# Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

Authors: Chen Qian; Jing Wang
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.


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## Body

## 1. Introduction

The theory of Nichols algebras is relatively young, but it has interesting applications to other research fields such as Kac-Moody Lie superalgebras [1] and conformal field theory [2–4]. Besides, it plays an important role in quantum groups [5–8].The theory of Nichols algebras is motivated by the Hopf algebra theory. In any area of mathematics, the classification of all objects is very important. In Hopf algebra theory, the classification of all finite dimensional Hopf algebras is a tough question [5]. The structure of Nichols algebras appears naturally in the classification of pointed Hopf algebras in the following way. Given a Hopf algebra H, consider its coradical filtration
(1)H0⊂H1⊂⋯,such that H0 is a Hopf subalgebra of H and the associated graded coalgebra
(2)grH=⊕iHi/Hi−1.Then,grH is a graded Hopf algebra, since the coradical H0 of H is a Hopf subalgebra. In addition, consider a projection π:grH⟶H0; let R be the algebra of coinvariants of π. Then, by a result of Radford and Majid, R is a braided Hopf algebra and grH is the bosonization (or biproduct) of R and H0: grH≃R#H0. The principle of the “lifting method” introduced in [7, 9] is first to study R, then to transfer the information to grH via bosonization, and finally to lift to H. The braided Hopf algebra R is generated by the vector space V of H0-coinvariants of H1/H0, namely, Nichols algebra [9] BV generated by V in commemoration of W. Nichols [10] who started to study these objects as bialgebras of type one in 1978. Nichols algebras can be described in many different but alternative ways (see, for example, [11–15]).The methods developed in the study of the generalizations of Lie algebras are also useful to analyze Nichols algebras [16]. Kharchenko proved that any Hopf algebra generated by skew-primitive and group-like elements has a restricted Poincaré-Birkhoff-Witt basis ([17], Theorem 2). Note that Kharchenko results can be applied to Nichols algebras of diagonal type. Motivated by the close relation to Lie theory, Heckenberger [18] defined the arithmetic root system and Weyl groupoid for Nichols algebras BV of diagonal type. Later, Cuntz and Heckenberger [19] and Heckenberger and Yamane [20] developed the combinatorial theory of these two structures. Then, the theory of root systems and Weyl groupoids was carried out in more general Nichols algebras [21–23]. Further, all finite Weyl groupoids were classified in [24, 25]. Those constructions are very important theoretical tools for the classification of Nichols algebra BV.The crucial step to classify pointed Hopf algebras is to determine that all Nichols algebrasBV is finite dimensional. Andruskiewitsch stated the following question.Question 1 (Andruskiewitsch [5]).
Given a braiding matrixqij1≤i,j≤θ whose entries are roots of 1, when BV is finite-dimensional, where V is a vector space with basis x1,⋯,xθ and braiding cxi⊗xj=qijxj⊗xi? If so, compute dimkBV, and give a “nice” presentation by generators and relations.Several authors obtained the classification result for infinite and finite dimensional Nichols algebra of Cartan type (see [13, 18, 26]). Heckenberger determined all finite dimensional Nichols algebra of diagonal type [27–30], that is, when V is the direct sum of 1-dimensional Yetter-Drinfel’d modules over H0. The generators and relations of such Nichols algebras were also given [31, 32]. With the classification result, Andruskiewitsch and Schneider [33] obtained a classification theorem about finite-dimensional pointed Hopf algebras under some technical assumptions.Based on such successful applications, the analysis to Nichols algebras over arbitrary fields is crucial and has also potential applications. Towards this direction, new examples of Nichols algebras in positive characteristic and a combinatorial formula to study the relations in Nichols algebras were found [34]. Over fields of positive characteristic, rank 2, rank 3, and rank 4 finite dimensional Nichols algebras of diagonal type were listed in [35–37].In this paper, we illustrate the main properties of the finite Cartan graphs of rank6 attached to the Nichols algebras of diagonal type in Theorem 39. As a consequence, a subclass of rank 6 Nichols algebras of diagonal type with a finite root system is given in Section 5. The main algorithm in Section 3 is essential for our main goal.Besides, the notations and conventions in [35–37] are followed, and several results from these papers will be used.


## 2. Nichols Algebra of Diagonal Type

In this section, we recall Yetter-Drinfel’d modules, braided vector spaces, and their relations. The main object of this paper is also presented. For further details on these topics, we refer to [1, 5, 6].


### 2.1. Yetter-Drinfel’d Modules

Letk be a field of characteristic p>0. Let k∗=k\0, ℕ0=ℕ∪0, θ∈ℕ0, and I=1,⋯,θ. We start by recalling the main objects.Definition 2.
LetV be a θ-dimensional vector space over k. The pair V,c is called a braided vector space, if c∈AutV⊗V is a solution of the braid equation, that is,
(3)c⊗idid⊗cc⊗id=id⊗cc⊗idid⊗c.
A braided vector spaceV,c is termed of diagonal type if V admits a basis xi∣i∈I such that for all i,j∈I one has
(4)cxi⊗xj=qijxj⊗xiforsomeqij∈k∗.
The matrixqiji,j∈I is termed of the braiding matrix of V. We say that the braiding matrix qiji,j∈I is indecomposable if for any i≠j there exists a sequence i1=i,i2,⋯,it=j, of elements of I such that qisis+1qis+1is≠1, where 1≤s≤t−1.
In this paper, we mainly concern the braided vector spaces of diagonal type with an indecomposable braiding matrix.Definition 3.
LetH be a Hopf algebra. A Yetter-Drinfel’d module V over H is a left H-module with left action: H⊗V⟶V and a left H-comodule with left coaction δL:V⟶H⊗V satisfying the compatibility condition
(5)δLh·v=h1v−1κh3⊗h2·v0,h∈H,v∈V.
We say thatV is of diagonal type if H=k’G and V is a direct sum of one-dimensional Yetter-Drinfel’d modules over the group algebra k’G, where G is abelian.
We denote byHHYD the category of Yetter-Drinfel’d modules over H, where morphisms preserve both the action and the coaction of H. The category HHYD is braided with braiding
(6)cV,Wv⊗w=v−1·w⊗v0
for allV,W∈YHHD, v∈V, and w∈W. Actually, the category YHHD ia a braided monoidal category, where the monoidal structure is given by the tensor product over k. Then, any Yetter-Drinfel’d module V∈HHYD over H admits a braiding cV,V, and hence, V,cV,V is a braided vector space. Conversely, any braided vector space can be realized as a Yetter-Drinfel’d module over some Hopf algebras if and only if the braiding is rigid ([38], Section 2). Notice that Yetter-Drinfel’d module structures on V with different Hopf algebras can give the same braiding and not all braidings of V are induced by the above Equation (1).
IfH=k’G, then we write YGGD for the category of Yetter-Drinfel’d modules over k’G and say that V∈YHHD is a Yetter-Drinfel’d module over G. Notice that if V∈YGGD is of diagonal type, then V,cV,V is a braided vector space of diagonal type. Any braided vector space of diagonal type is also a Yetter-Drinfel’d module of diagonal type. Indeed, assume that V,c is a braided vector space of diagonal type with an indecomposable braiding matrix qiji,j∈I of a basis xi∣i∈I. Let G0 be a free abelian group generated by elements gi∣i∈I. Define the left coaction and left action by
(7)δLxi=gi⊗xi∈G0⊗V,gi·xj=qijxj∈V.Then,V=⊕i∈Ikxi, and each kxi is one-dimensional Yetter-Drinfel’d modules over G0. Hence, V is a Yetter-Drinfel’d module of diagonal type over G0.


