γ=δUγ ⊗ (Vβ ⊗Wα).
Therefore the associativity constraint in Vectk defines an associativity constraint
αUVW : (U ⊗ V )⊗W → U ⊗ (V ⊗W )
in Vectk(π). The tensor unit in Vectk(π) is 1 = (1γ)γ∈π with 1a = k for identity arrows a ∈ A
and 1γ = 0 for all other arrows. Then for every object V of Vectk(π), (1 ⊗ V )γ = k ⊗ Vγ and
(V ⊗ 1)γ = Vγ ⊗ k. Thus the structure isomorphisms Vγ ∼= Vγ ⊗ k ∼= k ⊗ Vγ of Vectk define
natural isomorphisms λ : V ∼= V ⊗ 1, ρ : V ∼= 1⊗ V in Vectk(π).
2.7 Duality
The dual of a π-graded vector space V ∈ Vectk(π) is the π-graded vector space V ∗ with com-
ponents (V ∗)γ = Homk(Vγ−1 , k). Recall that an object V of a monoidal category admits a left
dual of an object V if there is an object V ∨, called left dual of V , together with morphisms
δ : 1→ V ⊗ V ∨, ev : V ∨ ⊗ V → 1 such that the compositions
V ∼= 1⊗ V →
(
V ⊗ V ∨
)
⊗ V ∼= V ⊗
(
V ∨ ⊗ V
)
→ V ⊗ 1 ∼= V,
V ∨ ∼= V ∨ ⊗ 1→ V ∨ ⊗
(
V ⊗ V ∨
) ∼= (V ∨ ⊗ V )⊗ V ∨ → 1⊗ V ∨
are equal to the identity morphism. Similarly one has the notion of right dual object ∨V with
morphisms 1→ ∨V ⊗ V , V ⊗ ∨V → 1. Right and left duals of finite dimensional vector spaces
coincide.
Lemma 2.7. Let V ∈ Vectk(π). Then V ∨ with components
(
V ∨
)
γ
= (Vγ−1)∗ = Homk(Vγ−1 , k),
and structure morphisms induced by those of the category of finite dimensional vector spaces is
both left and right dual to V .
For example the morphism δ : 1→ V ⊗V ∨ is the collection of maps 1a = k → ⊕γ∈s−1(a)Vγ ⊗(
V ∨
)
γ−1 , defined by the canonical map k → Vγ ⊗ (Vγ)∗.
Monoidal categories admitting left and right duals for all objects, which are then uniquely
determined up to unique isomorphism, are called rigid. Left and right dualities are monoidal
functors to the opposite categories with opposite tensor product. Rigid monoidal categories with
coinciding left and right dual functors are called pivotal, see [45, Sections 1.6 and 1.7] for more
details.
Theorem 2.8. The k-additive category Vectk(π) with the tensor product ⊗, the tensor unit 1,
associativity constraint α, left and right multiplication by the tensor unit λ, ρ, and duality ( )∨
is an abelian pivotal monoidal category.
This is an immediate consequence of the fact that Vectk is a k-additive abelian monoidal
category and Lemma 2.7.
10 G. Felder and M. Ren
Remark 2.9. Contrary to the case of finite dimensional vector spaces, the monoidal category
Vectk(π) is not symmetric or braided, so that V ⊗W is not isomorphic to W ⊗ V in general.
As we will see presently, the Grothendieck ring is not commutative in general.
Remark 2.10. The above construction works for any k-additive rigid monoidal category C over
a commutative ring k instead of Vectk. The resulting category of π-graded objects of C of finite
type is a k-additive monoidal category. For example, if we view a ring as a monoidal category
with one object, the convolution ring Z(π) is the category of π-graded objects of finite type of Z.
One can also replace π by a general small category, at the cost of giving up duality.
2.8 Characters
The character chV ∈ Z(π) of V ∈ Vectk(π) is the map γ 7→ dim(Vγ).
Lemma 2.11. Let V,W ∈ Vectk(π). Then
ch1 = 1, chV ∨ = σ(chV ),
chV⊕W = chV + chW ,
chV⊗W = chV ∗ chW
Since exact sequences of vector spaces split, the character map ch: V 7→ chV descends to
a ring homomorphism from the Grothendieck ring K(Vectk(π)) to the convolution ring Z(π).
Proposition 2.12. The map K(Vectk(π))→ Z(π) is an isomorphism of involutive unital rings.
The inverse map sends n = n+ − n− with n±(γ) ≥ 0 for all γ to the formal difference[
(kn+(γ))γ∈π
]
−
[
(kn−(γ))γ∈π
]
.
2.9 Subgroupoids
Let i : π′ ↪→ π be a subgroupoid. Then we have an exact fully faithful functor i∗ : Vectk(π
′) →
Vectk(π) so that
(i∗V )γ =
{
Vγ , if γ ∈ π′,
0, otherwise.
We can thus view Vectk(π
′) as a full subcategory of Vectk(π) of π-graded vector spaces V such
that Vγ = 0 for γ 6∈ π′.
2.10 Convolution algebras with coefficients in π-graded algebras
In the setting of π-graded vector spaces the natural home for transfer matrices is convolution
algebras with coefficients in π-graded algebras over a field (or commutative ring) k.
Definition 2.13. Let π be a groupoid. A π-graded algebra R over k is a collection (Rγ)γ∈π of
k-vector spaces labeled by arrows of π with bilinear products Rα × Rβ → Rβ◦α, (x, y) 7→ xy,
defined for composable arrows α, β and units 1a ∈ Ra, for a ∈ A such that (i) (xy)z = x(yz)
whenever defined and (ii) x1b = x = 1ax for all x ∈ Rα of degree α ∈ π(a, b).
Remark 2.14. Am algebra object in Vectπ defines a π-graded algebra, but we will need to
consider more general examples which do not necessarily fulfill the finite type condition, such
as End 1 below.
Quantum Groups for Restricted SOS Models 11
Example 2.15. Let V ∈ Vectk(π) and let EndV be the π-graded vector space with (EndV )α =
⊕γ∈π(a,a) Homk(Vα◦γ◦α−1 , Vγ), where a = s(α). Then EndV with the product given by the
composition of linear maps
Homk(Vαγα−1 , Vγ)⊗Homk(Vβαγα−1β−1 , Vαγα−1)→ Homk(Vβαγ(βα)−1 , Vγ)
and unit 1a = ⊕γ∈π(a,a)IdVγ is a π-graded algebra.
