Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic
The paper introduces a new method to determine all rank two Nichols algebras of diagonal type over fields of positive characteristic.
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Wang, J. Heckenberger, I. 2019-02-12T18:06:58Z 2019-02-12T18:06:58Z 2015 Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic / J. Wang, I. Heckenberger // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16T05 DOI:10.3842/SIGMA.2015.011 https://nasplib.isofts.kiev.ua/handle/123456789/146995 The paper introduces a new method to determine all rank two Nichols algebras of diagonal type over fields of positive characteristic. It is a pleasure to thank N. Andruskiewitsch and H.-J. Schneider for a very fruitful discussion on some details of this topic. The authors thank the referees for their helpful comments and suggestions. J. Wang is supported by China Scholarship Council. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic Article published earlier |
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Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic |
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Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic Wang, J. Heckenberger, I. |
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Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic |
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Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic |
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The paper introduces a new method to determine all rank two Nichols algebras of diagonal type over fields of positive characteristic.
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Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic / J. Wang, I. Heckenberger // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 29 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 011, 24 pages
Rank 2 Nichols Algebras of Diagonal Type
over Fields of Positive Characteristic
Jing WANG and István HECKENBERGER
Philipps-Universität Marburg, FB Mathematik und Informatik,
Hans-Meerwein-Straße, 35032 Marburg, Germany
E-mail: jing@mathematik.uni-marburg.de, heckenberger@mathematik.uni-marburg.de
URL: http://www.mathematik.uni-marburg.de/~jing/,
http://www.mathematik.uni-marburg.de/~heckenberger/
Received August 01, 2014, in final form February 02, 2015; Published online February 07, 2015
http://dx.doi.org/10.3842/SIGMA.2015.011
Abstract. The paper introduces a new method to determine all rank two Nichols algebras
of diagonal type over fields of positive characteristic.
Key words: Nichols algebra; Cartan graph; Weyl groupoid; root system
2010 Mathematics Subject Classification: 16T05
1 Introduction
The theory of Nichols algebras was dominated and motivated by Hopf algebra theory. In 1978,
W. Nichols first introduced the structure of Nichols algebra in the paper “Bialgebras of type
one” [22], where he studied certain pointed Hopf algebras. S.L. Woronowicz rediscovered this
structure in his approach to “quantum differential calculus” [28, 29]. M. Rosso and G. Lusztig
defined and used them to present quantum groups in a different language [21, 23]. In fact,
Nichols algebra turns out to be very important in Hopf algebras and quantum groups [2, 6, 24]
and has applications in conformal field theory and mathematical physics [25, 26, 27]. Nichols al-
gebras play an important role in the classification of pointed Hopf algebras with certain finiteness
properties in papers by N. Andruskiewitsch and H.-J. Schneider, see for example [4, 6]. The
crucial step to classify pointed Hopf algebras is the computation of the Nichols algebras. The
explicit presentations by generators and relations of a finite-dimensional Nichols algebra of
a braided vector space in suitable classes are crucial for the classification lifting method in [4, 6].
Several authors have already classified both infinite and finite dimensional Nichols algebra
of Cartan type, see [5, 12, 23]. Further, N. Andruskiewitsch [1] stated the following question.
Question 5.9. Given a braiding matrix (qij)1≤i,j≤θ whose entries are roots of 1, when B(V )
is finite-dimensional, where V is a vector space with basis x1, . . . , xθ and braiding c(xi ⊗ xj) =
qij(xj ⊗ xi)? If so, compute dimk B(V ), and give a “nice” presentation by generators and
relations.
The first half of Question 5.9 was answered by the first named author in [14] when the charac-
teristic of the field is 0. The crucial theoretical tools of the classification were the Weyl groupoid
of a braided vector space of diagonal type and the root system associated to a Nichols algebra
of diagonal type, see [12]. From V. Kharchenko [20, Theorem 2] any Nichols algebra B(V ) of di-
agonal type has a (restricted) Poincaré–Birkhoff–Witt basis consisting of homogenous elements
with respect to the Zn-grading of B(V ). In [12], the root system and the Weyl groupoid of B(V )
for a Nichols algebra B(V ) of diagonal type was defined. This Weyl groupoid plays a similar
role as the Weyl group does for ordinary root systems. Based on these results, in [18] and [10]
the abstract combinatorial theory of Weyl groupoids and generalized root systems was initiated.
mailto:jing@mathematik.uni-marburg.de
mailto:heckenberger@mathematik.uni-marburg.de
http://www.mathematik.uni-marburg.de/~jing/
http://www.mathematik.uni-marburg.de/~heckenberger/
http://dx.doi.org/10.3842/SIGMA.2015.011
2 J. Wang and I. Heckenberger
Later, the theory of root systems and Weyl groupoids was extended to more general Nichols
algebras in [3, 16, 17]. With the classification result in [14], N. Andruskiewitsch and H.-J. Schnei-
der [7] obtained a classification theorem about finite-dimensional pointed Hopf algebras under
some technical assumptions. On the other hand, it is natural and desirable to analyze the clas-
sification of Nichols algebras of diagonal type for arbitrary fields. The authors in [8] constructed
new examples of Nichols algebras in positive characteristic by applying a combinatorial formula
for products in Hopf quiver algebras.
In this paper, all rank 2 Nichols algebras of diagonal type with a finite root system over fields
of positive characteristic are classified. We introduce some properties of rank two Cartan graphs,
see Theorem 4.15. Theorem 4.15 characterizes finite connected Cartan graphs of rank two in
terms of certain integer sequences. This theorem simplifies substantially the calculations needed
to check that the Weyl groupoids of the Nichols algebras in Tables 5.1–5.6 of our classification
are finite. Indeed, using the theorem, these calculations can be done by hand within a very
short time, in contrast to the calculations based on the definition of the Weyl groupoid W or
using a longest element of W. The main result of this paper is Theorem 5.1. Table 7 illustrates
all the exchange graphs of the corresponding Cartan graphs in Theorem 5.1.
The structure of the paper is as follows. In Section 2 we recall the definition of Cartan graphs,
their Weyl groupoids and root systems. Some well-known results are also recalled. In Section 3,
Theorem 3.1 associates a semi-Cartan graph C(M) of rank θ to a tuple M of finite-dimensional
irreducible Yetter–Drinfel’d-modules. Further, C(M) is a Cartan graph if the set of real roots
of M is finite. The Dynkin diagram for a braided vector space of diagonal type is recalled
in Section 3.1 and some corollaries are also obtained, see Lemma 3.4 and Proposition 3.6. In
Section 4 we recall certain integer sequences and prove a local property of them in Theorem 4.6.
This is another main technical part for the classification. Finally the main result of this paper
is formulated in Section 5, see Theorem 5.1. In this section, all rank two Nichols algebras of
diagonal type with a finite root system over fields of positive characteristic are classified. Since
many subcases have to be considered, this is the largest part of the paper. To simplify the
results, this paper ends with 5 tables containing all the Dynkin diagrams of rank two braided
vector spaces with a finite root system for all fields of positive characteristic.
Throughout the paper k denotes a field of characteristic p > 0. Let k∗ = k \ {0}. The set
of natural numbers not including 0 is denoted by N and we write N0 = N ∪ {0}. For n ∈ N,
let G′n denote the set of primitive n-th roots of unity in k, that is G′n = {q ∈ k∗ | qn = 1,
qk 6= 1 for all 1 ≤ k < n}.
2 Cartan graphs and root systems
Let I be a non-empty finite set. Recall from [19, § 1.1] that a generalized Cartan matrix is
a matrix A = (aij)i,j∈I with integer entries such that
• aii = 2 and ajk ≤ 0 for any i, j, k ∈ I with j 6= k,
• if aij = 0 for some i, j ∈ I, then aji = 0.
Let X be a non-empty set and let r : I × X → X be a map. For all i ∈ I, let ri : X → X ,
X 7→ r(i,X). Let AX = (aXij )i,j∈I be a generalized Cartan matrix in ZI×I for all X ∈ X . The
quadruple C = C(I,X , r, (AX)X∈X ) is called a semi-Cartan graph if
• r2i = idX for all i ∈ I,
• aXij = a
ri(X)
ij for all X ∈ X and i, j ∈ I.
The cardinality of I is called the rank of C, and the elements of I the labels of C. The
elements of X are called the points of C. Semi-Cartan graphs are called Cartan schemes in [9].
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 3
We change the terminology in order to increase the recognizability of our structures, to avoid
possible confusion with other mathematical concepts, and in order to shorten the notation for
our main objects, the Cartan graphs. We thank N. Andruskiewitsch and H.-J. Schneider for
a very fruitful discussion on this issue.
The exchange graph of C is a labeled non-oriented graph with vertices corresponding to points
of C, and edges marked by labels of C, where two vertices X, Y are connected by an edge i if
and only if X 6= Y and ri(X) = Y (and ri(Y ) = X). For simplification, instead of several edges
we display only one edge with several labels. A semi-Cartan graph is called connected if its
exchange graph is a connected graph.
Let C = C(I,X , r, (AX)X∈X ) be a semi-Cartan graph. We fix once and for all the notation
(αi)i∈I for the standard basis of ZI . Then there exists a unique category D(X , I) with objects
ObD(X , I) = X and morphisms Hom(X,Y ) = {(Y, f,X) | f ∈ End(ZI)} for X,Y ∈ X , such
that the composition is defined by
(Z, g, Y ) ◦ (Y, f,X) = (Z, gf,X)
for all X,Y, Z ∈ X , f, g ∈ End(ZI). For all X ∈ X and all i ∈ I, let
sXi ∈ Aut
(
ZI
)
, sXi αj = αj − aXijαi (2.1)
for all j ∈ I. We write W(C) for the smallest subcategory of D(X , I) which contains all mor-
phisms (ri(X), sXi , X), where i ∈ I and X ∈ X . The morphisms (ri(X), sXi , X) are usually
abbreviated by sXi , or by si, if no confusion is possible. Since all generators are invertible W(C)
is a groupoid.
Let C = C(I,X , r, (AX)X∈X ) be a semi-Cartan graph and (∆X)X∈X a family of sets ∆X ⊂ ZI .
We say that R = R(C, (∆X)X∈X ) is a root system of type C if and only if
• ∆X =
(
∆X ∩ NI0
)
∪ −
(
∆X ∩ NI0
)
,
• ∆X ∩ Zαi = {αi,−αi} for all i ∈ I,
• sXi
(
∆X
)
= ∆ri(X) for all i ∈ I,
• (rirj)
mXij (X) = X for any i, j ∈ I such that i 6= j where mX
ij :=
∣∣∆X ∩ (N0αi + N0αj)
∣∣ is
finite.
For any category D and any object X in D, let Hom(D, X) = ∪Y ∈D Hom(Y,X). For all X ∈ X ,
the set
∆Xre =
{
wαi ∈ ZI
∣∣w ∈ Hom(W(C), X)
}
is called the set of real rootsof C at X. The elements of ∆Xre
+ = ∆Xre ∩ NI0 are called positive
roots and those of ∆Xre ∩ −NI0 negative roots. A semi-Cartan graph is called finite if ∆Xre is
a finite set for all X ∈ X .
Let tXij =
∣∣∆Xre ∩ (N0αi + N0αj)
∣∣. We say that C is a Cartan graph if the following hold:
• For all X ∈ X the set ∆Xre consists of positive and negative roots.
• Let X ∈ X and i, j ∈ I. If tXij <∞ then (rirj)
tXij (X) = X.
In that case, W(C) is called the Weyl groupoid of C.
Remark 2.1.
• A semi-Cartan graph C is a Cartan graph if and only if R = R
(
C, (∆Xre)X∈X
)
is a root
system of type C.
• For any finite Cartan graph C, there is a unique root system R = R
(
C, (∆Xre)X∈X
)
of
type C, see [10, Propositions 2.9, 2.12].