### 2.2. Nichols Algebras of Diagonal Type

LetV,c be a θ-dimensional braided vector space of diagonal type. In this section, we recall a definition of the Nichols algebra BV generated by V,c. In order to do that, we introduce one more notion in the category YHHD.Definition 4.
LetH be a Hopf algebra. A braided Hopf algebra in HHYD is a 6-tuple B=B,μ,1,δ,ε,κB, where B,μ,1 is an algebra in YHHD and a coalgebra B,δ,ε in YHHD, and κB:B⟶B is a morphism in YHHD such that δ, ε, and κB satisfy κBb1b2=b1κBb2=εb1, where we define δb=b1⊗b2 as the coproduct of B to avoid the confusion.Definition 5.
The Nichols algebra generated byV∈YHHD is defined as the quotient
(8)BV=TV/IV=⊕n=0∞V⊗n/IV,where IV is the unique maximal coideal among all coideals of TV which are contained in ⊕n≥2V⊗n. Nichols algebra BV is said to be of diagonal type if V is a Yetter-Drinfel’d module of diagonal type. The dimension of V is the rank of Nichols algebra BV.Note that ifB∈YHHD and B is an algebra in YHHD, then B⊗B is an algebra in HHYD with the product given by
(9)a⊗bc⊗d=ab−1·c⊗b0d,for alla,b,c,d∈B, where · denotes the left action of H on B.Thetensor algebraTV admits a natural structure of a Yetter-Drinfel’d module and an algebra structure in YHHD. It is then a braided Hopf algebra in YHHD with coproduct δv=1⊗v+v⊗1∈TV⊗TV and counit εv=0 for all v∈V such that δ and ε are the algebra morphisms. The antipode of TV exists (see [7], Section 2.1). Notice that the product defined by Equation (9) on TV is the unique algebra structure such that δv=1⊗v+v⊗1∈TV⊗TV for all v∈V. The coproduct can be extended from V to TV. For example, for all v,w∈V, one gets (we write the elements of TV without the tensor product sign for brevity)
(10)δvw=δvδw=1⊗v+v⊗11⊗w+w⊗1=1⊗vw+v−1·w⊗v0+v⊗w+vw⊗1.LetIii∈I be the family of all ideals of TV contained in ⊕n≥2V⊗n, i.e.,
(11)δIi⊂Ii⊗TV+TV⊗Ii.Then, the coidealIV≔∑i∈IIi is the largest element in Iii∈I. Hence, BV is a braided Hopf algebra in YHHD. As proved in [6] (Proposition 3.2.12), Nichols algebra BV is the unique ℕ0-graded braided Hopf algebra generated by V in YHHD with homogenous components BV0=k, BV1=V, and PBV=V, where PBV is the space of primitive elements of BV.