Definition 2.16. Let R be a π-graded algebra. The convolution algebra Γ(π,R) with coefficients
in R is the k-algebra of maps f : π → tα∈πRα such that
(i) f(α) ∈ Rα for all arrows α ∈ π,
(ii) for every a ∈ A, there are finitely many α ∈ s−1(a) ∪ t−1(a) such that f(α) 6= 0.
The product is the convolution product
f ∗ g(γ) =
∑
β◦α=γ
f(α)g(β).
Example 2.17. Let R = End 1 be the π-graded algebra of Example 2.15 for the tensor unit 1.
Then Rα = k for all arrows α and Γ(π,R) = k(π) = k ⊗ Z(π) is the extension of scalars of the
convolution ring of π, see Remark 2.1.
Lemma 2.18. Let π be a groupoid with object set A. The convolution algebra Γ(π,R) with
coefficients in a π-graded algebra R is an associative unital k-algebra. The unit is the map
a 7→ 1a for identity arrows a ∈ A and α 7→ 0 for other arrows.
Let ϕ : R → R′ be a morphism of π-graded algebras. Then ϕ∗ : Γ(π,R) → Γ(π,R′) given
ϕ∗(f) = ϕ ◦ f is an algebra homomorphism. This defines a functor from the category of π-
graded k-algebras to associative unital algebras. In particular we have a morphism of algebras
k(π)→ Γ(π,R) for any R.
3 Yang–Baxter equation and RLL relations
3.1 Yang–Baxter equation
Let k = C. A Yang–Baxter operator on V ∈ Vectk(π) is a meromorphic function z 7→ Ř(z) ∈
End(V ⊗V ) of the spectral parameter z ∈ C with values in the endomorphisms of V ⊗V , obeying
the Yang–Baxter equation2
Ř(z − w)(23)Ř(z)(12)Ř(w)(23) = Ř(w)(12)Ř(z)(23)Ř(z − w)(12) (3.1)
in End(V ⊗V ⊗V ) for all generic values of the spectral parameters z, w, and the inversion (also
called unitarity) relation
Ř(z)Ř(−z) = idV⊗V ,
for generic z. The restriction of Ř(z) to Vα⊗ Vβ for composable arrows α, β has components in
each direct summand of the decomposition (2.1):
Ř(z)|Vα⊗Vβ = ⊕γ,δW(z;α, β, γ, δ).
2The Yang–Baxter equation is usually formulated for the operator R(z) = p ◦ Ř(z) obtained by composition
with the flip p : u ⊗ v 7→ v ⊗ u. Since p is not a morphism of π-graded vector spaces in general, it is better to
use Ř(z).
12 G. Felder and M. Ren
∑
ρ,σ,τ
α
ρ δ
β
σ
ε
γτ
ζ
=
∑
ρ,σ,τ
α ρ
δ
β
σ
ε
γ
τζ
Figure 1. The Yang–Baxter equation. The arrows α, . . . , ζ form a commutative hexagon and the sum
is over arrows ρ, σ, τ making the squares commutative.
The sum is over γ, δ such that β ◦ α = δ ◦ γ, and W is the component
W(z;α, β, γ, δ) ∈ Homk(Vα ⊗ Vβ, Vγ ⊗ Vδ).
The Yang–Baxter equation translates to its IRF (interaction-round-a-face) version, called the
star-triangle relation∑
ρ,σ,τ
W(z − w; ρ, σ, ε, δ)(23)W(z;α, τ, ζ, ρ)(12)W(w;β, γ, τ, σ)(23)
=
∑
ρ,σ,τ
W(w;σ, τ, ζ, ε)(12)W(z; ρ, γ, τ, δ)(23)W(z − w;α, β, σ, ρ)(12).
There is one such equation for all α, . . . , ζ so that γ ◦β ◦α = δ ◦ ε ◦ ζ and the sum is over arrows
ρ, σ, τ for which all factors are defined, namely such that the diagrams are commutative in π.
It is convenient to have a graphical representation for these morphisms:
W(z;α, β, γ, δ) =
a
γ
d
δα
b
β
c
The dashed lines are associated with the vector spaces between which W acts: moving from
the southwest to the northeast according to the orientation of the dashed lines we move from
Vα⊗Vβ to Vγ ⊗Vδ. We have also displayed in the corners the objects between which the arrows
α, . . . , δ are defined. For example α is an arrow from a to b. In the literature one often considers
the case where there is at most one arrow from one object to any other object, as is the case in
the original Andrews–Baxter–Forrester RSOS models, and it is then customary to label W by
the four objects a, b, c, d instead of the morphisms.
3.2 RLL relations
The machinery of the quantum inverse scattering method [18] can be applied: given a solution
Ř(z) ∈ End(V ⊗ V ) of the dynamical Yang–Baxter equation for a π-graded vector space V , an
L-operator on W ∈ Vectk is a meromorphic function z 7→ L(z) ∈ Hom(V ⊗W,W ⊗V ) such that
(i) L(z) is invertible for generic z,
Quantum Groups for Restricted SOS Models 13
(ii) L obeys the RLL relations
Ř(z − w)(23)L(z)(12)L(w)(23) = L(w)(12)L(z)(23)Ř(z − w)(12),
in Hom(V ⊗ V ⊗W,W ⊗ V ⊗ V ).
For example Ř(z) is an L-operator on V thanks to the Yang–Baxter equation.
Given a basis of V the RLL relations may be written as relations for the matrix entries
Lij(z) ∈ End(W ). Thus L-operators may be understood as π-graded meromorphic representa-
tions of the quadratic algebra AR with generators Lij(z), and RLL relations. Here meromorphic
refers to the required meromorphic dependence on z ∈ C.
3.3 The monoidal category M(R, π)
The L-operators form an abelian monoidal category M(R, π) of π-graded meromorphic repre-
sentations of AR: an object (W,LW ) is a π-graded vector space W ∈ Vectk(π) endowed with
an L-operator LW on W . A morphism from (W,LW ) to (Z,LZ) is a morphism f : W → Z of
π-graded vector spaces such that
(f ⊗ idV )LW (z) = LZ(z)(idV ⊗ f),
for all z.
The tensor product (W ⊗ Z,LW⊗Z) is the tensor product in Vectk(π) endowed with the
composition
LW⊗Z(z) : V ⊗W ⊗ Z LW (z)⊗idZ−−−−−−−→W ⊗ V ⊗ Z idW⊗LZ(z)−−−−−−−→W ⊗ Z ⊗ V.