4 J. Wang and I. Heckenberger
3 Cartan graphs for Nichols algebras
Let H be a Hopf algebra over k with bijective antipode. Let H
HYD denote the category of
Yetter–Drinfel’d modules over H and FHθ the set of θ-tuples of finite-dimensional irreducible
objects in H
HYD for all θ ∈ N. For all V ∈ H
HYD, let δ denote the left coaction of H on V
and · the left action of H on V . Let θ ∈ N, I = {1, . . . , θ}, and M = (M1, . . . ,Mθ) ∈ FHθ .
Write [M ] = ([M1], . . . , [Mθ]) ∈ XHθ , where XHθ denotes the set of θ-tuples of isomorphism
classes of finite-dimensional irreducible objects in H
HYD. Let B(M) denote the Nichols algebra
B(M1⊕ · · ·⊕Mθ). The Nichols algebra B(V ) is known to be a Nθ0-graded algebra and coalgebra
in H
HYD such that degMi = αi for all i ∈ I.
The adjoint action in the braided category H
HYD is given by adc x(y) = xy − (x(−1) · y)x(0)
for all x ∈M1 ⊕ · · · ⊕Mθ, y ∈ B(M), where δ(x) = x(−1) ⊗ x(0).
By [17, Definition 6.8], the Nichols algebra B(M) is called decomposable if there exists a totally
ordered index set (L,≤) and a family (Wl)l∈L of finite-dimensional irreducible Nθ0-graded objects
in H
HYD such that
B(M) '
⊗
l∈L
B(Wl). (3.1)
Decomposability of B(M) is known under several assumptions on M . In particular, if H is
a group algebra of an abelian group and dimMi = 1 for all 1 ≤ i ≤ θ, then B(M) is decomposable
by a theorem of V. Kharchenko [20, Theorem 2].
Assume that B(M) is decomposable. One defines for any decomposition (3.1) the set of
positive roots ∆
[M ]
+ ⊂ ZI and the set of roots ∆[M ] ⊂ ZI of [M ] by
∆
[M ]
+ = {deg(Wl) | l ∈ L}, ∆[M ] = ∆
[M ]
+ ∪ −∆
[M ]
+ .
The set of roots of [M ] does not depend on the choice of the decomposition (3.1).
Let i ∈ I. Following [16, Definition 6.4] we say that M is i-finite, if for any j ∈ I \ {i},
(adcMi)
m(Mj) = 0 for some m ∈ N. Assume that M is i-finite. Let (aMij )j∈I ∈ ZI and
Ri(M) = (Ri(M)j)j∈I , where
aMij =
{
2 if j = i,
−max
{
m ∈ N0 | (adcMi)
m(Mj) 6= 0
}
if j 6= i,
Ri(M)i = Mi
∗, Ri(M)j = (adcMi)
−aMij (Mj). (3.2)
Then Ri(M)j is irreducible by [16, Theorem 7.2(3)]. If M is not i-finite, then let Ri(M) = M .
Let
XHθ (M) =
{
[Ri1 · · ·Rin(M)] ∈ XHθ |n ∈ N0, i1, . . . , in ∈ I
}
,
FHθ (M) =
{
Ri1 · · ·Rin(M) ∈ FHθ |n ∈ N0, i1, . . . , in ∈ I
}
.
We say that M ∈ FHθ admits all reflections if N is i-finite for all N ∈ FHθ (M).
Theorem 3.1. Let M ∈ FHθ . Assume that M admits all reflections. Let r : I × XHθ (M) →
XHθ (M), (i, [N ]) 7→ [Ri(N)] for all i ∈ I. Then
C(M) =
{
I,XHθ (M), r,
(
A[N ]
)
[N ]∈XHθ (M)
}
is a semi-Cartan graph. If moreover ∆[M ]re is finite, then C(M) is a Cartan graph.
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 5
Proof. The first claim follows from [3, Theorem 3.12], see [16, Theorem 6.10] for details.
If ∆[M ]re is finite and M admits all reflections, then
R(M) = R
(
C(M),
(
∆Xre
)
X∈XHθ (M)
)
is a root system of type C(M) by [17, Corollary 6.16]. Hence C(M) is a Cartan graph because
of [10, Proposition 2.9]. �
Therefore if M admits all reflections then we can attach the groupoid W(M) := W(C(M))
to M .
3.1 Small Cartan graphs for Nichols algebras of diagonal type
Let G be an abelian group and let V be a Yetter–Drinfel’d module over kG of rank θ with a basis
{xi|i ∈ I}. Let . : kG⊗ V → V and δ : V → kG⊗ V denote the left action and the left coaction
of kG on V , respectively. Assume that V is of diagonal type. More precisely, let {gi | i ∈ I} be
a subset of G and qij ∈ k∗ for all i, j ∈ I, such that
δ(xi) = gi ⊗ xi, gi.xj = qijxj
for i, j ∈ I. Then V is a braided vector space of dimension θ [1, Definition 5.4] and the braiding
c ∈ End(V ⊗V ) is of diagonal type, that is c(xi⊗xj) = qijxj ⊗xi for all i, j ∈ I. Then (qij)i,j∈I
is the braiding matrix of V with respect to the basis {xi|i ∈ I}.
The braiding matrix is known to be independent of the basis {xi|i ∈ I}, up to permutation
of I. It can be obtained for example from [15, Proposition 1.3] using the arguments in its proof.
The Nichols algebra B(V ) generated by V is said to be of diagonal type [1, Definition 5.8].
For ρ ∈ B(V ), the braided commutator adc takes the form adc xi(ρ) = xiρ− (gi.ρ)xi.
Lemma 3.2. Let M = (Mi)i∈I ∈ FkG
θ be a tuple of one-dimensional Yetter–Drinfel’d modules
over kG and let xi be a basis of Mi, for all i ∈ I. Let m ∈ N0. For any i, j ∈ I with i 6= j, the
following are equivalent:
(a) (m+ 1)qii(q
m
ii qijqji − 1) = 0 and (k + 1)qii(q
k
iiqijqji − 1) 6= 0 for all 0 ≤ k < m,
(b) (adc xi)
m+1(xj) = 0 and (adc xi)
m(xj) 6= 0 in B(V ),
(c) −aMij = m.
Here (n)q := 1 + q + · · ·+ qn−1, which is 0 if and only if qn = 0 for q 6= 1 or p|n for q = 1.
Proof. (a)⇔(b) follows from [6, Lemma 3.7] and (b)⇔(c) holds by the definition of aMij . �
Lemma 3.3. Let i ∈ I. Then M = (Mj)j∈I ∈ FkG
θ is i-finite if and only if for any j ∈ I \ {i}
there is a non-negative integer m satisfying (m+ 1)qii(q
m
ii qijqji − 1) = 0.
Proof. The claim follows from Lemma 3.2. �
Let V be a θ-dimensional braided vector space of diagonal type. Let (qij)i,j∈I be a braiding
matrix of V . The Dynkin diagram [13, Definition 4] of V is denoted by D. It is a non-directed
graph with the following properties:
• there is a bijective map φ from I to the vertices of D,
• for all i ∈ I the vertex φ(i) is labeled by qii,
• for any i, j ∈ I with i 6= j, the number nij of edges between φ(i) and φ(j) is either 0 or 1.
If qijqji = 1 then nij = 0, otherwise nij = 1 and the edge is labeled by qijqji.
6 J. Wang and I. Heckenberger
Let M = (Mi)i∈I ∈ FkG
θ be a tuple of one-dimensional Yetter–Drinfel’d modules over kG.
The Dynkin diagram of M is the Dynkin diagram of the braided vector space M1 ⊕ · · · ⊕Mθ.
Let i ∈ I. Assume that M is i-finite. By definitions of Ri(M) and M , the tuple (yj)j∈I is
a basis of Ri(M), where
yj :=
{
yi if j = i,
(adc xi)
−aMij (xj) if j 6= i,
where yi ∈M∗i \ {0}.
From the method in [13, Example 1], one can obtain the labels of the Dynkin diagram of
Ri(M) = (Ri(M)j)j∈I . In more detail, we have the following lemma.
Lemma 3.4. Let i ∈ I. Assume that M is i-finite and let aij := aMij for all j ∈ I. Let (q′jk)j,k∈I
be the braiding matrix of Ri(M) with respect to (yj)j∈I . Then the labels of the Dynkin diagram
of Ri(M) = (Ri(M)j)j∈I are
q′jj =
qii if j = i,
qjj if j 6= i, qijqji = q
aij
ii ,
qiiqjj(qijqji)
−aij if j 6= i, qii ∈ G′1−aij ,
qjj(qijqji)
−aij if j 6= i, qii = 1,
q′ijq
′
ji =
qijqji if j 6= i, qijqji = q
aij
ii ,
q2ii(qijqji)
−1 if j 6= i, qii ∈ G′1−aij ,
(qijqji)
−1 if j 6= i, qii = 1,
and if j, k 6= i, j 6= k, then
q′jkq
′
kj =
qjkqkj if qirqri = qairii , r ∈ {j, k},
qjkqkj(qikqkiq
−1
ii )−aij if qijqji = q
aij
ii , qii ∈ G′1−aik ,
qjkqkj(qijqji)
−aik(qikqki)
−aij if qii = 1,
qjkqkjq
2
ii(qijqjiqikqki)
−aij if qii ∈ G′1−aik , qii ∈ G′1−aij .
Assume that M admits all reflections. By Theorem 3.1, we are able to construct a semi-
Cartan graph C(M) of M
C(M) =
(
I,X kG
θ (M), (ri)i∈I ,
(
AX
)
X∈X kG
θ (M)
)
,
where X kG
θ (M) = {[Ri1 · · ·Rin(M)] ∈ FkG
θ |n ∈ N0, i1, . . . , in ∈ I}. Note that any X ∈ X kG
θ (M)
has a well-defined braiding matrix given by the braiding matrix of any representative of X.
Definition 3.5. Assume that M admits all reflections. For all X ∈ X kG
θ (M) let
[X]sθ =
{
Y ∈ X kG
θ (M)
∣∣Y and X have the same Dynkin diagram
}
.
Let Ysθ (M) = {[X]sθ|X ∈ X kG
θ (M)} and A[X]sθ = AX for all X ∈ X kG
θ (M). Let t : I × Ysθ (M)→
Ysθ (M), (i, [X]sθ) 7→ [ri(X)]sθ. Then the tuple
Cs(M) =
{
I,Ysθ (M), t,
(
AY
)
Y ∈Ysθ (M)
}
is called the small semi-Cartan graph of M .
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 7
Proposition 3.6. Assume that M admits all reflections. Then the tuple
Cs(M) =
{
I,Ysθ (M), t,
(
AY
)
Y ∈Ysθ (M)
}
is a semi-Cartan graph. Moreover, if C(M) is a finite Cartan graph, then Cs(M) is a finite
Cartan graph.
Proof. The map t and A[X]sθ are well-defined for all [X]sθ ∈ Ysθ (M). Indeed, if X,X ′ ∈ [X]sθ,
then AX = AX
′
by Lemma 3.2. Thus A[X]sθ is well-defined. Further, Lemma 3.4 implies that
ri(X) and ri(X
′) have the same Dynkin diagram. Hence t is well-defined. Since ti([X]sθ) =
t(i, [X]sθ) = [ri(X)]sθ and r2i = id, then for all i ∈ I, t2i = idYsθ (M). Moreover, a
[X]sθ
ij = a
ti([X]sθ)
ij .
Hence the first claim holds.
The second claim follows from the definitions of ∆Xre and Cs(M). �
4 Finite Cartan graphs of rank two
In this section we simplify slightly the fundaments of the theory presented in [9] and give
a characterization of finite Cartan graphs of rank two.