### 2.3. Weyl Groupoids

In this section, we recall the notations of semi-Cartan graphs, root systems, and Weyl groupoids. We mainly follow the terminology from [20, 39] (see also [35–37]).Definition 6.
A generalized Cartan matrix is a matrixA=aiji,j∈I with integer entries such that
(i)
aii=2 and ajk≤0 for any i,j,k∈I with j≠k(ii)
ifaij=0 for some i,j∈I, then aji=0A generalized Cartan matrixA∈ℤI×I is decomposable if there exists a nonempty proper subset I1⊂I such that aij=0 for any i∈I1 and j∈I\I1. We say that A is indecomposable if A is not decomposable.Definition 7.
LetX be a nonempty set and AX=aijXi,j∈I a generalized Cartan matrix for all X∈X. For any i∈I, letri:X⟶X, X↦ri,X, where r:I×X⟶X is a map. The quadruple
(12)C=CI,X,r,AXX∈Xis called a semi-Cartan graph if ri2=idX for all i∈I, and aijX=aijriX for all X∈X and i,j∈I. We say that a semi-Cartan graph C is indecomposable if AX is indecomposable for all X∈X.For the sake of simplicity, the elements of the setriX,i∈I are termed the neighbors of X for all X∈X. The cardinality of I is termed the rank of C, and the elements of X are the points of C.Definition 8.
A semi-Cartan graphC is standard, if AX=AY for all X,Y∈ObWC.Definition 9.
The exchange graph ofC is a labeled nonoriented graph with vertices set X and edges set I, where two vertices X,Y are connected by an edge i if and only if X≠Y and riX=Y (and riY=X). We display one edge with several labels instead of several edges for simplification. A semi-Cartan graph C is said to be connected if its exchange graph is connected.
For the remaining part of this section, we assume thatC=CI,X,r,AXX∈X is a connected semi-Cartan graph. Let αii∈I be the standard basis of ℤI. For all X∈X, let
(13)siX∈AutℤI,siXαj=αj−aijXαi,
for allj∈I. Let DX,I be the category such that ObDX,I=X and morphisms HomX,Y=Y,f,X∣f∈EndℤI for X,Y∈X with the composition Z,g,Y∘Y,f,X=Z,gf,X for all X,Y,Z∈X, f,g∈EndℤI. Let WC be the smallest subcategory of DX,I, where the morphisms are generated by riX,siX,X, with i∈I, X∈X. From now on, we write siX instead of riX,siX,X, if no confusion is possible. Notice that all generators siX are reflections and hence are invertible. Then, WC is a groupoid.
For any categoryD and any object X in D, let HomD,X=∪Y∈DHomY,X.Definition 10.
For allX∈X, the set
(14)ΔXre=ωαi∈ℤI∣ω∈HomWC,X
is called the set of real roots ofC at X. The elements of Δ+Xre=ΔXre∩ℕ0I are called positive roots and those of ΔXre∩−ℕ0I negative roots, denoted by Δ−Xre. If the set ΔXre is finite for all X∈X, then we say that C is finite.Definition 11.
We say thatR=RC,ΔXX∈X is a root system of type C if for all X∈X, the sets ΔX are the subsets of ℤI such that
(15)ΔX=ΔX∩ℕ0I∪−ΔX∩ℕ0I.(i)
ΔX∩ℤαi=αi,−αi for all i∈I(ii)
siXΔX=ΔriX for all i∈I(iii)
rirjmijXX=X for any i,j∈I with i≠j where mijX=∣ΔX∩ℕ0αi+ℕ0αj∣ is finite
We say thatWR≔WC is the groupoid of R. As in [13] (Definition 4.3), we say that R is reducible if there exist nonempty disjoint subsets of I′,I′′⊂I such that I=I′∪I′′ and aij=0 for all i∈I′, j∈I′′ and
(16)ΔX=ΔX∩∑i∈I′ℤαi∪ΔX∩∑j∈I′′ℤαjforallX∈X.
In this case, we writeR=RI1⊕RI2. If R≠RI1⊕RI2 for all nonempty disjoint subsets I1,I2⊂I, then R is termed irreducible.Definition 12.
LetR=RC,ΔXX∈X be a root system of type C. We say that R is finite if ΔX is finite for all X∈X.LetR=RC,ΔXX∈X be a root system of type C. We recall some properties of R from [19, 24].Lemma 13.
LetX∈X, k∈ℤ, and i,j∈I such that i≠j. Then αj+kαi∈ΔXre if and only if 0≤k≤−aijX.Notice that the finiteness ofR does not mean that WR is also finite, since the set X might be infinite.Lemma 14.
LetC=CI,X,r,AXX∈X be a connected semi-Cartan graph and R=RC,ΔXX∈X be a root system of type C. Then, the following are equivalent.
(1)
R is finite(2)
ΔX is finite for some X∈X(3)
C is finite(4)
WR is finiteRecall thatC is a connected semi-Cartan graph. Then, we get the following.Proposition 15.
LetR=RC,ΔXX∈X be a root system of type C. Then, the following are equivalent.
(1)
There existsX∈X such that AX is indecomposable(2)
The semi-Cartan graphC is indecomposable
IfR is finite, then the semi-Cartan graph C is indecomposable if and only if the root system R is irreducible.Definition 16.
We sayC is a Cartan graph if the following hold:
(i)
For allX∈X, the set ΔXre=Δ+Xre∪Δ−Xre(ii)
IflmnY≔ΔYre∩ℕ0αm+ℕ0αn is finite, then rmrnlmnYY=Y, where m,n∈I, Y∈X
In this case,WC is called the Weyl groupoid of C.LetRre≔RC,ΔXreX∈X. Then, C is a Cartan graph if and only if Rre is a root system of type C. Indeed, we get that siXΔXre=ΔriXre by Equation (14). For all X∈X, we obtain that ΔXre=Δ+Xre∪Δ−Xre, since ωsiriXαi=−ωαi for any ω∈HomWC,X.The following proposition implies that ifR is a finite root system of type C, then R=Rre; namely, all roots are real and R is uniquely determined by C.Proposition 17.
LetR=RC,ΔXX∈X be a root system of type C. Let X∈X, m∈ℕ0, and i1,⋯,im∈I such that
(17)ω=idXsi1si2⋯sim∈HomWC,Xand ℓω=m. Then, the elements
(18)βn=idXsi1si2⋯sin−1αin∈ΔX∩ℕ0Iare pairwise different, where n∈1,2,⋯,m (and β1=αi1). Here,
(19)ℓω=minm∈ℕ0∣ω=idXsi1si2⋯sim,i1,i2,⋯,im∈Iis the length of ω∈HomWC,X. In particular, if R is finite and ω∈HomWC is the longest element, then
(20)βn∣1≤n≤ℓω=ΔX2=ΔX∩ℕ0I.Remark 18.
IfC is a finite Cartan graph, then R is finite, and hence,
(21)Rre=RreC,ΔXreX∈Xis the unique root system of type C by Proposition 17; that is, R is uniquely determined by C.