The fact that LW⊗Z is an L-operator is a straightforward consequence of the definitions. We
have an action of C on the category M(R, π): for each u ∈ C let tu be the endofunctor sending an
object (W,LW ) to (W,LW (·+u)) and a morphism f to f . Clearly t0 is the identity endofunctor
and tutv = tu+v. Moreover tu is a monoidal functor: the obvious map tu(W )⊗tu(Z)→ tu(W⊗Z)
is a natural isomorphism.
Example 3.1. Let V ∈M(R, π) be the representation with L-operator Ř. Then for each u ∈ C,
the representation V (u) = tuV has L operator LV (u)(z) = Ř(z + u). This object of M(R, π) is
called the vector representation with evaluation point u.
Example 3.2. Let 1 ∈ Vectk(π) be the tensor unit, see Section 2.6 and let L1(z) be the
composition ρλ−1 : V ⊗ 1 → V → 1 ⊗ V of the structure isomorphisms. Then 1 with this
L-operator is a representation, called the trivial representation. It is fixed by the action of tu.
Example 3.3. The dual representation of a representation (W,LW ) admitting a dual is the
representation
(
W∨, LW∨
)
on the π-graded dual vector space W (see Section 2.7). Its L-operator
LW∨(z) = L̃W (z)−1 is the inverse of the dual operator L̃W (z) : W∨ ⊗ V → V ⊗W∨ defined as
the composition
W∨ ⊗ V →W∨ ⊗ V ⊗W ⊗W∨ LW (z)(23)−−−−−−→W∨ ⊗W ⊗ V ⊗W∨ → V ⊗W∨
with the structure maps defining the duality in the category of π-graded vector spaces, see
Section 2.7. It exists whenever L̃W (z) is invertible for generic z.
14 G. Felder and M. Ren
3.4 R-matrices
Let W,Z ∈M(R, π). An isomorphism
ŘW,Z : W ⊗ Z → Z ⊗W
in M(R, π) is called an R-matrix. This means that ŘW,Z is a morphism of π-graded vector
spaces obeying the intertwining relation
Ř
(12)
W,ZLZ(z)(23)LW (z)(12) = LW (z)(23)LZ(z)(12)Ř
(23)
W,Z
in Hom(V ⊗W ⊗ Z,Z ⊗W ⊗ V ).
Example 3.4. Let V (u) be the vector representation with evaluation point u. Then the Yang–
Baxter equation implies that Ř(u − v) is a morphism V (u) ⊗ V (v) → V (v) ⊗ V (u). It is an
R-matrix except if u− v or v − u is a pole of Ř.
Proposition 3.5.
(i) If ŘW,Z is an R-matrix for W,Z ∈M(R, π) and u ∈ C, then the same isomorphism ŘW,Z
of π-graded vector spaces is an R-matrix for tuV , tuW .
(ii) If ŘW,Z , ŘW,Z′ are R-matrices then ŘW,Z⊗Z′ = Ř
(23)
W,Z′Ř
(12)
W,Z is an R-matrix for W , Z⊗Z ′.
(iii) If ŘW,Z , ŘW ′,Z are R-matrices then ŘW⊗W ′,Z =Ř
(12)
W,ZŘ
(23)
W ′,Z is an R-matrix for W⊗W ′, Z.
3.5 Partial traces and transfer matrices
The partial trace over V is the map
trV : HomVectk(π)(V ⊗W,W ⊗ V )→ Γ(π,EndW )
defined as follows.
For f ∈ Hom(V ⊗W,W ⊗ V ) and α ∈ π(a, b), γ ∈ π(a, a), let f(α, γ) be the component of f
mapping
f(α, γ) : Vα ⊗Wαγα−1 →Wγ ⊗ Vα.
Define
trVα f(α, γ) =
∑
i
(id⊗ e∗i )f(α, γ)(ei ⊗ id) ∈ Hom(Wαγα−1 ,Wγ).
for any basis ei of Vα and dual basis e∗i of the dual vector space (Vα)∗.
Definition 3.6. The partial trace trV f ∈ Γ(π,EndW ) of f ∈ HomVectπ(V ⊗W,W ⊗ V ) over
V is the section
trV f : α 7→ ⊕γ∈π(a,a) trVα f(α, γ) ∈ (EndW )α.
Example 3.7. Let W = 1 be the tensor unit, with nonzero components Wa = k, indexed
by identity arrows a ∈ A. For α ∈ π(a, b), we have (EndW )α = Homk(Wa,Wb) = k, see
Example 2.17.
The convolution algebra Γ(π,End 1) is the extension of scalars of the convolution ring of π.
The partial trace of the identity V ⊗ 1 ∼= V → V ∼= 1⊗ V is
trV (id) : α 7→ dimVα,
which is the (image in k of the) character chV of V .
Quantum Groups for Restricted SOS Models 15
Lemma 3.8.
(i) If ϕ : V → V ′ is an isomorphism of π-graded vector spaces then
trV ′
(
(id⊗ ϕ)f
(
ϕ−1 ⊗ id
))
= trV f.
(ii) Let fi ∈ Hom(Vi ⊗W,W ⊗ Vi), i = 1, 2, be homomorphisms of π-graded vector spaces and
let f
(12)
1 f
(23)
2 be the composition
V1 ⊗ V2 ⊗W
id⊗f2−−−→ V1 ⊗W ⊗ V2
f1⊗id−−−→W ⊗ V1 ⊗ V2.
Then
trV1⊗V2 f
(12)
1 f
(23)
2 = trV1 f1 trV2 f2.
Proof. Recall that a morphism ϕ is a collection of linear maps ϕα : Vα → V ′α, so (i) is the
standard property of the trace on each Vα.
As for (ii) to compute trV1⊗V2 we need to select for each γ ∈ π(a, b) and µ ∈ π(b, b) the
component g(α) of g = f
(12)
1 f
(23)
2 sending (V1⊗V2)γ⊗Wµ to Wµ′⊗(V1⊗V2)γ with µ′ = γ−1◦µ◦γ.
A basis of (V1 ⊗ V2)γ is given by choosing a basis of each component V1α ⊗ V2β with γ = βα
with α ∈ π(a, c) and β ∈ π(c, b), Thus the trace is non-trivial on the components of f mapping
V1α ⊗ V2β ⊗Wµ to Wµ′ ⊗ V1α ⊗ V2β. These components factor as
V1α ⊗ V2β ⊗Wµ
id⊗f2(β)−−−−−→ V1α ⊗Wβ−1◦µ◦β ⊗ V2β
f1(α)⊗id−−−−−→Wµ′ ⊗ V1α ⊗ V2β.