Definition 4.1. Let A+ denote the smallest subset of ∪n≥2Nn0 such that
• (0, 0) ∈ A+,
• if (c1, . . . , cn) ∈ A+ and 1 < i ≤ n, then (c1, . . . , ci−2, ci−1 + 1, 1, ci + 1, . . . , cn) ∈ A+.
Remark 4.2. Note that our definition of A+ is different from the one in [11].
From the definition of A+, we get the following lemma.
Lemma 4.3. Let n ≥ 2 and (c1, . . . , cn) ∈ A+. Then
n∑
i=1
ci = 3n− 6.
The definition of A+ implies the following.
Proposition 4.4. Let n ≥ 2. Enumerate the vertices of a convex n-gon by 1, . . . , n such that
consecutive integers correspond to neighboring vertices. Let Tn be the set of triangulations of
a convex n-gon with non-intersecting diagonals. Let T = ∪n≥2Tn. For any triangulation t ∈ Tn
and any i ∈ {1, . . . , n}, let ci be the number of triangles meeting at the i-th vertex. Then the
map ψ : T → A+, t 7→ (c1, . . . , cn) is a bijection.
Proof. We proceed by induction on n. For n = 2, a triangulation of a convex 2-gon is itself.
Then (c1, c2) = (0, 0). Hence the claim is true for n = 2. For n ≥ 3, the definition of A+
corresponds bijectively to the construction of a triangulation of a convex (n+ 1)-gon by adding
a new triangle between two consecutive vertices of a convex n-gon, but not at the edge between
the first and the last vertex. By adding one triangle between two consecutive vertices of a con-
vex n-gon, one increases the number of triangles at the two adjacent vertices and the number of
triangles at the new vertex becomes 1. �
Corollary 4.5. Let n ≥ 2 and let (c1, . . . , cn) ∈ A+.
(1) (cn, cn−1, . . . , c1) ∈ A+ and (c2, c3, . . . , cn, c1) ∈ A+.
(2) If n ≥ 3, then there is 1 < i < n satisfying ci = 1. For any such i, (c1, . . . , ci−2, ci−1 − 1,
ci+1 − 1, ci+2, . . . , cn) ∈ A+.
(3) If n ≥ 3, then ci ≥ 1 for all 1 ≤ i ≤ n.
(4) If ci = 1, ci+1 = 1 for some 1 ≤ i ≤ n− 1, then n = 3 and c = (1, 1, 1).
8 J. Wang and I. Heckenberger
Table 4.1. Sequences in A+ containing exactly one subsequence from Theorem 4.6.
subsequences sequences in A+
(1, 1) (1, 1, 1)
(1, 2, 1) (1, 2, 1, 2)
(1, 2, 2) (1, 2, 2, 2, 2, 2, 1, 6)
(1, 2, 3) (1, 2, 3, 1, 6, 1, 2, 3, 1, 6, 1, 2, 3, 1, 6)
(2, 1, 3) (2, 1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4)
(2, 1, 4) (2, 1, 4, 2, 1, 4, 2, 1, 4)
(2, 1, 5) (2, 1, 5, 1, 2, 4, 2, 1, 5, 1, 2, 4)
(1, 3, 1, 3) (1, 3, 1, 3, 1, 3)
(1, 3, 1, 4) (1, 3, 1, 4, 1, 3, 1, 4)
(1, 3, 1, 5) (1, 3, 1, 5, 1, 3, 1, 5, 1, 3, 1, 5)
Proof. (1) and (2) follow directly from the bijection betweenA+ and triangulations of convex n-
gons in Proposition 4.4. (3) follows from the definition of A+. (4) follows from (2) and (3). �
We say that two consecutive entries of a sequence in A+ are neighbors.
Theorem 4.6. Let n ≥ 3. Then any sequence (c1, . . . , cn) ∈ A+ contains a subsequence
(ck)i≤k≤j, where 1 ≤ i ≤ j ≤ n, of the form
(1, 1), (1, 2, a), (2, 1, b), (1, 3, 1, b)
or their transpose, where 1 ≤ a ≤ 3 and 3 ≤ b ≤ 5.
Remark 4.7. We record that it is natural to exclude the cases b = 1 and b = 2 since (1, 3, 1, 1)
contains the subsequence (1, 1) and (1, 3, 1, 2) contains the transpose of (2, 1, 3).
Remark 4.8. The claim becomes false by omitting one of the sequences from the theorem. In
Table 4.1 we list sequences in A+ which contain precisely one of the sequences in Theorem 4.6.
Proof. Let c = (c1, . . . , cn) ∈ A+ such that the claim does not hold for c. Then n ≥ 5 and c
has no subsequence (2, 1, 2). Otherwise c = (1, 2, 1, 2) or c = (2, 1, 2, 1) by Corollary 4.5(2),(4).
We define E = {νij | i, j ∈ {1, 2}}, where the sequences νij are given by
ν11 = (1), ν12 = (2, 1), ν21 = (1, 2), ν22 = (1, 3, 1).
Now we decompose c by the following steps.
Replace all subsequences (2, 1) by ν12, then all subsequences (1, 2) by ν21, then all subse-
quences (1, 3, 1) by ν22, and finally all entries 1 by ν11. By this construction, (ν11, 3, ν11) is not
a subsequence of d. Hence we get a decomposition d = (d1, . . . , dk), where k ≥ 2, of c into
subsequences of the form (a) and ν, where a ≥ 2 and ν ∈ E.
Since (1, 1), (1, 2, a), (2, 1, b), (1, 3, 1, b) and their transposes are not subsequences of c, where
1 ≤ a ≤ 3, 2 ≤ b ≤ 5, we obtain the following conditions on the entries of d:
– no entry νij of d, where i, j ∈ {1, 2}, has 2 or νkl with k, l ∈ {1, 2} as a neighbor.
– if (ν21, a) or (a, ν12) is a subsequence of d, then a ≥ 4.
– if (νi2, b) or (b, ν2i) is a subsequence of d, where i ∈ {1, 2}, then b ≥ 6.
By applying Corollary 4.5(2) we get further reductions of d:
(. . . , dm−1, νij , dm+1, . . . )→ (. . . , dm−1− i, dm+1− j), (νi2, d2, . . . )→ (νi1, d2 − 1, . . . ),
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 9
where i, j ∈ {1, 2}. Thus we can perform such reductions at all places in d, where an entry νij
with i, j ∈ {1, 2} appears. After decreasing them, we get dm ≥ 2, where 1 < m < k. Indeed, we
get the following conditions.
– If d = (. . . , dm−1, dm, dm+1, . . . ), where dm ≥ 6, then dm can be reduced at most by 4.
Hence the value of dm after reduction is at least 2.
– If 4 ≤ dm ≤ 5, then neither (νi2, dm) nor (dm, ν2i) is a subsequence of d, where i ∈ {1, 2}.
Hence d can be reduced by at most 2.
– If dm = 3, then dm−1, dm+1 /∈ {ν12, ν21, ν22}. Further, (dm−1, dm+1) 6= (ν11, ν11). Hence
dm decreases by at most 1.
– If dm = 2, then it has no neighbour νij with i, j ∈ {1, 2}. Hence dm does not change.
Thus one can reduce c to a sequence (c′1, . . . , c
′
l) with l ≥ 1, where c′m ≥ 2 for all 1 < m < l and
c′1, c
′
l ≥ 1. This is a contradiction to Corollary 4.5(2). �
Recall that (α1, α2) is the standard basis of Z2. We define a map
η : Z→ SL(2,Z), a 7→
(
a −1
1 0
)
. (4.1)
Lemma 4.9. Let n ∈ N and (ck)1≤k≤n ∈ Zn. For all 1 ≤ k ≤ n + 1, let β0 = −α2 and
βk = η(c1) . . . η(ck−1)(α1). Then the following hold:
(1) βk+1 = ckβk − βk−1 for all 1 ≤ k ≤ n,
(2) if c1 ≥ 1 and ck ≥ 2 for all 1 < k < n, then βk ∈ N2
0 for all 1 ≤ k ≤ n and βk − βk−1 ∈
N2
0 \ {0} for 1 < k ≤ n.
Proof. (1) By definition, β1 = α1 and β2 = η(c1)(α1) = c1α1 + α2. Thus the claim holds for
k = 1. Since η(ck−1)(α2) = −α1, then
βk+1 = η(c1) . . . η(ck)(α1) = η(c1) . . . η(ck−1)(ckα1 + α2) = ckβk − βk−1
for all k ≥ 2.
(2) For all 0 ≤ k ≤ n, let ak, bk ∈ Z such that βk = akα1 + bkα2. By induction on k, we get
the following.
• If ck ≥ 2 for 1 ≤ k < n, then
ak > bk ≥ 0, ak > ak−1, bk > bk−1, ak − bk − (ak−1 − bk−1) ≥ 0
for all 1 ≤ k ≤ n.
• If c1 = 1 and ck ≥ 2 for 2 ≤ k < n, then
bk ≥ ak > 0, ak ≥ ak−1, bk > bk−1, ak − bk − (ak−1 − bk−1) < 0.
for all 2 ≤ k ≤ n.
Thus βk ∈ N2
0 for all 1 ≤ k ≤ n and βk − βk−1 ∈ N2
0 \ {0} for 1 < k ≤ n. �
The following theorem will be used in the proof of Theorem 4.15. It was proven partially in [9,
Propositon 5.3]. Notice that the definition of A+ is different from the one in [9].
Theorem 4.10. Let n ≥ 2 and (ci)1≤i≤n ∈ Zn. Then the following are equivalent:
(1) (ci)1≤i≤n ∈ A+,
(2) η(c1) · · · η(cn) = −id and βk = η(c1) · · · η(ck−1)(α1) ∈ N2
0 for all 1 ≤ k ≤ n.
10 J. Wang and I. Heckenberger
Proof. (1)⇒(2). We apply induction on n. If n = 2, then (c1, c2) = (0, 0), η(0)2 = −id, β1 = α1,
and β2 = α2. Assume that n ≥ 3. By the definition of A+, there is a (c′1, . . . , c
′
n−1) ∈ A+ and
1 < i ≤ n− 1 such that
(c1, . . . , cn) = (c′1, . . . , c
′
i−1 + 1, 1, c′i + 1, c′i+1, . . . , c
′
n−1).
By calculation,
η(a)η(b) = η(a+ 1)η(1)η(b+ 1) (4.2)
for all a, b ∈ Z. Then
η(c1) · · · η(cn) = η(c′1) · · · η(c′n−1) = −id.
Let β′i = η(c′1) · · · η(c′i−1)(α1) for all 1 ≤ i ≤ n−1. Then βk = β′k for all 1 ≤ k < i and βk = β′k−1
for all i+ 1 ≤ k ≤ n+ 1. Finally
βi = η(c1) · · · η(ci−1)(α1) = η(c′1) · · · η(c′i−2)η(c′i−1 + 1)(α1)
= η(c′1) · · · η(c′i−2)(η(c′i−1)(α1) + α1) = β′i + β′i−1 ∈ N2
0.
Then (2) follows.
(2)⇒(1). Again we proceed by induction. If n = 2, then
η(c1)η(c2) =
(
c1c2 − 1 −c1
c2 −1
)
= −id
implies that (c1, c2) = (0, 0) ∈ A+. Assume that n ≥ 3. Set β0 = −α2. One has βk+1 =
ckβk − βk−1 for all 1 ≤ k < n. By assumption, the condition βk−1, βk, βk+1 ∈ N2
0 implies
ck > 0 for 2 ≤ k < n and c1 ≥ 0. If c1 = 0 then β2 = α2 and β3 = c2α2 − α1 /∈ N2
0.