### 2.4. Cartan Graphs for Nichols Algebras of Diagonal Type

In this section, we attach a semi-Cartan graph to a tuple of finite-dimensional Yetter-Drinfel’d modules under some finiteness conditions. LetG be an abelian group. Let FθG be the set of θ-tuples of finite-dimensional irreducible objects in YGGD and XθG be the set of θ-tuples of isomorphism classes of finite-dimensional irreducible objects in GGYD. For any M1,⋯,Mθ∈FθG, write M≔M1,⋯,Mθ∈XθG the corresponding isomorphism class of M1,⋯,Mθ.Assume thatVM=⊕i∈Ikxi∈GGYD is a Yetter-Drinfel’d module of diagonal type over G, where xi∣i∈I is a basis of V. Then, there exists a matrix qiji,j∈I such that δxi=gi⊗xi and gi·xj=qijxj for all i,j∈I. We fix that M=kx1,kx2,⋯,kxθ∈FθG is a tuple of one-dimensional Yetter-Drinfel’d over G and M∈XθG. We say that the matrix qiji,j∈I is the braiding matrix ofM. Recall that the matrix is independent of the basis xi∣i∈I up to permutation of I. We say BVM=B⊕i=1nkxi is the Nichols algebra of the tuple M, denoted by BM.Recall that the adjoint representationad of a Nichols algebra BV from [9] is the linear map adc:V⟶EndBV(22)adcxy=μid−cx⊗y=xy−x−1.yx0for all x∈V, y∈BV, where μ is the multiplication map of BV and c is defined by Equation (6). In particular, the braided commutator adc of BM takes the form
(23)adcxiy=xiy−gi·yxiforalli∈I,y∈BM.In order to construct a semi-Cartan graph toM, we recall some finiteness conditions from [7, 22].Definition 19.
Leti∈I. We say that M is i-finite, if for any j∈I\i, adcximxj=0 for some m∈ℕ.Lemma 20.
For anyi,j∈I with i≠j,s, the following are equivalent.
(i)
m+1qiiqiimqijqji−1=0 and k+1qiiqiikqijqji−1≠0 for all 0≤k<m(ii)
adcxim+1xj=0 and adcximxj≠0 in BV
Here,nq≔1+q+⋯+qn−1, which is 0 if and only if qn=1 for q≠1 or p∣n for q=1. Notice that 1q≠0 for any q∈ℕ.Hence, we get the following from Lemma20.Lemma 21.
Leti∈I. Then, M=kxjj∈I is i-finite if and only if for any j∈I\i there is a nonnegative integer m satisfying m+1qiiqiimqijqji−1=0.Leti∈I. Assume that M is i-finite. Let aijMj∈I∈ℤI and RiM=RiMjj∈I, where
(24)aijM=2,ifj=i,−maxm∈ℕ0∣adcximxj≠0,ifj≠i,RiMi=kyi,RiMj=kadcxi−aijMxj,whereyi∈kxi∗\0. If M is not i-finite, then let RiM=M. Then, RiM is a θ-tuple of one-dimensional Yetter-Drinfel’d modules over G.Let(25)FθGM=Ri1⋯RinM∈FθG∣n∈ℕ0,i1,⋯,in∈IXθGM=Ri1⋯RinM∈XθG∣n∈ℕ0,i1,⋯,in∈I.Definition 22.
We say thatM admits all reflections if N is i-finite for all N∈FθGM.Notice that the reflections depend only on the braiding matrixqiji,j∈I. We recall the notion of generalized Dynkin diagram for a braided vector space of diagonal type [40].Definition 23.
LetV be a θ-dimensional braided vector space of diagonal type with the braiding matrix qiji,j∈I. The generalized Dynkin diagram of V is a nondirected graph D with the following properties:
(i)
There is a bijective mapϕ from I to the vertices of D(ii)
For alli∈I, the vertex ϕi is labeled by qii(iii)
For alli,j∈I with i≠j, the number nij of edges between ϕi and ϕj is either 0 or 1. If qijqji=1, then nij=0; otherwise, nij=1 and the edge is labeled by qijqjiWe say that thegeneralized Dynkin diagram ofM is the generalized Dynkin diagram of braided vector space ⊕i∈IMi. Notice that the generalized Dynkin diagram of M is connected if the braiding matrix of M is indecomposable.In more details, one can obtain the labels of the generalized Dynkin diagram ofRiM=RiMjj∈I by the following lemma.Lemma 24.
Leti∈I. Assume that M is i-finite and let aij≔aijM for all j∈I. Let q′jkj,k∈I be the braiding matrix of RiM with respect to yjj∈I. Then,
(26)qjj′=qii,ifj=i,qjj,ifj≠i,qijqji=qiiaij,qiiqjjqijqji−aij,ifj≠i,qii∈G′1−aij,qjjqijqji−aij,ifj≠i,qii=1,qij′qji′=qijqji,ifj≠i,qijqji=qiiaij,qii2qijqji−1,ifj≠i,qii∈G′1−aij,qijqji−1,ifj≠i,qii=1,qjk′qkj′=qjkqkj,ifqirqri=qiiair,r∈j,k,qjkqkjqikqkiqii−1−aij,ifqijqji=qiiaij,qii∈G′1−aik,qjkqkjqijqji−aikqikqki−aij,ifqii=1,qjkqkjqii2qijqjiqikqki−aij,ifqii∈G′1−aik,qii∈G′1−aij,
forj≠k, j−i=1, k−i=1. For j−i>1, k−i>1, qjk′qkj′=qjkqkj. Here, Gn′ denotes the set of primitive nth roots of unity in k, that is, Gn′=q∈k∗∣qn=1,qk≠1forall1≤k<n for n∈ℕ.IfM admits all reflections, then we are able to construct a semi-Cartan graph CM of M by ([36], Proposition 1.5).Theorem 25.
Assume thatM admits all reflections. For all X∈XθGM, let Xθ=Y∈XθGMYandXhavethesamegeneralizedDynkindiagram.LetYθM=Xθ∣X∈XθGM and AXθ=AX for all X∈XθGM. Let t:I×YθM⟶YθM, i,Xθ↦RiXθ. Then, the tuple
(27)CM=I,YθM,t,AYY∈YθMis a connected semi-Cartan graph. We say that CM is the semi-Cartan graph attached to M.Furthermore, one can attach a groupoidWM≔WCM to M if M admits all reflections.Notice that Nichols algebraBM is ℕ0θ-graded with degMi=αi for all i∈I. Following the terminology in [22], we say that the Nichols algebra BM is decomposable if there exists a totally ordered index set L,≤ and a sequence Wll∈L of finite-dimensional irreducible ℕ0θ-graded objects in GGYD such that
(28)BM≃⊗l∈LBWl.For each decomposition (28), we define the set of positive rootsΔ+M⊂ℤI and the set of rootsΔM⊂ℤI of M by
(29)Δ+M=degWl∣l∈L,ΔM=Δ+M∪−Δ+M.By [22] (Theorem 4.5), we obtain that the set of roots ΔM of M does not depend on the choice of the decomposition.Remark 26.
IfdimMi=1 for all i∈I, then the Nichols algebra BM is decomposable based on the theorem of Kharchenko ([17], Theorem 2). The set of roots of Nichols algebra BM can be always defined, and it is denoted by ΔM. If the set of roots ΔM is finite, then we can check that M admits all reflections by [22] (Corollary 6.12).
IfM admits all reflections and ΔM is finite, then we can define a finite root system RMCM,ΔNN∈FθGM of type CM.Theorem 27.
Assume thatM admits all reflections. Then, the following are equivalent.
(1)
ΔM is finite(2)
CM is a finite Cartan graph(3)
WM is finite(4)
RM≔RMCM,ΔNN∈FθGM is finite
In all cases,RM is the unique root system of type CM.Proof.
SinceM is a θ-tuple of one-dimensional Yetter-Drinfel’d modules, we obtain that the Nichols algebra BM generated by VM is decomposable, and hence, ΔM is defined. Then, RM=RMCM,ΔNN∈FθGM is the root system of type CM by [22] (Theorem 6.11). Hence, the claim is true by Lemma 14.☐