The claim follows by taking the tensor product of bases of V1α and V2β. �
Corollary 3.9. Suppose that ŘViVj ∈ Hom(Vi ⊗ Vj , Vj ⊗ Vi), (1 ≤ i < j ≤ 3) are a solution of
the Yang–Baxter equation
Ř
(23)
V1V2
Ř
(12)
V1V3
Ř
(23)
V2V3
= Ř
(12)
V2V3
Ř
(23)
V1V3
Ř
(12)
V1V2
with invertible ŘV1V2. Then the transfer matrices
Ti = trVi ŘViV3 ∈ Γ(π,End(V3)), i = 1, 2.
commute: T1T2 = T2T1.
Proof. We can write the Yang–Baxter equation as
(id⊗ ϕ)Ř
(12)
V2V3
Ř
(23)
V1V3
(
ϕ−1 ⊗ id
)
= Ř
(12)
V1V3
Ř
(23)
V2V3
with ϕ = Ř−1
V1V2
. By Lemma 3.8 (i) we deduce that
trV2⊗V1 Ř
(12)
V2V3
Ř
(23)
V1V3
= trV1⊗V2 Ř
(12)
V1V3
Ř
(23)
V2V3
.
The claim then follows from Lemma 3.8(ii). �
Remark 3.10. For W = 1 the R-matrix RV,1 is the tautological map V ⊗ 1 → 1 ⊗ V of
Example 3.7. Thus the transfer matrix generalizes the notion of character.
Remark 3.11. In particular Corollary 3.9 applies to the case of the category M(R, π): for each
object (W,LW ) we can interpret the RLL relations of Section 3.2 as a Yang–Baxter equation
with V1 = V (z), V2 = V (w) (see Example 3.1), V3 = W and ŘVW = LW (z). One gets the
basic statement of the quantum inverse scattering method that the transfer matrices trV LW (z)
commute for different values of the spectral parameter z. In that language Corollary is the
generalization of this statement to the case of arbitrary “auxiliary spaces” V1, V2.
16 G. Felder and M. Ren
3.6 Action groupoids and dynamical Yang–Baxter equation
If π = A o G is an action groupoid, R-matrices for π-graded vector spaces are expressed in
terms of the graded components as dynamical R-matrices. Then the tensor product of π-graded
modules is
(V ⊗W )(a,g) =
∑
h∈G
V(a,h) ⊗W(ah,h−1g).
Let Ř(z) be a solution of the Yang–Baxter equation (3.1) and let
Ř(z, a) ∈ ⊕g∈G Endk((V ⊗ V )(a,g))
be the restriction of Ř(z) to the graded components with fixed a ∈ A. Then the Yang–Baxter
equation can be written as
Ř(23)
(
z − w, ah(1)
)
Ř(12)(z, a)Ř(23)
(
w, ah(1)
)
= Ř(12)(w, a)Ř(23)
(
z, ah(1)
)
Ř(12)(z − w, a).
Here we use the “dynamical” notation with the placeholder h(i):
Ř(23)
(
ah(1)
)
(u⊗ v ⊗ w) = u⊗ Ř(ag)(v ⊗ w) if u ∈ V(a,g).
If we compose on the left with the product p(23)p(13)p(12) = p(12)p(13)p(23) of flips p : v⊗w 7→ w⊗v
we get the YBE for R = p ◦ Ř in the form
R(12)
(
z − w, ah(3)
)
R(13)(z, a)R(23)
(
w, ah(1)
)
= R(23)(w, a)R(13)
(
z, ah(2)
)
R(12)(z − w, a).
3.7 Transfer matrices in the case of action groupoids
In the case of action groupoid π = AoG we can identify convolution algebras with coefficients in
π-graded endomorphisms with algebras of difference operators (or discrete connections) acting
on the sections of sheaves over A. Let G(a) = {g ∈ G | ag = a} denote the stabilizer subgroup
of an object a ∈ A and for W ∈ Vectk(π) let WG(a) = ⊕g∈G(a)W(a,g). Let Γ(A,W ) be the space
of maps ψ : A → ta∈AWG(a) such that ψ(a) ∈ WG(a) for all a ∈ A. Then Γ(A,W ) is naturally
a module over Γ(π,EndW ) and the transfer matrices can be realized as linear operators on
Γ(A,W ).
Explicitly, we have (EndW )(a,g) = ⊕h∈G(a) Hom(W(ag,g−1hg),W(a,h)). For ψ ∈ Γ(A,W ) let
(tgψ)(a) = ψ(ag) ∈WG(ag). Then f ∈ Γ(π,EndW ) acts on ψ ∈ Γ(A,W ) as
(fψ)(a) =
∑
g∈G
f(a, g)(tgψ)(a).
Let ŘUW [a, g] be the component U(a,g) ⊗ WG(ag) → WG(a) ⊗ U(a,g) of a dynamical R-matrix
U ⊗W → W ⊗ U . Then rg(a) = trU(a,g)
ŘUW [a, g] maps WG(ga) to WG(a). The transfer matrix
is
trU ŘUW =
∑
g∈G
rgtg,
where rg is understood as a multiplication operator.
4 Elliptic quantum groups
As a class of examples of the above construction let us work out the case of the dynamical
R-matrix defining the elliptic quantum group in the gln-case, see [19]. Here k = C.
Quantum Groups for Restricted SOS Models 17
4.1 Action groupoids of the weight lattice
Let h ∼= Cn be the Lie subalgebra of diagonal matrices in gln and h0 ⊂ h the subalgebra of
traceless diagonal matrices. The weight lattice is the lattice P =
∑n
i=1 Zεi ⊂ h∗ spanned by the
coordinate functions εi : x 7→ xi. It acts on h∗0 = h∗/C(1, . . . , 1) by translations on h∗ composed
with the canonical projection h∗ → h∗0. For each orbit Ob = b + P , b ∈ h∗0 we have a groupoid
Ob o P , consisting of pairs (a, µ) ∈ Ob × P with composition (a1, µ1) ◦ (a2, µ2) = (a2, µ1 + µ2),
defined if a1 = a2 + µ2.
We consider the following standard subsets of h∗0:
� The set of sln-dominant weights P+ = {λ ∈ Zn/Z(1, . . . , 1) |λ1 ≥ · · · ≥ λn}.
� The set of regular dominant weights P++ = {λ ∈ P+ |λ1 > · · · > λn}.
� The set of dominant affine weights P r+ = {λ ∈ P+ |λ1 − λn ≤ r}, of level r ∈ Z≥0.
� The set of regular dominant affine weights P r++ = {λ ∈ P++ |λ1−λn < r}, of level r ∈ Z≥n.