Hence ck ≥ 1 for all 1 ≤ k < n. Moreover, there is 1 < i < n satisfying ci = 1. Indeed,
βn+1 = cnβn − βn−1 = (cn − 1)βn + (βn − βn−1) by Lemma 4.9(1). Assume that ci ≥ 2 for all
1 < i < n. Then βn+1 ∈ N2
0 if cn ≥ 1 and −βn+1 ∈ N2
0 \ {0, α1} if cn ≤ 0 by Lemma 4.9(2), since
n ≥ 3. This is a contradiction to βn+1 = η(c1) · · · η(cn)(α1) = (−id)(α1) = −α1.
Hence there is (c′1, . . . , c
′
n−1) ∈ Zn−1 such that
(c1, . . . , cn) = (c′1, . . . , c
′
i−1 + 1, 1, c′i + 1, c′i+1, . . . , c
′
n−1).
Then η(c1) · · · η(cn) = η(c′1) · · · η(c′n−1) = −id by equation (4.2) and β′k = η(c′1) . . . η(c′k−1)(α1) ∈
N2
0 for all 1 ≤ k ≤ n− 1. Hence (c′1, . . . , c
′
n−1) ∈ A+ by induction hypothesis. Then
(c1, . . . , cn) ∈ A+. �
Definition 4.11. Let C = C(I,X , r, (AX)X∈X ) be a semi-Cartan graph of rank two and let
X ∈ X and i ∈ I. The characteristic sequence of C with respect to X and i is the infinite
sequence (cX,ik )k≥1 of non-negative integers, where
cX,i2k+1 = −a(rjri)
k(X)
ij = −ari(rjri)
k(X)
ij , cX,i2k+2 = −ari(rjri)
k(X)
ji = −a(rjri)
k+1(X)
ji
for all k ≥ 0 and j ∈ I\{i}.
By the definition of a characteristic sequence, we get the following remark.
Remark 4.12. Let C = C(I,X , r, (AX)X∈X ) be a semi-Cartan graph of rank two and let X ∈ X
and i, j ∈ I with i 6= j. Let (ck)k≥1 be the characteristic sequence of C with respect to X and i.
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 11
– The characteristic sequence of C with respect to ri(X) and j is (ck+1)k≥1.
– Suppose that (rjri)
n(X) = X for some n ≥ 1. Then the characteristic sequence of C with
respect to X and j is (c2n+1−k)k≥1.
Definition 4.13. Let C = C(I,X , r, (AX)X∈X ) be a semi-Cartan graph of rank two and let
X ∈ X and i ∈ I. Let (ck)k≥1 be the characteristic sequence of C with respect to X and i.
The root sequence of C with respect to X and iis the infinite sequence (βk)k≥1 of elements of Z2,
where
βk = η(c1) · · · η(ck−1)(α1)
for all k ≥ 1. In particular, β1 = α1.
Let C = C(I = {1, 2},X , r, (AX)X∈X ) be a semi-Cartan graph. For allX ∈ X , the maps sX1 , sX2
are defined by equation (2.1). Recall that (α1, α2) is a basis of Z2 and η is a map defined by
equation (4.1). Define a map
τ : Z2 → Z2, aα1 + bα2 7→ bα1 + aα2
for any a, b ∈ Z. One obtains
sX1 = η
(
−aX12
)
τ, sX2 = τη
(
−aX21
)
(4.3)
for all X ∈ X .
Lemma 4.14. Let C = C(I = {1, 2},X , r, (AX)X∈X ) be a semi-Cartan graph of rank two and
let X ∈ X . Let (βk)k≥1 be the root sequence of C with respect to X and 1 and let (γk)k≥1 be the
root sequence of C with respect to X and 2. Then
β2k+1 = idX(s1s2)
kα1, β2k+2 = idX(s1s2)
ks1α2,
τγ2k+1 = idX(s2s1)
kα2, τγ2k+2 = idX(s2s1)
ks2α1
for all k ≥ 0. Hence ∆Xre = {±βk,±τγk|k ≥ 1}.
Proof. Let (ck)k≥1 be the characteristic sequence with respect to X and i = 1. By equa-
tion (4.3) and the definition of the root sequence, one obtains that
β2k+1 = η(c1)η(c2) · · · η(c2k−1)η(c2k)(α1)
= η
(
−aX12
)
η
(
−ar1(X)
21
)
· · · η
(
−a(r2r1)
k−1(X)
12
)
η
(
−ar1(r2r1)
k−1(X)
21
)
(α1)
= η
(
−aX12
)
ττη
(
−ar1(X)
21
)
· · · η
(
−a(r2r1)
k−1(X)
12
)
ττη
(
−ar1(r2r1)
k−1(X)
21
)
(α1)
= sX1 s
r1(X)
2 · · · s(r2r1)
k−1(X)
1 s
r1(r2r1)k−1(X)
2 (α1) = idX(s1s2)
kα1,
τγ2k+1 = τη
(
−aX21
)
η
(
−ar2(X)
12
)
· · · η
(
−a(r1r2)
n−2(X)
21
)
η
(
−ar2(r1r2)
k−1(X)
12
)
(α1)
=
(
τη
(
−aX21
)
η
(
−ar2(X)
12
)
τ
)
(τ · · · τ)
(
τη
(
−a(r1r2)
k−2(X)
21
)
η
(
−ar2(r1r2)
k−1(X)
12
)
τ
)
τ(α1)
= idX(s2s1)
kα2.
The claims β2k+2 = idX(s1s2)
ks1α2, τγ2k+2 = idX(s2s1)
ks2α1 hold by a similar argument.
Thus ∆Xre = {±βk,±τγk|k ≥ 1} follows from the definition of ∆Xre. �
For a finite sequence (v1, . . . , vn) of integers or vectors, where n ≥ 1, let (v1, . . . , vn)∞ =
(uk)k≥1 be the sequence where umn+i = vi for all 1 ≤ i ≤ n, m ≥ 0.
12 J. Wang and I. Heckenberger
Theorem 4.15. Let C = C(I = {1, 2},X , r, (AX)X∈X ) be a connected semi-Cartan graph of
rank two such that |X | is finite. Let X ∈ X and let n be the smallest positive integer with
(r2r1)
n(X) = X. Let (ck)k≥1 be the characteristic sequence of C with respect to X and 1, and
let l = 6n−
2n∑
i=1
ci. The following are equivalent:
(1) C is a finite Cartan graph,
(2) l > 0, l|12, (c1, c2, . . . , c12n/l) ∈ A+, and (ck)k≥1 = (c1, c2, . . . , c12n/l)
∞.
In this case 12n/l = |∆Xre
+ | = tX12.
Proof. Let (βk)k≥1 be the root sequence of C with respect to X and 1 and (γk)k≥1 the root
sequence of C with respect to X and 2.
(1)⇒(2). Let q = tX12. Then ∆Y re ⊂ N2
0 ∪ −N2
0 for all Y ∈ X since C is a Cartan
graph. By [16, Lemmas 3, 4] and Lemma 4.14, we have βk ∈ N2
0 for all 1 ≤ k ≤ q and
βq = η(c1) · · · η(cq−1)(α1) = α2. By the same reason we obtain that η(c2) · · · η(cq)(α1) = α2.
Then we have
−βq+1 = −η(c1) · · · η(cq)(α1) = −η(c1)(α2) = α1.
Thus −η(c1) · · · η(cq) = id. Indeed, if we set w := −η(c1) . . . η(cq) and w(α2) := aα1 + bα2, then
b = 1 since det(w) = 1. If q is odd then −wτ ∈ Hom(Y,X) by equation (4.1) and equation (4.3),
where Y = r1(r2r1)
(q−1)/2(X). In the same way, one gets −w ∈ Hom((r2r1)
q/2(X), X) if q is
even. Hence w(α1), w(α2) ∈ ∆Xre. Then a ≥ 0 since w(α2) = η(c1) · · · η(cq−1)(α1) = βq ∈
∆Xre ⊂ N2
0 ∪ −N2
0. Moreover, a ≤ 0 since w−1(α2) = α2 − aα1 ∈ N2
0 ∪ −N2
0. Hence a = 0 and
w(α2) = α2. Then −η(c1) · · · η(cq) = id. Hence (c1, . . . , cq) ∈ A+ by Theorem 4.10. Therefore
q∑
i=1
ci = 3q − 6 by Lemma 4.3.
Further, we apply the first part of the proof for r1(X) and the label 2 instead of X and the
label 1, respectively. Then (c2, . . . , cq+1) ∈ A+ by Remark 4.12 and
q+1∑
i=2
ci = 3q − 6. Hence
cq+1 = c1. By induction, (ck)k≥1 = (c1, c2, . . . , cq)
∞. In particular, we obtain that
2qn∑
i=1
ci =
2n(3q − 6) = q
( 2n∑
i=1
ci
)
. Therefore
2n∑
i=1
ci = 6n − 12n/q. Hence q|12n and l = 6n −
2n∑
i=1
ci =
12n/q > 0. Further, n|q since C is a Cartan graph. Thus l|12.
(2)⇒(1). Set q = 12n/l. Then η(c1) . . . η(cq) = −id and βk ∈ N2
0 for 1 ≤ k ≤ q by
Theorem 4.10. Then (cq, cq−1, . . . , c1) ∈ A+ by Corollary 4.5(1) since (c1, c2, . . . , cq) ∈ A+.
Since l|12, q is a multiple of n. Hence (cq, cq−1, . . . , c1)
∞ is the characteristic sequence of C
with respect to X and 2. By Lemma 4.14, we get that γk ∈ N2
0 for all 1 ≤ k ≤ q. Therefore,
since (ck)k≥1 = (c1, . . . , cq)
∞, ∆Xre = {±βk,±τγk | 1 ≤ k ≤ q} ⊆ N2
0 ∪ −N2
0 for all X ∈ X by
Lemma 4.14. Hence C is finite.
By the definition of tX12 and [16, Lemma 4], we obtain that tX12 = q = |∆Xre
+ |. Hence n|tX12 by
assumption and (r2r1)
tX12(X) = X. Therefore, C is a Cartan graph. �
5 Classif ication of rank two Nichols algebras of diagonal type
over fields of positive characteristic
In this section, we classify all the two-dimensional braided vector spaces V of diagonal type over
fields of positive characteristic such that the Nichols algebra of V has a finite root system. The
proof uses the characterization of the finite Cartan graphs of rank two.
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 13
Table 5.1. Dynkin diagrams in characteristic p = 2.
Dynkin diagrams fixed parameters
1 e eq r
q, r ∈ k∗
2 e eq q−1 q
q ∈ k∗ \ {1}
3 e eq q−1 1 e e1 q 1
q ∈ k∗ \ {1}
4 e eq q−2 q2
q ∈ k∗ \ {1}
5 e eq q−2 1 e eq−1
q2 1
q ∈ k∗ \ {1}
6 e eζ q−1 q e eζ ζ−1q ζq
−1
ζ ∈ G′3, q ∈ k∗ \ {1, ζ, ζ2}
7 e eζ ζ 1 e eζ−1
ζ−1 1
ζ ∈ G′3
10 e eζ2 ζ 1 e eζ3 ζ−1 1 e eζ3 ζ−2 ζ
ζ ∈ G′9
11 e eq q−3 q3
q ∈ k∗ \ {1}, q /∈ G′3
14 e eζ ζ2 1 e eζ−2
ζ−2 1
ζ ∈ G′5
16 e eζ5 ζ−3 ζ e eζ5 ζ−2 1 e eζ3 ζ2 1 e eζ3 ζ4 ζ−4
ζ ∈ G′15
17 e eζ ζ−3 1 e eζ−2
ζ3 1
ζ ∈ G′7
Let V be a braided vector space of diagonal type with a basis {x1, x2} and the braiding
c(xi ⊗ xj) = qijxj ⊗ xi, where qij ∈ k, i, j ∈ {1, 2}. We choose an abelian group G and the set
{gi | gi ∈ G, i ∈ {1, 2}} such that the assignments δ(xi) = gixi, gi.xj = qijxj for i, j ∈ {1, 2}
define a Yetter–Drinfel’d module structure on V over kG. Let B(V ) denote the Nichols algebra
of V .