## 2.1. Yetter-Drinfel’d Modules

Letk be a field of characteristic p>0. Let k∗=k\0, ℕ0=ℕ∪0, θ∈ℕ0, and I=1,⋯,θ. We start by recalling the main objects.Definition 2.
LetV be a θ-dimensional vector space over k. The pair V,c is called a braided vector space, if c∈AutV⊗V is a solution of the braid equation, that is,
(3)c⊗idid⊗cc⊗id=id⊗cc⊗idid⊗c.
A braided vector spaceV,c is termed of diagonal type if V admits a basis xi∣i∈I such that for all i,j∈I one has
(4)cxi⊗xj=qijxj⊗xiforsomeqij∈k∗.
The matrixqiji,j∈I is termed of the braiding matrix of V. We say that the braiding matrix qiji,j∈I is indecomposable if for any i≠j there exists a sequence i1=i,i2,⋯,it=j, of elements of I such that qisis+1qis+1is≠1, where 1≤s≤t−1.
In this paper, we mainly concern the braided vector spaces of diagonal type with an indecomposable braiding matrix.Definition 3.
LetH be a Hopf algebra. A Yetter-Drinfel’d module V over H is a left H-module with left action: H⊗V⟶V and a left H-comodule with left coaction δL:V⟶H⊗V satisfying the compatibility condition
(5)δLh·v=h1v−1κh3⊗h2·v0,h∈H,v∈V.
We say thatV is of diagonal type if H=k’G and V is a direct sum of one-dimensional Yetter-Drinfel’d modules over the group algebra k’G, where G is abelian.
We denote byHHYD the category of Yetter-Drinfel’d modules over H, where morphisms preserve both the action and the coaction of H. The category HHYD is braided with braiding
(6)cV,Wv⊗w=v−1·w⊗v0
for allV,W∈YHHD, v∈V, and w∈W. Actually, the category YHHD ia a braided monoidal category, where the monoidal structure is given by the tensor product over k. Then, any Yetter-Drinfel’d module V∈HHYD over H admits a braiding cV,V, and hence, V,cV,V is a braided vector space. Conversely, any braided vector space can be realized as a Yetter-Drinfel’d module over some Hopf algebras if and only if the braiding is rigid ([38], Section 2). Notice that Yetter-Drinfel’d module structures on V with different Hopf algebras can give the same braiding and not all braidings of V are induced by the above Equation (1).
IfH=k’G, then we write YGGD for the category of Yetter-Drinfel’d modules over k’G and say that V∈YHHD is a Yetter-Drinfel’d module over G. Notice that if V∈YGGD is of diagonal type, then V,cV,V is a braided vector space of diagonal type. Any braided vector space of diagonal type is also a Yetter-Drinfel’d module of diagonal type. Indeed, assume that V,c is a braided vector space of diagonal type with an indecomposable braiding matrix qiji,j∈I of a basis xi∣i∈I. Let G0 be a free abelian group generated by elements gi∣i∈I. Define the left coaction and left action by
(7)δLxi=gi⊗xi∈G0⊗V,gi·xj=qijxj∈V.Then,V=⊕i∈Ikxi, and each kxi is one-dimensional Yetter-Drinfel’d modules over G0. Hence, V is a Yetter-Drinfel’d module of diagonal type over G0.


## 2.2. Nichols Algebras of Diagonal Type

LetV,c be a θ-dimensional braided vector space of diagonal type. In this section, we recall a definition of the Nichols algebra BV generated by V,c. In order to do that, we introduce one more notion in the category YHHD.Definition 4.
LetH be a Hopf algebra. A braided Hopf algebra in HHYD is a 6-tuple B=B,μ,1,δ,ε,κB, where B,μ,1 is an algebra in YHHD and a coalgebra B,δ,ε in YHHD, and κB:B⟶B is a morphism in YHHD such that δ, ε, and κB satisfy κBb1b2=b1κBb2=εb1, where we define δb=b1⊗b2 as the coproduct of B to avoid the confusion.Definition 5.
The Nichols algebra generated byV∈YHHD is defined as the quotient
(8)BV=TV/IV=⊕n=0∞V⊗n/IV,where IV is the unique maximal coideal among all coideals of TV which are contained in ⊕n≥2V⊗n. Nichols algebra BV is said to be of diagonal type if V is a Yetter-Drinfel’d module of diagonal type. The dimension of V is the rank of Nichols algebra BV.Note that ifB∈YHHD and B is an algebra in YHHD, then B⊗B is an algebra in HHYD with the product given by
(9)a⊗bc⊗d=ab−1·c⊗b0d,for alla,b,c,d∈B, where · denotes the left action of H on B.Thetensor algebraTV admits a natural structure of a Yetter-Drinfel’d module and an algebra structure in YHHD. It is then a braided Hopf algebra in YHHD with coproduct δv=1⊗v+v⊗1∈TV⊗TV and counit εv=0 for all v∈V such that δ and ε are the algebra morphisms. The antipode of TV exists (see [7], Section 2.1). Notice that the product defined by Equation (9) on TV is the unique algebra structure such that δv=1⊗v+v⊗1∈TV⊗TV for all v∈V. The coproduct can be extended from V to TV. For example, for all v,w∈V, one gets (we write the elements of TV without the tensor product sign for brevity)
(10)δvw=δvδw=1⊗v+v⊗11⊗w+w⊗1=1⊗vw+v−1·w⊗v0+v⊗w+vw⊗1.LetIii∈I be the family of all ideals of TV contained in ⊕n≥2V⊗n, i.e.,
(11)δIi⊂Ii⊗TV+TV⊗Ii.Then, the coidealIV≔∑i∈IIi is the largest element in Iii∈I. Hence, BV is a braided Hopf algebra in YHHD. As proved in [6] (Proposition 3.2.12), Nichols algebra BV is the unique ℕ0-graded braided Hopf algebra generated by V in YHHD with homogenous components BV0=k, BV1=V, and PBV=V, where PBV is the space of primitive elements of BV.