We have a bijection P r+ → P r+n+ given by λ 7→ λ+ ρ, ρ = (n− 1, . . . , 1, 0).
4.2 Dynamical R-matrix
Fix two complex numbers τ , γ such that Im τ > 0 and γ 6∈ Z + τZ. Let
θ(z, τ) = −
∑
n∈Z
eiπ(n+ 1
2
)2τ+2πi(n+ 1
2
)(z+ 1
2
)
be the odd Jacobi theta function and [z] = θ(γz, τ)/(γθ′(0, τ)) is normalized to have derivative 1
at z = 0. The function z 7→ [z] of one complex variable is an odd entire function with first order
zeros on the lattice
Λ = Z 1
γ + Z τ
γ .
The defining representation V̄ = Cn of gln has a weight decomposition V̄ = ⊕ni=1V̄εi where
V̄εi = Cei is the span of the i-th standard basis vector.
Let Eij be the n×n matrix such that Eijek = δjkei for all k ∈ {1, . . . , n}. The (unnormalized)
elliptic dynamical R-matrix with spectral parameter z ∈ C is
Ř(z, a) =
n∑
i=1
Eii ⊗ Eii −
n∑
i 6=j=1
[ai − aj + 1][z]
[ai − aj ][1− z]
Eij ⊗ Eji
+
n∑
i 6=j=1
[ai − aj + z][1]
[ai − aj ][1− z]
Eii ⊗ Ejj . (4.1)
It is a meromorphic function of z ∈ C, a ∈ Cn and solves the dynamical Yang–Baxter equation
in the additive form
Ř
(
z − w, a+ h(1)
)(23)
Ř(z, a)(12)Ř
(
w, a+ h(1)
)(23)
= Ř(w, a)(12)Ř
(
z, a+ h(1)
)(23)
Ř(z − w, a)(12),
and the inversion relation
Ř(z, a)Ř(−z, a) = idCn⊗Cn ,
valid for generic z, a. The relation to the R matrix presented in [19] is Ř(z, a) = PR(z, λ)
where λ = γa. By restricting a to take values in an orbit Ob we can construct R-matrices acting
18 G. Felder and M. Ren
on groupoid-graded vector spaces. To do this we need to avoid the poles on the hyperplanes
ai − aj ≡ 0 mod Λ, (i 6= j) of the R-matrix.
We consider the case where r = 1/γ is an integer > n. Then for a ∈ P the poles are at
ai − aj = mr, i 6= j ∈ {1, . . . , n}, m ∈ Z.
Let ∆ ⊂ P be the union of these hyperplanes. The affine Weyl group Wr is the group generated
by orthogonal reflections at these hyperplane in the Euclidean space P ⊗R = Rn. It acts freely
and transitively on the complement of ∆ ⊗ R and P r++ is the set of weights in a connected
component of the complement and is a fundamental domain for the action of Wr on P r ∆.
We distinguish two cases, named after the corresponding models of statistical mechanics.
1. Generalized SOS model. Let b ∈ h∗, π = Ob o P , V b = ⊕γ∈πV b
γ with
V b
(a,µ) =
{
V̄εi = Cei if µ = εi, i = 1, . . . , n and a ∈ Ob,
0, if µ 6∈ {ε1, . . . , εn}.
If b does not lie in ∆ then R(z) is a well-defined endomorphism of V b ⊗ V b and obeys the
Yang–Baxter equation.
2. Generalized RSOS model. Let b = 0. The groupoid π is the full subgroupoid of O0oP
on P r++: it consists of pairs (a, µ) ∈ O0 o P such that both a and a + µ lie in P r++. The
non-zero components are
V RSOS
(a,εi)
= V̄εi = Cei, a, a+ εi ∈ P r++.
To define Ř(z) in the RSOS case we view V RSOS as an O0 o P -graded subspace of V b=0 such
that V RSOS
γ = 0 for γ 6∈ π, see Section 2.9.
Proposition 4.1. Let V = V b=0 and let V RSOS ⊂ V be viewed as an O0 o P -graded subspace
of V . Then Ř(z) is well-defined on V RSOS ⊗ V RSOS and maps V RSOS ⊗ V RSOS to itself.
Proof. While Ř(z) is not defined on all vectors in V ⊗ V , it is well-defined on V RSOS ⊗ V RSOS
since by construction the denominators [ai − aj ] don’t vanish for a ∈ P r++. Suppose (a+ εj , εi)
and (a, εj) are composable arrows in the subgroupoid π, meaning that a, a+ εj and a+ εi + εj
belong to P r++. Then Ř(z) maps V(a,εj) ⊗ V(a+εj ,εi) to itself if i = j and to
V(a,εj) ⊗ V(a+εj ,εi) ⊕ V(a,εi) ⊗ V(a+εi,εj)
if i 6= j. The first summand is indexed by a pair of arrows in the subgroupoids but the second
is not if a + εi 6∈ P r++. Thus Ř(z) preserves V RSOS if and only if the component of the image
of Ř(z) in V(a,εi)⊗ V(a+εi,εj) for a+ εi 6∈ P r++ vanishes. So we need to check the vanishing of the
component
− [ai − aj + 1][z]
[ai − aj ][1− z]
Eij ⊗ Eji : V(a,εj) ⊗ V(a+εj ,εi) → V(a,εi) ⊗ V(a+εi,εj), i 6= j, (4.2)
of Ř(z, a) in the case where a + εi 6∈ P r++ and a, a + εj , a + εi + εj ∈ P r++. The condition for a
being in P r++ is a1 > · · · > an > a1 − r. For a ∈ P r++, a + εi violates an inequality if and only
if i ≥ 2 and ai−1 = ai + 1 or i = 1 and an = a1 − r + 1. The condition that a + εi + εj ∈ P r++
implies that j = i− 1 if i ≥ 2 and j = n if i = 1. In both cases aj ≡ ai + 1 mod r and thus (4.2)
vanishes. �
Quantum Groups for Restricted SOS Models 19
Corollary 4.2. The restriction of Ř(z) to V RSOS ⊗ V RSOS obeys the dynamical Yang–Baxter
equation.
Example 4.3. The case n = 2 is the original RSOS model of [3]. In this case we have a bijection
O0 = Z× Z/(Z(1, 1)) → Z sending the class of (a1, a2), to l = a1 − a2 under this identification
A = P r++ = {1, . . . , r − 1}.