The following theorem determines whether Weyl groupoidW(kx1, kx2) of (kx1, kx2) is finite
in terms of the Dynkin diagram of V .
Theorem 5.1. Let V be a two-dimensional braided vector space of diagonal type with the brai-
ding
c(xi ⊗ xj) = qijxj ⊗ xi,
where i, j ∈ {1, 2} and {x1, x2} is a basis of V . Let M := (kx1,kx2). Assume that the charac-
teristic p of k is positive. Then the following are equivalent:
(1) B(V ) is decomposable and ∆[M ] is finite,
(2) the Dynkin diagram D of V appears in Tables 5.1, 5.2, 5.3, 5.4 and 5.5, if p = 2, p = 3,
p = 5, p = 7 and p > 7, respectively,
(3) M admits all refections and W(M) is finite.
In this case, the row of Table 5.6 containing D consists precisely of the Dynkin diagrams of
the points of Cs(M). Further, the corresponding row of Table 5.7 contains the exchange graph
of Cs(M).
14 J. Wang and I. Heckenberger
Table 5.2. Dynkin diagrams in characteristic p = 3.
Dynkin diagrams fixed parameters
1 e eq r
q, r ∈ k∗
2 e eq q−1 q
q ∈ k∗ \ {1}
3 e eq q−1 −1 e e−1 q −1
q ∈ k∗ \ {−1, 1}
4 e eq q−2 q2
q ∈ k∗ \ {−1, 1}
5 e eq q−2 −1 e e−q−1
q2 −1
q ∈ k∗ \ {−1, 1}, q /∈ G′4
6′ e e1 q q−1 e e1 q−1 q
q ∈ k∗ \ {1,−1}
6′′′ e e1 −1 −1
9′ e eζ ζ −1 e e1 −ζ −1 e e1 ζ 1
ζ ∈ G′4
11 e eq q−3 q3
q ∈ k∗ \ {−1, 1}
12 e eζ −ζ −1 e eζ2 −ζ−1−1 e eζ2 ζ ζ−1
ζ ∈ G′8
13′ e e−ζ ζ−1 −1 e e1 ζ −1 e e1 ζ−1 −ζ2 e e−ζ−1
−ζ −ζ
2
ζ ∈ G′8
14 e eζ ζ2 −1 e e−ζ−2
ζ−2 −1
ζ ∈ G′5
15 e eζ ζ−3 −1 e e−ζ−2
ζ3 −1 e e−ζ−2
−ζ3 −1 e e−ζ−ζ−3−1
ζ ∈ G′20
16′ e e1 −ζ−1−ζ2 e e1 −ζ −1 e eζ −ζ−1−1 e eζ −ζ3−ζ
−3
ζ ∈ G′5
17 e e−ζ−ζ−3−1 e e−ζ−2
−ζ3 −1 ζ ∈ G′7
Remark 5.2. In order to illustrate the exchange graphs of the semi-Cartan graph Cs(M) in
Theorem 5.1, we use the following notation in Tables 5.1–5.7.
– In row n of Tables 5.1–5.6 let Dnl be the l-th Dynkin diagram for all l ≥ 1. Since the
exchange graph of the semi-Cartan graph is labeled, we write τDnl for the graph Dnl where
the two vertices of Dnl change the positions.
– We also use the notation (2, 1, 6, 1, 2, 3)2 = (2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3) and (3, 1, 5, 1)3 =
(3, 1, 5, 1, 3, 1, 5, 1, 3, 1, 5, 1) in Table 5.7.
Proof. (1)⇒(3). Since B(V ) is decomposable and ∆[M ] is finite, we obtain that M admits
all reflections and R(M) = (C(M), (∆[X])[X]∈X2(M)) is a root system of type C(M) by [16,
Corollary 6.12]. Then W(M) is finite by [16, Lemma 5.1] since ∆[M ] is finite.
(3)⇒(1). Since M admits all refections and W(M) is finite, the set ∆[M ] is finite by [16,
Lemma 5.1]. Moreover B(M) is decomposable by [17, Corollary 6.16].
(2)⇒(3). By Lemma 3.3 one obtains that M is i-finite for all i ∈ I. For i ∈ I, one can
determine the Dynkin diagram of Ri(M) by Lemma 3.4. One observes that it appears in the
same row of Table 5.6 as D. Doing the same for all the Dynkin diagrams in the same row of D
implies that M admits all reflections. Hence Cs(M) is well-defined by Proposition 3.6. Now, we
identify the objects of Cs(M) with their Dynkin diagrams.
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 15
Table 5.3. Dynkin diagrams in characteristic p = 5.
Dynkin diagrams fixed parameters
1 e eq r
q, r ∈ k∗
2 e eq q−1 q
q ∈ k∗ \ {1}
3 e eq q−1 −1 e e−1 q −1
q ∈ k∗ \ {−1, 1}
4 e eq q−2 q2
q ∈ k∗ \ {−1, 1}
5 e eq q−2 −1 e e−q−1
q2 −1
q ∈ k∗ \ {−1, 1}, q /∈ G′4
6 e eζ q−1 q e eζ ζ−1q ζq
−1 ζ ∈ G′3, qζ 6= −1
q ∈ k∗ \ {1, ζ, ζ2}
6′′ e eζ −ζ−ζ
−1
ζ ∈ G′3
7 e eζ −ζ −1 e eζ−1
−ζ−1−1
ζ ∈ G′3
8 e e−ζ2 ζ −1 e e−ζ2 ζ3 −ζ
−2 e e−1−ζ−1−ζ−2 e e−1 −ζ ζ3 e e−1 ζ−1 ζ3
ζ ∈ G′12
9 e e−ζ−1
−ζ3 −1 e e−ζ2 ζ3 −1 e e−ζ2 ζ −ζ
2
ζ ∈ G′12
10 e e−ζ2 ζ −1 e eζ3 ζ−1 −1 e eζ3 ζ−2 −ζ
ζ ∈ G′9
11 e eq q−3 q3
q ∈ k∗ \ {−1, 1}, q /∈ G′3
12 e eζ −ζ −1 e eζ2 −ζ−1−1 e eζ2 ζ ζ−1
ζ ∈ G′8
13 e eζ ζ−5 −1 e e−ζ−4
ζ5 −1 e e−ζ−4
−ζ−1 ζ6 e eζ−1
ζ ζ6
ζ ∈ G′24
15′ e eζ ζ −1 e e1 −ζ −1 e e1 ζ −1 e e−ζ −ζ −1 ζ ∈ G′4
16′′ e eζ2 −1 −ζ e eζ2 −ζ −1 e e1 −ζ−1−1 e e1 −ζ−ζ
−1
ζ ∈ G′3
17 e e−ζ−ζ−3−1 e e−ζ−2
−ζ3−1 ζ ∈ G′7
Assume that D appears in row r of one of Tables 5.1–5.6. Then by the above calculations,
the exchange graph of Cs(M) appears in row r of Table 5.7. Then we calculate the smallest
integer n with (r2r1)
n(D) = D and we observe that it appears in the third column of row r of
Table 5.7.
Then we compute the characteristic sequence (ck)k≥1 with respect to the first Dynkin diagram
in row r and the label 1. We observe that (ck)k≥1 is the infinite power of the sequence in the fifth
column of row r of Table 5.7. Further, we get the numbers l = 6n−
2n∑
i=1
ci. They appear in the
fourth column of Table 5.7. One checks that l|12 and (c1, c2, . . . , c12n/l) ∈ A+ by Corollary 4.5(2).
The detailed calculations are skipped at this point here. Then Theorem 4.15 implies that Cs(M)
is a finite Cartan graph. Hence C(M) is a finite Cartan graph and W(M) is a finite Weyl
groupoid by [16, Lemma 5.1].
(3)⇒(2). Since M admits all reflections, the tuple
C(M) =
{
I,X2(M), (ri)i∈I ,
(
AX
)
X∈X2(M)
}
16 J. Wang and I. Heckenberger
Table 5.4. Dynkin diagrams in characteristic p = 7.
Dynkin diagrams fixed parameters
1 e eq r
q, r ∈ k∗
2 e eq q−1 q
q ∈ k∗ \ {1}
3 e eq q−1 −1 e e−1 q −1
q ∈ k∗ \ {−1, 1}
4 e eq q−2 q2
q ∈ k∗ \ {−1, 1}
5 e eq q−2 −1 e e−q−1
q2 −1
q ∈ k∗ \ {−1, 1}, q /∈ G′4
6 e eζ q−1 q e eζ ζ−1q ζq
−1 ζ ∈ G′3, qζ 6= −1
q ∈ k∗ \ {1, ζ, ζ2}
6′′ e eζ −ζ−ζ
−1
ζ ∈ G′3
7 e eζ −ζ −1 e eζ−1
−ζ−1−1
ζ ∈ G′3
8 e e−ζ2 ζ −1 e e−ζ2 ζ3 −ζ
−2 e e−1−ζ−1−ζ−2 e e−1 −ζ ζ3 e e−1 ζ−1 ζ3
ζ ∈ G′12
9 e e−ζ−1
−ζ3 −1 e e−ζ2 ζ3 −1 e e−ζ2 ζ −ζ
2
ζ ∈ G′12
10 e e−ζ2 ζ −1 e eζ3 ζ−1 −1 e eζ3 ζ−2 −ζ
ζ ∈ G′9
11 e eq q−3 q3
q ∈ k∗ \ {−1, 1}, q /∈ G′3
12 e eζ −ζ −1 e eζ2 −ζ−1−1 e eζ2 ζ ζ−1
ζ ∈ G′8
13 e eζ ζ−5 −1 e e−ζ−4
ζ5 −1 e e−ζ−4
−ζ−1 ζ6 e eζ−1
ζ ζ6
ζ ∈ G′24
14 e eζ ζ2 −1 e e−ζ−2
ζ−2 −1
ζ ∈ G′5
15 e eζ ζ−3 −1 e e−ζ−2
ζ3 −1 e e−ζ−2
−ζ3 −1 e e−ζ−ζ−3−1
ζ ∈ G′20
16 e eζ5 −ζ−3−ζ e eζ5 −ζ−2−1 e eζ3 −ζ2 −1 e eζ3 −ζ4−ζ
−4
ζ ∈ G′15
18 e eζ−1
−1 −ζ e eζ−1
−ζ −1 e e1 −ζ−1−1
ζ ∈ G′3
e e1 −ζ −1 e eζ −ζ−1−1 e eζ −1−ζ
−1
defined in Theorem 3.1 is a semi-Cartan graph. In particular,
X2(M) = {[Ri1 · · ·Rin(M)] ∈ X2 |n ∈ N0, i1, . . . , in ∈ I}
and Ri(M) = (Ri(M)j)j∈I is defined by equation (3.2). Moreover, AX = (aXij )i,j∈I for all
X ∈ X2(M), where
−aXij = min
{
m ∈ N0 | (m+ 1)q′ii
(
q′ii
m
q′ijq
′
ji − 1
)
= 0
}
by Lemma 3.2 and (q′ij)i,j∈I is the braiding matrix of X. Then ∆[M ] is finite since W(M) is
finite. Hence all roots are real by [10, Proposition 2.12]. Then C(M) is a finite Cartan graph
by Theorem 3.1. Hence Cs(M) is a finite Cartan graph by Proposition 3.6. We can apply
Theorem 4.15 to Cs(M).
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 17
Table 5.5. Dynkin diagrams in characteristic p > 7.