## 2.3. Weyl Groupoids

In this section, we recall the notations of semi-Cartan graphs, root systems, and Weyl groupoids. We mainly follow the terminology from [20, 39] (see also [35–37]).Definition 6.
A generalized Cartan matrix is a matrixA=aiji,j∈I with integer entries such that
(i)
aii=2 and ajk≤0 for any i,j,k∈I with j≠k(ii)
ifaij=0 for some i,j∈I, then aji=0A generalized Cartan matrixA∈ℤI×I is decomposable if there exists a nonempty proper subset I1⊂I such that aij=0 for any i∈I1 and j∈I\I1. We say that A is indecomposable if A is not decomposable.Definition 7.
LetX be a nonempty set and AX=aijXi,j∈I a generalized Cartan matrix for all X∈X. For any i∈I, letri:X⟶X, X↦ri,X, where r:I×X⟶X is a map. The quadruple
(12)C=CI,X,r,AXX∈Xis called a semi-Cartan graph if ri2=idX for all i∈I, and aijX=aijriX for all X∈X and i,j∈I. We say that a semi-Cartan graph C is indecomposable if AX is indecomposable for all X∈X.For the sake of simplicity, the elements of the setriX,i∈I are termed the neighbors of X for all X∈X. The cardinality of I is termed the rank of C, and the elements of X are the points of C.Definition 8.
A semi-Cartan graphC is standard, if AX=AY for all X,Y∈ObWC.Definition 9.
The exchange graph ofC is a labeled nonoriented graph with vertices set X and edges set I, where two vertices X,Y are connected by an edge i if and only if X≠Y and riX=Y (and riY=X). We display one edge with several labels instead of several edges for simplification. A semi-Cartan graph C is said to be connected if its exchange graph is connected.
For the remaining part of this section, we assume thatC=CI,X,r,AXX∈X is a connected semi-Cartan graph. Let αii∈I be the standard basis of ℤI. For all X∈X, let
(13)siX∈AutℤI,siXαj=αj−aijXαi,
for allj∈I. Let DX,I be the category such that ObDX,I=X and morphisms HomX,Y=Y,f,X∣f∈EndℤI for X,Y∈X with the composition Z,g,Y∘Y,f,X=Z,gf,X for all X,Y,Z∈X, f,g∈EndℤI. Let WC be the smallest subcategory of DX,I, where the morphisms are generated by riX,siX,X, with i∈I, X∈X. From now on, we write siX instead of riX,siX,X, if no confusion is possible. Notice that all generators siX are reflections and hence are invertible. Then, WC is a groupoid.
For any categoryD and any object X in D, let HomD,X=∪Y∈DHomY,X.Definition 10.
For allX∈X, the set
(14)ΔXre=ωαi∈ℤI∣ω∈HomWC,X
is called the set of real roots ofC at X. The elements of Δ+Xre=ΔXre∩ℕ0I are called positive roots and those of ΔXre∩−ℕ0I negative roots, denoted by Δ−Xre. If the set ΔXre is finite for all X∈X, then we say that C is finite.Definition 11.
We say thatR=RC,ΔXX∈X is a root system of type C if for all X∈X, the sets ΔX are the subsets of ℤI such that
(15)ΔX=ΔX∩ℕ0I∪−ΔX∩ℕ0I.(i)
ΔX∩ℤαi=αi,−αi for all i∈I(ii)
siXΔX=ΔriX for all i∈I(iii)
rirjmijXX=X for any i,j∈I with i≠j where mijX=∣ΔX∩ℕ0αi+ℕ0αj∣ is finite
We say thatWR≔WC is the groupoid of R. As in [13] (Definition 4.3), we say that R is reducible if there exist nonempty disjoint subsets of I′,I′′⊂I such that I=I′∪I′′ and aij=0 for all i∈I′, j∈I′′ and
(16)ΔX=ΔX∩∑i∈I′ℤαi∪ΔX∩∑j∈I′′ℤαjforallX∈X.
In this case, we writeR=RI1⊕RI2. If R≠RI1⊕RI2 for all nonempty disjoint subsets I1,I2⊂I, then R is termed irreducible.Definition 12.
LetR=RC,ΔXX∈X be a root system of type C. We say that R is finite if ΔX is finite for all X∈X.LetR=RC,ΔXX∈X be a root system of type C. We recall some properties of R from [19, 24].Lemma 13.
LetX∈X, k∈ℤ, and i,j∈I such that i≠j. Then αj+kαi∈ΔXre if and only if 0≤k≤−aijX.Notice that the finiteness ofR does not mean that WR is also finite, since the set X might be infinite.Lemma 14.
LetC=CI,X,r,AXX∈X be a connected semi-Cartan graph and R=RC,ΔXX∈X be a root system of type C. Then, the following are equivalent.
(1)
R is finite(2)
ΔX is finite for some X∈X(3)
C is finite(4)
WR is finiteRecall thatC is a connected semi-Cartan graph. Then, we get the following.Proposition 15.
LetR=RC,ΔXX∈X be a root system of type C. Then, the following are equivalent.
(1)
There existsX∈X such that AX is indecomposable(2)
The semi-Cartan graphC is indecomposable
IfR is finite, then the semi-Cartan graph C is indecomposable if and only if the root system R is irreducible.Definition 16.
We sayC is a Cartan graph if the following hold:
(i)
For allX∈X, the set ΔXre=Δ+Xre∪Δ−Xre(ii)
IflmnY≔ΔYre∩ℕ0αm+ℕ0αn is finite, then rmrnlmnYY=Y, where m,n∈I, Y∈X
In this case,WC is called the Weyl groupoid of C.LetRre≔RC,ΔXreX∈X. Then, C is a Cartan graph if and only if Rre is a root system of type C. Indeed, we get that siXΔXre=ΔriXre by Equation (14). For all X∈X, we obtain that ΔXre=Δ+Xre∪Δ−Xre, since ωsiriXαi=−ωαi for any ω∈HomWC,X.The following proposition implies that ifR is a finite root system of type C, then R=Rre; namely, all roots are real and R is uniquely determined by C.Proposition 17.
LetR=RC,ΔXX∈X be a root system of type C. Let X∈X, m∈ℕ0, and i1,⋯,im∈I such that
(17)ω=idXsi1si2⋯sim∈HomWC,Xand ℓω=m. Then, the elements
(18)βn=idXsi1si2⋯sin−1αin∈ΔX∩ℕ0Iare pairwise different, where n∈1,2,⋯,m (and β1=αi1). Here,
(19)ℓω=minm∈ℕ0∣ω=idXsi1si2⋯sim,i1,i2,⋯,im∈Iis the length of ω∈HomWC,X. In particular, if R is finite and ω∈HomWC is the longest element, then
(20)βn∣1≤n≤ℓω=ΔX2=ΔX∩ℕ0I.Remark 18.
IfC is a finite Cartan graph, then R is finite, and hence,
(21)Rre=RreC,ΔXreX∈Xis the unique root system of type C by Proposition 17; that is, R is uniquely determined by C.