The dynamical R-matrix in the basis e1 ⊗ e1, e1 ⊗ e2, e2 ⊗ e1, e2 ⊗ e2 is
Ř(z, a) =
1 0 0 0
0
[l + z][1]
[l][1− z]
− [l + 1][z]
[l][1− z]
0
0 − [l − 1][z]
[l][1− z]
[l − z][1]
[l][1− z]
0
0 0 0 1
=
W1 0 0 0
0 W5 W4 0
0 W3 W6 0
0 0 0 W2
.
The Boltzmann weights Wj correspond to the local configurations of Fig. 2.
l + 1 l + 2
l + 1l
W1
l − 1 l − 2
l − 1l
W2
l + 1 l
l − 1l
W3
l − 1 l
l + 1l
W4
l + 1 l
l + 1l
W5
l − 1 l
l − 1l
W6
Figure 2. Six possible types of Boltzmann weight in the eight-vertex SOS model.
The hard hexagon model is part of the family of RSOS models with r = 5. In this case the
allowed values of l are 1, 2, 3, 4 and, because of the identity [u] = [5−u], u ∈ C, the Boltzmann
weights are invariant under l 7→ 5− l. We can map the model to a lattice gas model on a square
lattice. A configuration of the RSOS model, i.e., a map from the vertices of a square lattice
to {1, 2, 3, 4} is mapped to a configuration of particles on the lattice: there is a particle at the
vertices with value 1 or 4 and no particle at the vertices with value 2 or 3. To each configuration
of particles there correspond two RSOS configuration related by l 7→ 5−l and thus with the same
Boltzmann weight. The rule that the value at neighbouring lattice sites differs by 1 translate
to the rule that no two particles of the lattice gas can sit at nearest neighbouring sites. We
can think of this model as a hard square model: the squares whose vertices are the nearest
neighbours of the positions of the particles are required not to overlap (i.e., to have disjoint
interiors). The row-to-row transfer matrices of these models commute among themselves and
in particular with the transfer matrix with spectral parameter z = −1, which is the transfer
matrix of the hard hexagon model. In this case W5 vanishes for l = 1 and W6 vanishes for l = 4,
implying that in a configuration with non-zero Boltzmann weight no two particles can sit at the
endpoints of a NW-SE diagonal. We can translate these rules by thinking of the particles as
midpoints of lattice hexagons which are not allowed to overlap.
4.3 Characters
Here we consider the RSOS case and write V instead of V RSOS for the vector representation.
Out of the vector evaluation representation V (z) one can construct several new representa-
tions by the fusion or reproduction method [35, 37], which admit pairwise R-matrices for generic
values of the evaluation parameters. It follows that their characters form a unital commutative
algebra of difference operators with integer coefficients. The unit element is the character of the
trivial representation.
20 G. Felder and M. Ren
Here are some examples of character calculations. The groupoid πr(P ) of the restricted
elliptic quantum group with weight lattice P is the full subgroupoid of O0 o P on the set
A = P r++. Its convolution ring is isomorphic to the subring χADP (O0)χA of the crossed product
DP (O0) = ZO0 oZP , see Lemma 2.3 and Proposition 2.4. The subring ZP = Z
[
t±1
1 , . . . , t±1
n
]
is
the ring of Laurent polynomials with generators ti = tεi , i = 1, . . . , n.
� The character of the trivial representation is the multiplication operator by the character-
istic function of A = P r++:
ch1 = χA.
� The character of the vector representation V (z).
chV (z) =
n∑
i=1
χAtiχA =
n∑
i=1
χA∩(A−εi)ti.
� The R-matrix (4.1) has a pole at z = 1 and is not invertible at z = −1. We set Řreg(1, a) =
resz=1 Ř(z, a). Then we have an exact sequence (see Appendix A)
· · · Ř(−1)−−−−→ V (z + 1)⊗ V (z)
Řreg(1)−−−−→ V (z)⊗ V (z + 1)
Ř(−1)−−−−→ · · · .
The analogue of the symmetric square of the vector representation, is S2V (z) = Ker Řreg(1)
∼= Coker Ř(−1). Its character can be computed from the explicit basis of Lemma A.1
chS2V (z) =
n∑
i=1
χA∩(A−2εi)t
2
i +
∑
1≤i<j≤n
χA∩(A−εi)∩(A−εj)titj .
Similarly, the second exterior power
∧2 V (z) = Coker Řreg(1) ∼= Ker Ř(−1) has character
ch∧2 V (z) =
∑
1≤i<j≤n
χA∩(A−εi−εj)titj .
From the exact sequence 0 → S2V (z) → V (z + 1) ⊗ V (z) →
∧2 V (z) → 0, follows the
identity
ch2
V (z) = ch∧2 V (z) + chS2V (z) .
� For n = 2 we have the bijection O0 = Z × Z/Z(1, 1) → Z sending the class of (a1, a2)
to a1 − a2. Under this identification, A = [1, r − 1] = {1, 2, . . . , r − 1}. Then V (z + p −
1)⊗ · · · ⊗ V (z + 1)⊗ V (z) has a subrepresentation SpV (z) = ∩p−1
i=1 Ker Řreg(1)(i,i+1) with
character
chSpV (z) = χ[1,r−p−1]t
p
1 + χ[2,r−p]t
p−1
1 t2 + · · ·+ χ[p+1,r−1]t
p
2,
p = 0, . . . , r − 2. The characters Lp = chSpV (z) obey the fusion rules
LpLq =
∑
s≡p+q mod 2
N s
pqu
p+q−s
2 Ls,
where u is the central element t1t2 and
N s
pq =
{
1, |p− q| ≤ s ≤ min(p+ q, 2r − 4− p− q),
0, otherwise.
These are the famous fusion rules that first appeared in conformal field theory [6, 46]. The
algebra with generators Lp and relations above is called the Verlinde algebra.
Quantum Groups for Restricted SOS Models 21
4.4 Exterior powers
The kth exterior power
∧k V (z) is defined as the quotient of V (z+k−1)⊗· · ·⊗V (z+1)⊗V (z)
by the sum of the images of Řreg(1)(j,j+1) for j = 1, . . . , k − 1. Its character is
ch∧k V (z) = χAek(t1, . . . , tn)χA,
with A = P r++. Here ek(t1, . . . , tn) =
∑
1≤i1<···<ik≤n ti1 · · · tik is the kth elementary symmetric
polynomial. It follows from the existence of R-matrices for pairs of exterior powers that these
characters commute.
The convolution ring Z[πr(P )] ∼= χADP (O0)χA acts naturally on functions on A = P r++. The
characters can be simultaneously diagonalized.
Theorem 4.4. Let q = e2πi/r and pick any n-th root q1/n of q. For each λ ∈ A = P r++ let ψλ
be the function on A given by
ψλ(a) = q−
1
n
∑
i ai
∑
i λi det
(
qλiaj
)
.