Dynkin diagrams fixed parameters
1 e eq r
q, r ∈ k∗
2 e eq q−1 q
q ∈ k∗ \ {1}
3 e eq q−1 −1 e e−1 q −1
q ∈ k∗ \ {−1, 1}
4 e eq q−2 q2
q ∈ k∗ \ {−1, 1}
5 e eq q−2 −1 e e−q−1
q2 −1 q ∈ k∗ \ {−1, 1}
q /∈ G′4
6 e eζ q−1 q e eζ ζ−1q ζq
−1 ζ ∈ G′3, qζ 6= −1
q ∈ k∗ \ {1, ζ, ζ2}
6′′ e eζ −ζ −ζ
−1
ζ ∈ G′3
7 e eζ −ζ −1 e eζ−1
−ζ−1−1
ζ ∈ G′3
8 e e−ζ2 ζ −1 e e−ζ2 ζ3 −ζ
−2 e e−1−ζ−1−ζ−2 e e−1 −ζ ζ3 e e−1 ζ−1 ζ3
ζ ∈ G′12
9 e e−ζ−1
−ζ3 −1 e e−ζ2 ζ3 −1 e e−ζ2 ζ −ζ
2
ζ ∈ G′12
10 e e−ζ2 ζ −1 e eζ3 ζ−1 −1 e eζ3 ζ−2 −ζ
ζ ∈ G′9
11 e eq q−3 q3 q ∈ k∗ \ {−1, 1}
q /∈ G′3
12 e eζ −ζ −1 e eζ2 −ζ−1−1 e eζ2 ζ ζ−1
ζ ∈ G′8
13 e eζ ζ−5 −1 e e−ζ−4
ζ5 −1 e e−ζ−4
−ζ−1 ζ6 e eζ−1
ζ ζ6
ζ ∈ G′24
14 e eζ ζ2 −1 e e−ζ−2
ζ−2 −1
ζ ∈ G′5
15 e eζ ζ−3 −1 e e−ζ−2
ζ3 −1 e e−ζ−2
−ζ3 −1 e e−ζ−ζ−3−1
ζ ∈ G′20
16 e eζ5 −ζ−3−ζ e eζ5 −ζ−2−1 e eζ3 −ζ2 −1 e eζ3 −ζ4−ζ
−4
ζ ∈ G′15
17 e e−ζ−ζ−3−1 e e−ζ−2
−ζ3 −1 ζ ∈ G′7
By the implication (2)⇒(3), it is enough to prove that the Dynkin diagram of at least one
point in Cs(M) is contained in Table 5.6.
Set X = [M ]s and m = tX12. Let (ck)k≥1 be the characteristic sequence of Cs(M) with respect
to X and the label 1. Then we have (c1, c2, . . . , cm) ∈ A+ and (ck)k≥1 = (c1, c2, . . . , cm)∞ by
Theorem 4.15.
If m = 2 then (c1, c2) = (0, 0). Hence aX12 = aX21 = 0. Then q12q21 = 1 and D = D11.
If m > 2, by Theorem 4.6, one of (1, 1), (1, 2, a), (2, 1, b), (1, 3, 1, b) or their transpose, where
1 ≤ a ≤ 3 and 3 ≤ b ≤ 5, is a subsequence of (c1, c2, . . . , cm). Let n be the smallest integer
with (r2r1)
n(X) = X. Then n|m by Theorem 4.15. Since cm+k = ck for all k ∈ N, we have
the freedom to assume any position in (ck)k≥1, where any of these subsequences is starting. Let
c0 = cm and q0 = q12q21.
We may assume that i = 1, j = 2, but change the labels if necessary. Next we proceed case
by case:
18 J. Wang and I. Heckenberger
Table 5.6. The Dynkin diagrams in Theorem 5.1.
Dynkin diagrams fixed parameters char k
1 e eq r
q, r ∈ k∗
2 e eq q−1 q
q ∈ k∗ \ {1}
3 e eq q−1 −1 e e−1 q −1
q ∈ k∗ \ {−1, 1}
4 e eq q−2 q2
q ∈ k∗ \ {−1, 1}
5 e eq q−2 −1 e e−q−1
q2 −1 q ∈ k∗ \ {−1, 1}
q /∈ G′4
6 e eζ q−1 q e eζ ζ−1q ζq−1 ζ ∈ G′3, qζ 6= −1
q ∈ k∗ \ {1, ζ, ζ2} p 6= 3
6′ e e1 q q−1 e e1 q−1 q
q ∈ k∗ \ {1,−1} p = 3
6′′ e eζ −ζ−ζ
−1
ζ ∈ G′3 p 6= 2, 3
6′′′ e e1 −1 −1 p = 3
7 e eζ −ζ −1 e eζ−1
−ζ−1−1
ζ ∈ G′3 p 6= 3
8 e e−ζ2 ζ −1 e e−ζ2 ζ3 −ζ
−2 e e−1−ζ−1−ζ−2 e e−1 −ζ ζ3 e e−1 ζ−1 ζ3
ζ ∈ G′12 p 6= 2, 3
9 e e−ζ−1
−ζ3 −1 e e−ζ2 ζ3 −1 e e−ζ2 ζ −ζ
2
ζ ∈ G′12 p 6= 2, 3
9′ e eζ ζ −1 e e1 −ζ −1 e e1 ζ 1
ζ ∈ G′4 p = 3
10 e e−ζ2 ζ −1 e eζ3 ζ−1 −1 e eζ3 ζ−2 −ζ
ζ ∈ G′9 p 6= 3
11 e eq q−3 q3 q ∈ k∗ \ {−1, 1}
q /∈ G′3
12 e eζ −ζ −1 e eζ2 −ζ−1−1 e eζ2 ζ ζ−1
ζ ∈ G′8 p 6= 2
13 e eζ ζ−5 −1 e e−ζ−4
ζ5 −1 e e−ζ−4
−ζ−1 ζ6 e eζ−1
ζ ζ6
ζ ∈ G′24 p 6= 2, 3
13′ e e−ζ ζ−1 −1 e e1 ζ −1 e e1 ζ−1 −ζ2 e e−ζ−1
−ζ −ζ
2
ζ ∈ G′8 p = 3
14 e eζ ζ2 −1 e e−ζ−2
ζ−2 −1
ζ ∈ G′5 p 6= 5
15 e eζ ζ−3 −1 e e−ζ−2
ζ3 −1 e e−ζ−2
−ζ3 −1 e e−ζ−ζ−3−1
ζ ∈ G′20 p 6= 2, 5
15′ e eζ ζ −1 e e1 −ζ −1 e e1 ζ −1 e e−ζ −ζ −1 ζ ∈ G′4 p = 5
16 e eζ5 −ζ−3−ζ e eζ5 −ζ−2−1 e eζ3 −ζ2 −1 e eζ3 −ζ4−ζ
−4
ζ ∈ G′15 p 6= 3, 5
16′ e e1 −ζ−1−ζ2 e e1 −ζ −1 e eζ −ζ−1−1 e eζ −ζ3−ζ
−3
ζ ∈ G′5 p = 3
16′′ e eζ−1
−1 −ζ e eζ−1
−ζ −1 e e1 −ζ−1−1 e e1 −ζ−ζ
−1
ζ ∈ G′3 p = 5
17 e e−ζ−ζ−3−1 e e−ζ−2
−ζ3 −1 ζ ∈ G′7 p 6= 7
18 e eζ−1
−1 −ζ e eζ−1
−ζ −1 e e1 −ζ−1−1 e e1 −ζ −1 e eζ −ζ−1−1 e eζ −1−ζ
−1
ζ ∈ G′3 p = 7
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 19
Table 5.7. The exchange graphs of Cs(M) in Theorem 5.1.
exchange graphs n l sequences in A+ char k
1 D11 1 6 (0, 0)
2 D21 1 4 (1, 1, 1)
3 D31
2
D32
1
τD31 3 12 (1, 1, 1)
4 D41 1 3 (2, 1, 2, 1)
5 D51
2
D52 2 6 (2, 1, 2, 1)
6 D61
1
D62 2 6 (2, 1, 2, 1) p 6= 3
6′ D6′,1
1
D6′,2 2 6 (2, 1, 2, 1) p = 3
6′′ D6′′,1 1 3 (2, 1, 2, 1) p 6= 2, 3
6′′′ D6′′′,1 1 3 (2, 1, 2, 1) p = 3
7 D71
2
D72 2 6 (2, 1, 2, 1) p 6= 3
8
D81
1
D82
2
2
D83
1
D84
2
D85
1
τD85
1
τD84
2
τD83
1
τD82
2
τD81 5 12 (2, 2, 1, 3, 1) p 6= 2, 3
9 D91
2
D92
1
D93
2
τD92
1
τD91 5 12 (3, 1, 2, 2, 1) p 6= 2, 3
9′ D9′,1
2
D9′,2
1
D9′,3
2
τD9′,2
1
τD9′,1 5 12 (3, 1, 2, 2, 1) p = 3
10 D10,1
2
D10,2
1
D10,3 3 6 (4, 1, 2, 2, 2, 1) p 6= 3
11 D11,1 1 2 (3, 1, 3, 1, 3, 1)
12 D12,1
2
D12,2
1
D12,3 3 6 (3, 1, 3, 1, 3, 1) p 6= 2
13 D13,1
2
D13,2
1
D13,3
2
D13,4 4 6 (5, 1, 2, 3, 1, 3, 2, 1) p 6= 2, 3
13′ D13′,1
2
D13′,2
1
D13′,3
2
D13′,4 4 6 (5, 1, 2, 3, 1, 3, 2, 1) p = 3
14 D14,1
2
D14,2 2 3 (3, 1, 4, 1, 3, 1, 4, 1) p 6= 5
15 D15,1
2
D15,2
1
D15,3
2
D15,4 4 6 (3, 1, 4, 1, 3, 1, 4, 1) p 6= 2, 5
15′ D15′,1
2
D15′,2
1
D15′,3
2
D15′,4 4 6 (3, 1, 4, 1, 3, 1, 4, 1) p = 5
16 D16,1
1
D16,2
2
D16,3
1
D16,4 4 6 (2, 1, 4, 1, 4, 1, 2, 3) p 6= 3, 5
16′ D16′,1
1
D16′,2
2
D16′,3
1
D16′,4 4 6 (2, 1, 4, 1, 4, 1, 2, 3) p = 3
16′′ D16′′,1
1
D16′′,2
2
D16′′,3
1
D16′′,4 4 6 (2, 1, 4, 1, 4, 1, 2, 3) p = 5
17 D17,1
2
D17,2 2 2 (3, 1, 5, 1)3 p 6= 7
18 D18,1
1
D18,2
2
D18,3
1
D18,4
2
D18,5
1
D18,6 6 6 (2, 1, 6, 1, 2, 3)2 p = 7
Step 1. If c0 = c1 = 1, then aX12 = aX21 = −1. Hence q0 6= 1. We distinguish four cases: 1aa,
1ab, 1ba and 1bb.
Case 1aa. If q11q0 = 1 and q22q0 = 1, then D = D21.
Case 1ab. If q11q0 = 1, q22 = −1, and q22q0 6= 1, then D = D31.
20 J. Wang and I. Heckenberger
Case 1ba. If q11 = −1, q22q0 = 1, and q11q0 6= 1, then D = τD31.
Case 1bb. If q11 = −1, q22 = −1, and q0 6= −1, then D = D32.
Step 2. Assume that (c0, c1, c2) = (1, 2, a′), where a′ ∈ {1, 2, 3}. Then we obtain that
aX21 = −1, aX12 = a
r1(X)
12 = −2, and a
r1(X)
21 = −a′. We distinguish four cases: 2aa, 2ab, 2ba
and 2bb.
Case 2aa. If q11
2q0 = 1 and q22q0 = 1, then D = D41.
Case 2ab. If q11
2q0 = 1, q22 = −1, and q22q0 6= 1, then D = D51.
Case 2ba. Assume that 1 + q11 + q11
2 = 0, q22q0 = 1, and q11
2q0 6= 1. If p = 3 then
1 + q11 + q11
2 = 0 yields q11 = 1. If q22 6= −1 then D = D6′,1 and if q22 = −1 then D = D6′′′,1.