## 2.4. Cartan Graphs for Nichols Algebras of Diagonal Type

In this section, we attach a semi-Cartan graph to a tuple of finite-dimensional Yetter-Drinfel’d modules under some finiteness conditions. LetG be an abelian group. Let FθG be the set of θ-tuples of finite-dimensional irreducible objects in YGGD and XθG be the set of θ-tuples of isomorphism classes of finite-dimensional irreducible objects in GGYD. For any M1,⋯,Mθ∈FθG, write M≔M1,⋯,Mθ∈XθG the corresponding isomorphism class of M1,⋯,Mθ.Assume thatVM=⊕i∈Ikxi∈GGYD is a Yetter-Drinfel’d module of diagonal type over G, where xi∣i∈I is a basis of V. Then, there exists a matrix qiji,j∈I such that δxi=gi⊗xi and gi·xj=qijxj for all i,j∈I. We fix that M=kx1,kx2,⋯,kxθ∈FθG is a tuple of one-dimensional Yetter-Drinfel’d over G and M∈XθG. We say that the matrix qiji,j∈I is the braiding matrix ofM. Recall that the matrix is independent of the basis xi∣i∈I up to permutation of I. We say BVM=B⊕i=1nkxi is the Nichols algebra of the tuple M, denoted by BM.Recall that the adjoint representationad of a Nichols algebra BV from [9] is the linear map adc:V⟶EndBV(22)adcxy=μid−cx⊗y=xy−x−1.yx0for all x∈V, y∈BV, where μ is the multiplication map of BV and c is defined by Equation (6). In particular, the braided commutator adc of BM takes the form
(23)adcxiy=xiy−gi·yxiforalli∈I,y∈BM.In order to construct a semi-Cartan graph toM, we recall some finiteness conditions from [7, 22].Definition 19.
Leti∈I. We say that M is i-finite, if for any j∈I\i, adcximxj=0 for some m∈ℕ.Lemma 20.
For anyi,j∈I with i≠j,s, the following are equivalent.
(i)
m+1qiiqiimqijqji−1=0 and k+1qiiqiikqijqji−1≠0 for all 0≤k<m(ii)
adcxim+1xj=0 and adcximxj≠0 in BV
Here,nq≔1+q+⋯+qn−1, which is 0 if and only if qn=1 for q≠1 or p∣n for q=1. Notice that 1q≠0 for any q∈ℕ.Hence, we get the following from Lemma20.Lemma 21.
Leti∈I. Then, M=kxjj∈I is i-finite if and only if for any j∈I\i there is a nonnegative integer m satisfying m+1qiiqiimqijqji−1=0.Leti∈I. Assume that M is i-finite. Let aijMj∈I∈ℤI and RiM=RiMjj∈I, where
(24)aijM=2,ifj=i,−maxm∈ℕ0∣adcximxj≠0,ifj≠i,RiMi=kyi,RiMj=kadcxi−aijMxj,whereyi∈kxi∗\0. If M is not i-finite, then let RiM=M. Then, RiM is a θ-tuple of one-dimensional Yetter-Drinfel’d modules over G.Let(25)FθGM=Ri1⋯RinM∈FθG∣n∈ℕ0,i1,⋯,in∈IXθGM=Ri1⋯RinM∈XθG∣n∈ℕ0,i1,⋯,in∈I.Definition 22.
We say thatM admits all reflections if N is i-finite for all N∈FθGM.Notice that the reflections depend only on the braiding matrixqiji,j∈I. We recall the notion of generalized Dynkin diagram for a braided vector space of diagonal type [40].Definition 23.
LetV be a θ-dimensional braided vector space of diagonal type with the braiding matrix qiji,j∈I. The generalized Dynkin diagram of V is a nondirected graph D with the following properties:
(i)
There is a bijective mapϕ from I to the vertices of D(ii)
For alli∈I, the vertex ϕi is labeled by qii(iii)
For alli,j∈I with i≠j, the number nij of edges between ϕi and ϕj is either 0 or 1. If qijqji=1, then nij=0; otherwise, nij=1 and the edge is labeled by qijqjiWe say that thegeneralized Dynkin diagram ofM is the generalized Dynkin diagram of braided vector space ⊕i∈IMi. Notice that the generalized Dynkin diagram of M is connected if the braiding matrix of M is indecomposable.In more details, one can obtain the labels of the generalized Dynkin diagram ofRiM=RiMjj∈I by the following lemma.Lemma 24.
Leti∈I. Assume that M is i-finite and let aij≔aijM for all j∈I. Let q′jkj,k∈I be the braiding matrix of RiM with respect to yjj∈I. Then,
(26)qjj′=qii,ifj=i,qjj,ifj≠i,qijqji=qiiaij,qiiqjjqijqji−aij,ifj≠i,qii∈G′1−aij,qjjqijqji−aij,ifj≠i,qii=1,qij′qji′=qijqji,ifj≠i,qijqji=qiiaij,qii2qijqji−1,ifj≠i,qii∈G′1−aij,qijqji−1,ifj≠i,qii=1,qjk′qkj′=qjkqkj,ifqirqri=qiiair,r∈j,k,qjkqkjqikqkiqii−1−aij,ifqijqji=qiiaij,qii∈G′1−aik,qjkqkjqijqji−aikqikqki−aij,ifqii=1,qjkqkjqii2qijqjiqikqki−aij,ifqii∈G′1−aik,qii∈G′1−aij,
forj≠k, j−i=1, k−i=1. For j−i>1, k−i>1, qjk′qkj′=qjkqkj. Here, Gn′ denotes the set of primitive nth roots of unity in k, that is, Gn′=q∈k∗∣qn=1,qk≠1forall1≤k<n for n∈ℕ.IfM admits all reflections, then we are able to construct a semi-Cartan graph CM of M by ([36], Proposition 1.5).Theorem 25.
Assume thatM admits all reflections. For all X∈XθGM, let Xθ=Y∈XθGMYandXhavethesamegeneralizedDynkindiagram.LetYθM=Xθ∣X∈XθGM and AXθ=AX for all X∈XθGM. Let t:I×YθM⟶YθM, i,Xθ↦RiXθ. Then, the tuple
(27)CM=I,YθM,t,AYY∈YθMis a connected semi-Cartan graph. We say that CM is the semi-Cartan graph attached to M.Furthermore, one can attach a groupoidWM≔WCM to M if M admits all reflections.Notice that Nichols algebraBM is ℕ0θ-graded with degMi=αi for all i∈I. Following the terminology in [22], we say that the Nichols algebra BM is decomposable if there exists a totally ordered index set L,≤ and a sequence Wll∈L of finite-dimensional irreducible ℕ0θ-graded objects in GGYD such that
(28)BM≃⊗l∈LBWl.For each decomposition (28), we define the set of positive rootsΔ+M⊂ℤI and the set of rootsΔM⊂ℤI of M by
(29)Δ+M=degWl∣l∈L,ΔM=Δ+M∪−Δ+M.By [22] (Theorem 4.5), we obtain that the set of roots ΔM of M does not depend on the choice of the decomposition.Remark 26.
IfdimMi=1 for all i∈I, then the Nichols algebra BM is decomposable based on the theorem of Kharchenko ([17], Theorem 2). The set of roots of Nichols algebra BM can be always defined, and it is denoted by ΔM. If the set of roots ΔM is finite, then we can check that M admits all reflections by [22] (Corollary 6.12).
IfM admits all reflections and ΔM is finite, then we can define a finite root system RMCM,ΔNN∈FθGM of type CM.Theorem 27.
Assume thatM admits all reflections. Then, the following are equivalent.
(1)
ΔM is finite(2)
CM is a finite Cartan graph(3)
WM is finite(4)
RM≔RMCM,ΔNN∈FθGM is finite
In all cases,RM is the unique root system of type CM.Proof.
SinceM is a θ-tuple of one-dimensional Yetter-Drinfel’d modules, we obtain that the Nichols algebra BM generated by VM is decomposable, and hence, ΔM is defined. Then, RM=RMCM,ΔNN∈FθGM is the root system of type CM by [22] (Theorem 6.11). Hence, the claim is true by Lemma 14.☐