Then
ch∧k V (z) ψλ = ek
(
qλ̄1 , . . . , qλ̄n
)
ψλ, λ̄i = λi −
1
n
∑
j
λj .
Proof. Let us first ignore the characteristic functions χA and consider ψλ as a function on the
whole weight lattice. Then for any symmetric polynomial P (t) in t1, . . . , tn,
P (t)ψλ = P
(
qλ̄1 , . . . , qλ̄n
)
ψλ.
The function ψλ(a) is a skew-symmetric r-periodic function of a1, . . . , an. It thus vanishes on the
hyperplanes ai−aj ≡ 0 mod r and in particular on the walls ai+1−ai = 0, a1−an = r forming
the complement of A = P r++ in Ā = P r+. Thus χAψλ = χĀψλ. The monomials tI = ti1 · · · tik
with i1 < · · · < ik map P r++ to P r+. Thus
χAek(t)χAψλ = χAek(t)χĀψλ =
∑
I⊂{1,...,n}, |I|=k
χA∩t−1
I Āek(t)ψλ
=
∑
I⊂{1,...,n}, |I|=k
χAek(t)ψλ = ek(q
λ̄)χAψλ. �
5 Quantum enveloping algebras at roots of unity
We are mainly concerned with dynamical R-matrices with non-trivial dependence on the spectral
parameter, but it is instructive to consider the case of constant dynamical R-matrices arising
from the representation theory of semisimple Lie algebras and their quantum versions. The
main simplification in this case is that the category of finite dimensional modules is braided,
namely for each pair of objects V , W there is an isomorphism τV,W : V ⊗W →W ⊗ V , obeying
compatibility conditions with the structure of a monoidal category, given by the evaluation of
the universal R-matrix composed with the permutation of factors. This property fails for some
pairs of objects in the case of quantum affine Lie algebras or Yangians, which is the case when
the R-matrices have a non-trivial dependence on the spectral parameter.
We focus on the case of quantum groups at root of unity which is a toy model for restricted
models. Strictly speaking the above has to be corrected in this case and one has to be more
careful in the definition of the category of finite dimensional modules. We consider the semisim-
ple quotient of the category of tilting modules [8, Section 11.3], [2, 41]. It is an abelian monoidal
22 G. Felder and M. Ren
C-linear ribbon category Cq(g) depending on a simple Lie algebra g and a primitive `-th root
of unity q. It has finitely many equivalence classes of simple objects Lλ labeled by dominant
weights in a scaled Weyl alcove P `+(g), and any object is isomorphic to a direct sum of simple
modules. The alcove P `+(g) is a finite subset of the cone P+ of dominant weight, bounded by
a hyperplane as in Section 4.1. See [41] for a description in the most general case of a Lie algebra
and any root of unity.
Let P be the weight lattice and π be the full subgroupoid of P oP on P `+(g). Its objects are
dominant weights and there is exactly one arrow a → b for any two weights a, b ∈ P `+(g). As
usual we denote by (a, b− a) this arrow.
Theorem 5.1. There is a faithful exact monoidal functor Uqg-mod → Vectk(π) sending an
object W to Ŵ = ⊕(a,λ)∈πŴa,λ with
Ŵa,λ = HomCq(g)(La,W ⊗ La+λ).
Proof. A morphism f : W →W ′ in Uqg-mod induces a morphism f̂ : ϕ 7→ (f ⊗ id) ◦ϕ from Ŵ
to Ŵ ′ and this assignment is compatible with compositions so that we get a well-defined functor.
The trivial module k which is the tensor unit in Uqg is mapped to the tensor unit k̂ = 1 in
Vectk(π), see Section 2.6. Moreover we have a natural transformation
Ŵ ⊗ Ẑ → Ŵ ⊗ Z,
whose restriction to Ŵa,µ ⊗ Ẑa+µ,λ−µ is the composition
Hom(La,W ⊗ La+µ)⊗Hom(La+µ, Z ⊗ La+λ)→ Hom(La,W ⊗ Z ⊗ La+λ),
ϕ⊗ ψ 7→ (id⊗ ψ) ◦ ϕ. �
Since Cq(g) is semisimple, by taking the direct sum over µ we get an isomorphism (Ŵ ⊗
Ẑ)a,λ → (Ŵ ⊗ Z)a,λ on each graded component.
Theorem 5.2. There is a system of isomorphisms (dynamical R-matrices)
ŘW,Z : Ŵ ⊗ Ẑ → Ẑ ⊗ Ŵ
obeying the quasitriangularity relations
ŘW⊗W ′,Z = Ř
(12)
W,ZŘ
(23)
W ′,Z , ŘW,Z⊗Z′ = Ř
(23)
W,Z′Ř
(12)
W,Z
and the dynamical Yang–Baxter equation
Ř
(12)
W,ZŘ
(23)
Y,Z Ř
(12)
Y,W = Ř
(23)
Y,W Ř
(12)
Y,Z Ř
(23)
W,Z ,
for any objects W , W ′, Z, Z ′, Y .
Let us compute the character of Ŵ . Let N c
ab = dim Hom(Lc, La ⊗ Lb) be the multiplicity
of Lc in the decomposition of La ⊗ Lb as a direct sum of simple objects. The numbers N c
ab
are called fusion coefficients. They are the structure constants of the fusion ring (or Verlinde
algebra) with generators ea and product eaeb =
∑
cN
c
abec. Then
chLµ =
∑
λ
nλ,µtλ,
where nλ,µ(a) = Na+λ
a,µ .
The commutativity of the characters is the associativity of the Verlinde algebra.
Quantum Groups for Restricted SOS Models 23
A R-matrix at special values
The R-matrix (4.1) has a pole at z = 1 mod Λ. Because of the inversion relation we see that it
is regular and invertible except for z ∈ ±1 + Λ. For z = −1 we have
Ř(−1, a) = −
n∑
i=1
Eii ⊗ Eii +
∑
i 6=j
[ai − aj + 1][1]
[ai − aj ][2]
(Eij ⊗ Eji + Ejj ⊗ Eii).
Let Řreg(1, a) = resz=1Ř(z, a). Then
Řreg(1, a) =
∑
i 6=j
[ai − aj + 1][1]
[ai − aj ]
(Eij ⊗ Eji − Eii ⊗ Ejj).
Lemma A.1. Let ei(a) denote the standard basis vector ei ∈ Cn in V(a,εi).