Assume that p 6= 3. Set ζ := q11 and q := q22. Then q0 = q−1 /∈ {1, ζ−1} since aX12 = −2, and
q0 6= ζ since q11
2q0 6= 1. Thus D = D61 or D6′′,1, p 6= 2.
Case 2bb. Consider the last case 1 + q11 + q11
2 = 0, q22 = −1 and q0 /∈ {1,−1, q11, q
2
11}.
Case 2bba. If p = 3 then q11 = 1. Set q := q0. By Lemma 3.4, the Dynkin diagrams of r1(X)
and X are
e e1 q−1 −q2 e e1 q −1
with q ∈ k∗ \ {−1, 1}. Then a
r1(X)
21 ≤ −2 since (−q2)q−1 = −q 6= 1, and −q2 6= −1.
Case 2bba1. If p = 3 and a′ = −ar1(X)
21 = 2 then one gets (−q2)2q−1 = 1 or 1 + (−q2) +
(−q2)2 = 0. If (−q2)2q−1 = 1 then q = 1, which is a contradiction. Hence −q2 = 1 from the
second equation since p = 3. Then D = D9′,2.
Case 2bba2. If p = 3 and a′ = −ar1(X)
21 = 3, then one has (−q2)3q−1 = 1 or 1 + (−q2) +
(−q2)2 + (−q2)3 = 0. The first equation (−q2)3q−1 = 1 yields (−q)5 = 1, hence D = D16′,2. If
1 + (−q2) + (−q2)2 + (−q2)3 = 0, then (1− q2)(1 + q4) = 0 and hence q ∈ G′8. Then D = D13′,2.
Case 2bbb. We now suppose that p 6= 3. Set ζ := q11 and q := q0. Hence the Dynkin diagram
of r1(X) is e eζ (ζq)−1−ζq2
with ζ ∈ G′3, q ∈ k∗ \ {1,−1, ζ, ζ−1}. Since a′ ∈ {1, 2, 3}, we distinguish
three cases:
(b1) p 6= 3, a
r1(X)
21 = 1,
(b2) p 6= 3, a
r1(X)
21 = 2,
(b3) p 6= 3, a
r1(X)
21 = 3.
Case 2bbb1. If the condition (b1) holds, then one gets (−ζq2)(ζq)−1 = 1 or 1 + (−ζq2) = 0.
If (−ζq2)(ζq)−1 = 1, then q = −1, which is a contradiction. If 1 + (−ζq2) = 0 then ζ2 = q2 and
hence q = −ζ. Then D = D71.
Case 2bbb2. If the condition (b2) holds, then (−ζq2)2(ζq)−1 = 1 or
2∑
i=0
(−ζq2)i = 0.
Case 2bbb2a. Consider the equation (−ζq2)2(ζq)−1 = 1. Then ζq3 = 1 and hence q ∈ G′9
since ζ ∈ G′3 and p 6= 3. Hence D = D10,2.
Case 2bbb2b. If
2∑
i=0
(−ζq2)i = 0, then −ζq2 ∈ {ζ, ζ−1}. Hence q2 = −1 or −q2 = ζ.
Case 2bbb2b1. If q2 = −1, then p 6= 2 and the Dynkin diagram of X is e eζ q −1
with q ∈ G′4,
ζ ∈ G′3. Set η := ζ2q−1. Then η ∈ G′12, ζ = −η2, and q = η3. Hence D = D92.
Case 2bbb2b2. If −q2 = ζ, then q ∈ G′12 and p 6= 2 since q 6= ζ−1. Hence D = D81.
Case 2bbb3. If the condition (b3) holds, then (−ζq2)3(ζq)−1= 1 or
3∑
i=0
(−ζq2)i= 0, 1−ζq2 6= 0.
Case 2bbb3a. Consider the equation (−ζq2)3(ζq)−1 = 1, that is −q5 = ζ. Hence q = −ζ−1
or −q ∈ G′15, p 6= 5.
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 21
Case 2bbb3a1. If −q ∈ G′15, p 6= 3, 5, and ζ = −q5, then D = D16,2.
Case 2bbb3a2. If q = −ζ−1 then p 6= 2 since q 6= ζ−1. Hence the Dynkin diagrams of
r1(X), X and r2(X), respectively, are
e eζ −1−ζ
−1 e eζ −ζ−1−1 e e1 −ζ −1
with ζ ∈ G′3. Then one obtains that a
r2(X)
12 = 1− p. We distinguish four cases.
Case 2bbb3a2a. If p = 5, then D = D16′′2.
Case 2bbb3a2b. If p = 7, then D = D18,2.
Case 2bbb3a2c. If p = 6s+1 (s ≥ 2), then the Dynkin diagrams of r1(X), X, r2(X), r1r2(X),
r2r1r2(X), and (r1r2)
2(X), respectively, are
e eζ −1−ζ
−1 e eζ −ζ−1−1 e e1 −ζ −1 e e1 −ζ−1−1 e eζ−1
−ζ −1 e eζ−1
−1 −ζ
with ζ ∈ G′3. Hence n = 6 in Theorem 4.15 and (ck)k≥0 = (2, 3, 2, 1, p− 1, 1, 2, 3, 2, 1, p− 1, 1)∞.
Then l = 20− 2p < 0, which is a contradiction to Theorem 4.15.
Case 2bbb3a2d. If p = 6s + 5, where s ≥ 1, then the Dynkin diagrams of r1(X), X, r2(X)
and r1r2(X), respectively, are
e eζ −1−ζ
−1 e eζ −ζ−1−1 e e1 −ζ −1 e e1 −ζ−1−ζ
with ζ ∈ G′3. Then n = 4 and (ck)k≥0 = (2, 3, 2, 1, p − 1, 1, p − 1, 1)∞. Hence l = 16 − 2p < 0.
Again, one gets a contradiction.
Case 2bbb3b. Consider the equation 0 =
3∑
i=0
(−ζq2)i = (1 − ζq2)(1 + ζ2q4), where ζq2 6= 1.
One gets ζ = −q4. If p = 2, then ζ4 = q4 and hence ζ = q, which is a contradiction to ζq2 6= 1.
Otherwise q ∈ G′24 and D = D13,2.
Step 3. Now we change the label. It means that (ck)k≥1 is the characteristic sequence
of Cs(M) with respect to X and the label 2.
Assume that (c0, c1, c2) = (2, 1, b′), where b′ ∈ {3, 4, 5}. Then we obtain that aX12 = −2, aX21 =
−1 and a
r2(X)
12 = −b′. If q11
2q0 = 1 and q22 = −1 then a
r2(X)
12 = −2, which is a contradiction.
If q0q22 = 1 then a
r2(X)
12 = aX12 = −2, which is again a contradiction. Suppose now that
1+q11 +q11
2 = 0, q22 = −1 and q0 /∈ {1,−1, q−211 }. Since aX12 = −2, we also obtain that q0 6= q−111 .
Case 3a. If p = 3, then by setting q := q0, the Dynkin diagrams of X and r2(X), respectively,
are
e e1 q −1 e e−q q−1 −1
q ∈ k∗ \ {−1, 1}. (5.1)
Case 3a1. If p = 3 and b′ = 3, then one gets (−q)3q−1 = 1 or
3∑
i=0
(−q)i = 0. Since q 6= 1, both
equations imply that q2 = −1. Hence D = D9′,2.
Case 3a2. If p = 3 and b′ = 4, then one has (−q)4q−1 = 1 or
4∑
i=0
(−q)i = 0. If (−q)4q−1 = 1
then q = 1, which is a contradiction to (5.1). If
4∑
i=0
(−q)i = 0 then q5 = −1 and D = D16′,2.
Case 3a3. If p = 3 and b′ = 5, then one obtains (−q)5q−1 = 1 or
5∑
i=0
(−q)i = 0. If (−q)5q−1 = 1
then q ∈ G′8 and D = D13′,2. Consider the equation 0 =
5∑
i=0
(−q)i = (1 − q)(1 + q2 + q4). Then
q2 = 1 since p = 3 and q 6= 1, which is a contradiction to (5.1).
22 J. Wang and I. Heckenberger
Case 3b. We now consider the cases in which the condition p 6= 3 holds. Set ζ := q11 and
q := q0. The Dynkin diagram of r2(X) is
e e−ζq q−1 −1
ζ ∈ G′3, q ∈ k∗ \
{
1,−1, ζ, ζ−1
}
. (5.2)
Case 3b1. If p 6= 3 and b′ = 3, then one gets (−ζq)3q−1 = 1 or
3∑
i=0
(−ζq)i = 0. If (−ζq)3q−1 = 1
then q ∈ G′4, p 6= 2 and D = D92. If 0 =
3∑
i=0
(−ζq)i = (1 − ζq)(1 + (ζq)2), then (ζq)2 = −1 and
p 6= 2 since q 6= ζ−1. Hence D = D81.
Case 3b2. If p 6= 3 and b′ = 4, then one gets (−ζq)4q−1 = 1 or
4∑
i=0
(−ζq)i = 0. If (−ζq)4q−1 = 1
then ζ = q−3. Since q /∈ G′3, one obtains q ∈ G′9 and D = D10,2. The equation
4∑
i=0
(−ζq)i = 0
gives ζ = −q5. Since ζ ∈ G′3, one gets −q ∈ G′3, ζ = −q−1, p = 5 or −q ∈ G′15, p 6= 3, 5. If
−q ∈ G′3 then D = D16′′,2 and if −q ∈ G′15 then D = D16,2.
Case 3b3. If p 6= 3 and b′ = 5, then one gets (−ζq)5q−1 = 1 or
5∑
i=0
(−ζq)i = 0, −ζq 6= 1. If
(−ζq)5q−1 = 1 and p = 2 then qζ−1 = 1, which is a contradiction to (5.2). If (−ζq)5q−1 = 1 and
p 6= 2 then ζ = −q4 and D = D13,2. Consider 0 =
5∑
i=0
(−ζq)i = (1 − ζq)(1 + (−ζq)2 + (−ζq)4).
Since q /∈ {1,−1, ζ, ζ−1}, one gets p 6= 2 and q3 = −1. Since −ζq 6= 1, one gets q = −ζ and
a
r2(X)
12 = −2, which is a contradiction.
Step 4. Again we use the same labeling as in steps 1 and 2. Assume that (c0, c1, c2, c3) =
(1, 3, 1, c′), where c′ ∈ {3, 4, 5}. Then aX21 = −1 and aX12 = −3. We distinguish four cases: 4aa,
4ab, 4ba and 4bb.
Case 4aa. If q11
3q0 = 1 and q22q0 = 1, then D = D11,1.
Case 4ab. Set q := q11. If q11
3q0 = 1 and q22 = −1, then the Dynkin diagrams of X, r1(X)
and r2r1(X), respectively, are
e eq q−3 −1 e eq q−3 −1 e e-q−2
q3 −1
q ∈ k∗ \ {1,−1}, q /∈ G′3.
Then −ar2r1(X)
12 = c′ ∈ {3, 4, 5}. Hence we distinguish three cases: 4ab1, 4ab2 and 4ab3.
Case 4ab1. If c′ = 3, then (−q−2)3q3 = 1 or
3∑
i=0
(−q−2)i = 0, 1 − q−2 6= 0. If (−q−2)3q3 = 1
then D = D11,1, where q3 = −1. If
3∑
i=0
(−q−2)i = 0, then D = D12,1.
Case 4ab2. If c′ = 4, then (−q−2)4q3 = 1 or
4∑
i=0
(−q−2)i = 0.
Case 4ab2a. The equation (−q−2)4q3 = 1 gives q5 = 1 and p 6= 5, since q 6= 1. Hence
D = D14,1.
Case 4ab2b. Consider the equation
4∑
i=0
(−q−2)i = 0. One gets q10 = −1. If p = 2 then
D = D14,1. If p = 5 then q2 = −1 and D = D15′,1. If p 6= 2, 5, then q ∈ G′20 and D = D15,1.