(i) The image of Ř(−1, a) coincides with the kernel of Řreg(1, a) and is spanned by the linearly
independent vectors
ei(a)⊗ ej(a+ εi) + ej(a)⊗ ei(a+ εj),
where a, i, j are such that i ≤ j and a, a+ εi, a+ εj , a+ εi + εj ∈ P r++.
(ii) The image of Řreg(1, a) coincides with the kernel of Ř(−1, a) and is spanned by the fol-
lowing linear independent vectors:
ei(a)⊗ ej(a+ εi)− ej(a)⊗ ei(a+ εj),
where a, i, j are such that i < j and a, a+ εi, a+ εj , a+ εi + εj ∈ P r++;
ei−1(a)⊗ ei(a+ εi−1),
where i = 2, . . . , n and ai−1 = ai + 1;
en(a)⊗ e1(a+ εn),
where an = a1 − r + 1.
Proof. We consider these operators on the weight spaces (V ⊗ V )(a,εi+εj), which are non-
vanishing for a, a + εi + εj ∈ P r++. Recall that a ∈ P r++ means a1 > · · · > an > a1 − r. Thus
the only case where the numerator [ai − aj + 1] vanishes is when j = i − 1 and ai−1 = ai + 1
for i = 1, . . . , n, where we set a0 = an + r. In this case a + εi 6∈ P r++. There are three cases to
consider (we may assume that i ≥ j):
(a) i = j. In this case the weight space is spanned by ei(a) ⊗ ei(a + εi), Ř(−1) acts by
multiplication by −[2]/[1] 6= 0 and Řreg(1) vanishes. Thus
Im Ř(−1) = Ker Řreg(1) = span(ei(a)⊗ ei(a+ εi)),
Im Řreg(1) = Ker Ř(−1) = 0.
(b) i > j and a + εi, a + εj ∈ P r++. A basis of the weight space consists of ej(a) ⊗ ei(a + εj)
and ei(a)⊗ ej(a+ εi). The matrices of Ř(−1) and Řreg(1) in this basis are (up to factors
of [1] and [2])
[ai − aj + 1]
[ai − aj ]
[ai − aj − 1]
[ai − aj ]
[ai − aj + 1]
[ai − aj ]
[ai − aj − 1]
[ai − aj ]
and
[ai − aj − 1]
[ai − aj ]
− [ai − aj + 1]
[ai − aj ]
− [ai − aj − 1]
[ai − aj ]
[ai − aj + 1]
[ai − aj ]
,
24 G. Felder and M. Ren
respectively. The matrix entries are all non-zero and we see that
Im Ř(−1) = Ker Řreg(1) = span(ej(a)⊗ ei(a+ εj) + ei(a)⊗ ej(a+ εi)),
Im Řreg(1) = Ker Ř(−1) = span(ej(a)⊗ ei(a+ εj)− ei(a)⊗ ej(a+ εi)).
(c) i > j and one of a + εi, a + εj 6∈ P r++. This happens only if j = i − 1 and ai−1 = ai + 1
(with a0 := an + r). Then the weight space is spanned by ei−1(a) ⊗ ei(a + εi−1); Ř(−1)
acts by zero and Řreg(1) acts by multiplication by [2]/[1] 6= 0. Thus
Im Ř(−1) = Ker Řreg(1) = 0,
Im Řreg(1) = Ker Řreg(−1) = span(ei−1(a)⊗ ei(a+ εi−1)), i = 1, . . . , n,
where for i = 1 the right-hand side is en(a)⊗ e1(a+ εn). �
Acknowlegments
The authors are supported in part by the National Centre of Competence in Research
SwissMAP – The Mathematics of Physics – of the Swiss National Science Foundation. They
are also supported by the grants 196892 and 178794 of the Swiss National Science Foundation,
respectively. We are grateful to the referees for their careful reading of the first version of this
paper and their many useful suggestions and corrections.
References
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1 Introduction
1.1 Quantum groups and exactly solvable models
1.2 Elliptic quantum groups and dynamical Yang–Baxter equation
1.3 Restricted SOS models
1.4 Categories of representations
1.5 Outline of the paper
2 Grading by groupoids
2.1 Groupoids
2.2 The convolution ring of a groupoid
2.3 Convolution rings of subgroupoids
2.4 Action groupoids
2.5 The category of pi-graded vector spaces of finite type
2.6 Tensor product
2.7 Duality
2.8 Characters
2.9 Subgroupoids
2.10 Convolution algebras with coefficients in pi-graded algebras
3 Yang–Baxter equation and RLL relations
3.1 Yang–Baxter equation
3.2 RLL relations
3.3 The monoidal category M(R,pi)
3.4 R-matrices
3.5 Partial traces and transfer matrices
3.6 Action groupoids and dynamical Yang–Baxter equation
3.7 Transfer matrices in the case of action groupoids
4 Elliptic quantum groups
4.1 Action groupoids of the weight lattice
4.2 Dynamical R-matrix
4.3 Characters
4.4 Exterior powers
5 Quantum enveloping algebras at roots of unity
A R-matrix at special values
References
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| id | nasplib_isofts_kiev_ua-123456789-211183 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T15:32:36Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Felder, Giovanni Ren, Muze 2025-12-25T13:24:37Z 2021 Quantum Groups for Restricted SOS Models. Giovanni Felder and Muze Ren. SIGMA 17 (2021), 005, 26 pages 1815-0659 2020 Mathematics Subject Classification: 17B37; 18M15 arXiv:2010.01060 https://nasplib.isofts.kiev.ua/handle/123456789/211183 https://doi.org/10.3842/SIGMA.2021.005 We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of π-graded vector spaces for a groupoid π. The authors are supported in part by the National Centre of Competence in Research, SwissMAP, and the Mathematics of Physics of the Swiss National Science Foundation. They are also supported by the grants 196892 and 178794 of the Swiss National Science Foundation, respectively. We are grateful to the referees for their careful reading of the first version of this paper and their many useful suggestions and corrections. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Groups for Restricted SOS Models Article published earlier |
| spellingShingle | Quantum Groups for Restricted SOS Models Felder, Giovanni Ren, Muze |
| title | Quantum Groups for Restricted SOS Models |
| title_full | Quantum Groups for Restricted SOS Models |
| title_fullStr | Quantum Groups for Restricted SOS Models |
| title_full_unstemmed | Quantum Groups for Restricted SOS Models |
| title_short | Quantum Groups for Restricted SOS Models |
| title_sort | quantum groups for restricted sos models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211183 |
| work_keys_str_mv | AT feldergiovanni quantumgroupsforrestrictedsosmodels AT renmuze quantumgroupsforrestrictedsosmodels |