Case 4ab3. If c′ = 5, then (−q−2)5q3 = 1 or
5∑
i=0
(−q−2)i = 0.
Case 4ab3a. Consider the equation (−q−2)5q3 = 1, which gives −q7 = 1. Since q 6= −1, one
gets p 6= 7 and −q ∈ G′7. Hence D = D17,1.
Rank 2 Nichols Algebras of Diagonal Type over Fields of Positive Characteristic 23
Case 4ab3b. Consider the equation 0 =
5∑
i=0
(−q−2)i = (1− q−2)(1 + q−4 + q−8). Since q2 6= 1,
one gets 1 + q−4 + q−8 = 0. If p = 3 then q ∈ G′4 since q2 6= 1. Hence a
r2r1(X)
12 = −2, which is
a contradiction. Then p 6= 3 and q−4 ∈ G′3. Since c′ = 5, one gets q ∈ G′6 or q ∈ G′12. If q ∈ G′6,
then a
r2r1(X)
12 = −3, which is a contradiction. If q ∈ G′12, then a
r2r1(X)
12 = −2, which is again
a contradiction.
Case 4ba. The conditions 1 + q11 + q11
2 + q11
3 = 0, q11 6= −1 and q22q0 = 1 hold. Then
q11 ∈ G′4 and p 6= 2 since aX12 = −3. Set ζ := q11 and q = q22. The Dynkin diagram of r1(X)
is e eζ −q ζq−2
with ζ ∈ G′4 and q ∈ k∗ \ {1,−1, ζ, ζ−1}. Since −ar1(X)
21 = c2 = 1, one gets
(ζq−2)(−q) = 1 or ζq−2 = −1. If (ζq−2)(−q) = 1 then q = −ζ, which is a contradiction. If
ζq−2 = −1 then q ∈ G′8 and D = D12,3.
Case 4bb. Consider the last case: 1 + q11 + q211 + q311 = 0, q22 = −1 and q11 6= −1. Then
q211 = −1 and p 6= 2. Set ζ := q11 and q = q0. The Dynkin diagram of r1(X) is e eζ −q−1−ζq3
with
q ∈ k∗ \ {1,−1, ζ, ζ−1} and ζ ∈ G′4. Since −ar1(X)
21 = c2 = 1, one has (−q−1)(−ζq3) = 1 or
ζq3 = 1.
Case 4bb1. If ζq3 = 1, then ζ = q−3 ∈ G′4. If p = 3, then ζ = q, which is a contradiction.
Hence p 6= 3 and D = τD85.
Case 4bb2. The condition (−q−1)(−ζq3) = 1 holds. Then ζ = q−2 ∈ G′4. Hence q ∈ G′8 and
D = D12,2.
By checking all cases in Theorem 4.6, the proof of Theorem 5.1 is completed. �
Remark 5.3. Assume that char k = p > 0. Let V be a two-dimensional braided vector space
of diagonal type. Let (x1, x2) be a basis of V and (qij)1≤i,j≤2 ∈ (k \ {0})2×2 satisfying
c(xi ⊗ xj) = qijxj ⊗ xi
for any i, j.
– By [13, Corollary 6], dimk B(V ) <∞ if and only if M = (kx1, kx2) admits all reflections,
W(M) is finite, and for all points D of Cs(M), the labels of the vertices of D are roots of
unity (including 1). Therefore with Theorem 5.1 one can easily decide whether dimk B(V )
is finite.
– If the Dynkin diagram of V appears in the row 18 of Table 5.6, then the Weyl groupoid
of B(V ) is not appearing for Nichols algebras in characteristic zero.
Acknowledgements
It is a pleasure to thank N. Andruskiewitsch and H.-J. Schneider for a very fruitful discussion
on some details of this topic. The authors thank the referees for their helpful comments and
suggestions. J. Wang is supported by China Scholarship Council.
References
[1] Andruskiewitsch N., About finite dimensional Hopf algebras, in Quantum Symmetries in Theoretical Physics
and Mathematics (Bariloche, 2000), Contemp. Math., Vol. 294, Amer. Math. Soc., Providence, RI, 2002,
1–57.
[2] Andruskiewitsch N., Graña M., Braided Hopf algebras over non-abelian finite groups, Bol. Acad. Nac. Cienc.
(Córdoba) 63 (1999), 45–78, math.QA/9802074.
[3] Andruskiewitsch N., Heckenberger I., Schneider H.-J., The Nichols algebra of a semisimple Yetter–Drinfeld
module, Amer. J. Math. 132 (2010), 1493–1547, arXiv:0803.2430.
http://dx.doi.org/10.1090/conm/294/04969
http://arxiv.org/abs/math.QA/9802074
http://arxiv.org/abs/0803.2430
24 J. Wang and I. Heckenberger
[4] Andruskiewitsch N., Schneider H.-J., Lifting of quantum linear spaces and pointed Hopf algebras of order p3,
J. Algebra 209 (1998), 658–691, math.QA/9803058.
[5] Andruskiewitsch N., Schneider H.-J., Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000),
1–45.
[6] Andruskiewitsch N., Schneider H.-J., Pointed Hopf algebras, in New Directions in Hopf Algebras, Math. Sci.
Res. Inst. Publ., Vol. 43, Cambridge University Press, Cambridge, 2002, 1–68, math.QA/0110136.
[7] Andruskiewitsch N., Schneider H.-J., On the classification of finite-dimensional pointed Hopf algebras, Ann.
of Math. 171 (2010), 375–417, math.QA/0502157.
[8] Cibils C., Lauve A., Witherspoon S., Hopf quivers and Nichols algebras in positive characteristic, Proc.
Amer. Math. Soc. 137 (2009), 4029–4041, arXiv:0901.4408.
[9] Cuntz M., Heckenberger I., Weyl groupoids of rank two and continued fractions, Algebra Number Theory 3
(2009), 317–340, arXiv:0807.0124.
[10] Cuntz M., Heckenberger I., Weyl groupoids with at most three objects, J. Pure Appl. Algebra 213 (2009),
1112–1128, arXiv:0805.1810.
[11] Cuntz M., Heckenberger I., Reflection groupoids of rank two and cluster algebras of type A, J. Combin.
Theory Ser. A 118 (2011), 1350–1363, arXiv:0911.3051.
[12] Heckenberger I., The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), 175–
188.
[13] Heckenberger I., Rank 2 Nichols algebras with finite arithmetic root system, Algebr. Represent. Theory 11
(2008), 115–132, math.QA/0412458.
[14] Heckenberger I., Classification of arithmetic root systems, Adv. Math. 220 (2009), 59–124,
math.QA/0605795.
[15] Heckenberger I., Schneider H.-J., Nichols algebras over groups with finite root system of rank two I, J. Al-
gebra 324 (2010), 3090–3114.
[16] Heckenberger I., Schneider H.-J., Root systems and Weyl groupoids for Nichols algebras, Proc. Lond. Math.
Soc. 101 (2010), 623–654, arXiv:0807.0691.
[17] Heckenberger I., Schneider H.-J., Right coideal subalgebras of Nichols algebras and the Duflo order on the
Weyl groupoid, Israel J. Math. 197 (2013), 139–187, arXiv:0909.0293.
[18] Heckenberger I., Yamane H., A generalization of Coxeter groups, root systems, and Matsumoto’s theorem,
Math. Z. 259 (2008), 255–276, math.QA/0610823.
[19] Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
[20] Kharchenko V.K., A quantum analogue of the Poincaré–Birkhoff–Witt theorem, Algebra and Logic 38
(1999), 259–276, math.QA/0005101.
[21] Lusztig G., Introduction to quantum groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York,
2010.
[22] Nichols W.D., Bialgebras of type one, Comm. Algebra 6 (1978), 1521–1552.
[23] Rosso M., Quantum groups and quantum shuffles, Invent. Math. 133 (1998), 399–416.
[24] Schauenburg P., A characterization of the Borel-like subalgebras of quantum enveloping algebras, Comm.
Algebra 24 (1996), 2811–2823.
[25] Semikhatov A.M., Virasoro central charges for Nichols algebras, arXiv:1109.1767.
[26] Semikhatov A.M., Tipunin I.Yu., The Nichols algebra of screenings, Commun. Contemp. Math. 14 (2012),
1250029, 66 pages, arXiv:1101.5810.
[27] Semikhatov A.M., Tipunin I.Yu., Logarithmic ŝ`(2) CFT models from Nichols algebras: I, J. Phys. A: Math.
Theor. 46 (2013), 494011, 53 pages, arXiv:1301.2235.
[28] Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.
[29] Woronowicz S.L., Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math.
Phys. 122 (1989), 125–170.
http://dx.doi.org/10.1006/jabr.1998.7643
http://arxiv.org/abs/math.QA/9803058
http://dx.doi.org/10.1006/aima.1999.1880
http://dx.doi.org/10.2977/prims/1199403805
http://dx.doi.org/10.2977/prims/1199403805
http://arxiv.org/abs/math.QA/0110136
http://dx.doi.org/10.4007/annals.2010.171.375
http://dx.doi.org/10.4007/annals.2010.171.375
http://arxiv.org/abs/math.QA/0502157
http://dx.doi.org/10.1090/S0002-9939-09-10001-1
http://dx.doi.org/10.1090/S0002-9939-09-10001-1
http://arxiv.org/abs/0901.4408
http://dx.doi.org/10.2140/ant.2009.3.317
http://arxiv.org/abs/0807.0124
http://dx.doi.org/10.1016/j.jpaa.2008.11.009
http://arxiv.org/abs/0805.1810
http://dx.doi.org/10.1016/j.jcta.2010.12.003
http://dx.doi.org/10.1016/j.jcta.2010.12.003
http://arxiv.org/abs/0911.3051
http://dx.doi.org/10.1007/s00222-005-0474-8
http://dx.doi.org/10.1007/s10468-007-9060-7
http://arxiv.org/abs/math.QA/0412458
http://dx.doi.org/10.1016/j.aim.2008.08.005
http://arxiv.org/abs/math.QA/0605795
http://dx.doi.org/10.1016/j.jalgebra.2010.06.021
http://dx.doi.org/10.1016/j.jalgebra.2010.06.021
http://dx.doi.org/10.1112/plms/pdq001
http://dx.doi.org/10.1112/plms/pdq001
http://arxiv.org/abs/0807.0691
http://dx.doi.org/10.1007/s11856-012-0180-3
http://arxiv.org/abs/0909.0293
http://dx.doi.org/10.1007/s00209-007-0223-3
http://arxiv.org/abs/math.QA/0610823
http://dx.doi.org/10.1017/CBO9780511626234
http://dx.doi.org/10.1007/BF02671731
http://arxiv.org/abs/math.QA/0005101
http://dx.doi.org/10.1007/978-0-8176-4717-9
http://dx.doi.org/10.1080/00927877808822306
http://dx.doi.org/10.1007/s002220050249
http://dx.doi.org/10.1080/00927879608825714
http://dx.doi.org/10.1080/00927879608825714
http://arxiv.org/abs/1109.1767
http://dx.doi.org/10.1142/S0219199712500290
http://arxiv.org/abs/1101.5810
http://dx.doi.org/10.1088/1751-8113/46/49/494011
http://dx.doi.org/10.1088/1751-8113/46/49/494011
http://arxiv.org/abs/1301.2235
http://dx.doi.org/10.1007/BF01219077
http://dx.doi.org/10.1007/BF01221411
http://dx.doi.org/10.1007/BF01221411
1 Introduction
2 Cartan graphs and root systems
3 Cartan graphs for Nichols algebras
3.1 Small Cartan graphs for Nichols algebras of diagonal type
4 Finite Cartan graphs of rank two
5 Classification of rank two Nichols algebras of diagonal type over fields of positive characteristic
References
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