## 3. Calculation of Finite Cartan Graphs

In this section, we obtain the properties of finite connected indecomposable Cartan graphs attached to Nichols algebras of diagonal type with a finite root system and present the results in Theorem39.Our algorithms described below are sufficiently powerful in finding the properties, which are extremely essential for our classification. The implementation in Shell terminates within a few seconds on a usual computer.


### 3.1. The Idea

For convenience, we copy the irreducible root systems of rank 6 in [25] (Appendix B) to Appendix 5.This is an overview of the main algorithm without any details.


### 3.2. Some Remarks

Remark 29.
Before starting the algorithms, we give several remarks on Algorithm28.
(1)
In step 1, we restrict the input root systems to Appendix 5. In details, since Cuntz and Heckenberger illustrate all the finite Weyl groupoids in [25], the finite root system of Nichols algebras of diagonal type must be contained in [25] (Cor. 3.13 Appendix B). The classification of Nichols algebras of diagonal type with a finite root system which appears in [25] (Cor. 3.13) is not difficult to achieve(2)
From steps 1 to 3, we obtain all the possible reflected root systems of the given root systemrs. By Lemma 14, the Weyl groupoid is finite. Hence, in step 3, the implementation of the reflections terminates(3)
From Table1 of Appendix A in [25], for each root system rs, there is an object such that the generalized Cartan matrix is of type A6. Hence, for simplification, we collect all objects with generalized Cartan matrixes of type A6 in step 4 of the algorithmAlgorithm 28:GoodNeighborhoods(rs).
Check the neighborhoods of root systems which have Cartan matrix of typeA6.Input: all root systems in Appendix 5.Output: All the A6 neighborhoods1. Copy the data in Appendix 5 to setrs and allrs.2. CallSortrs,cmpvect.3. CallReflectionrs,i, for i∈1,⋯,6 and sort all the reflected root systems. Check if the reflections yield new root systems. If yes, then sort them and add to the set allrs. Repeat above to all new found root systems and update the set allrs until the reflections yield no new root systems.4. Callcmallrsi, for i=1,⋯,Lengthallrs. Check if its Cartan matrix is of type A6 by calling IsTypeA6(cm(allrs[i])). Collect root systems which have Cartan matrix of finite type of A6 to set “A6roots”.5. Reflect all rank 6 root systems in set “A6roots” in 6 directions and remember all the positions of the reflected root systems in allrs.6. Return part of the Exchange graph including all objects with Cartan matrix of typeA6 and the Cartan matrixes of the reflected root systems.Table 1
Finite dimensional Nichols algebra of diagonal type overk (p≠3).
XGener. Dynkin diagramsFixed parametersX1ζ∈G′3X2X3X4X5X6X7


### 3.3. The Algorithms

The fundamental function in Algorithm28 is the function “Sort”:By Algorithm30, every root system could be sorted under the same given partial order “cmpvect.” Then, comparing and determining whether two root systems are equal are possible.Algorithm 30:Sort(rs,cmpvect).
Sort the input root system with the given partial order.Input: Every rank 6 finite root system in Appendix 5 and function “cmpvect”.Output: An ordered root system, where the order is given by function “cmpvect”.1. The function “cmpvect”: given two different rootsβi,βj in rs, the order of the two roots depends on the natural order of the two integers in the first place k where two roots differ (counting from the beginning of the roots): βi<βj if and only if βik<βjk. Return True, if βi<βj. Else, return False.2. For roots inrs, sort them with the partial order given by “cmpvect” in Step 1. Update rs and return.As a result, we could append new root systems to a container with an order, which is essential in the following function.Remark 32.
The reflection in step 2 of Algorithm31 is given in Equation (13).Algorithm 31:Reflection(rs,i).
Reflect a root system.Input: A root system and a direction.Output: Reflected root systems.1. Input a root systemrs in Appendix 5 and direction i, where i∈1,⋯,6.2. Calculate the matrix ofi-th reflection with respect to given standard basis αk,k∈1,⋯6, and denote it as si.3. For every positive rootα in the given root system rs, calculate si·α.4. Construct a setN consisting of αi and si·α, for all α in the given root system rs.5. ReturnN.Algorithm 33:cm(rs).
Calculate the generalized Cartan matrix of a root system.Input: A root system.Output: The generalized Cartan matrix of the root system.1. Input a root systemrs in Appendix 5.2. Create a matrixcm with entries aij, where aii=2, aij=−maxk∣kαi+αj∈rs, for i,j∈1,⋯,6 and i≠j.3. Returncm.Remark 34.
The entries of the generalized Cartan matrix of the given root system are defined in Lemma13.Finally, we still need a function to check which of the root system is indeed type ofA6.Algorithm 35:IsTypeA6(cm).
Check whether a matrix is type ofA6.Input: The generalized Cartan matrix of a root system.Output: True or False.1. Construct a matrixA6=2−10000−12−10000−12−10000−12−10000−12−10000−12.2. If there is a permutation of1,⋯,6 satisfying the input matrix cm is of type A6, then return True. Otherwise, return False.


## 3.1. The Idea

For convenience, we copy the irreducible root systems of rank 6 in [25] (Appendix B) to Appendix 5.This is an overview of the main algorithm without any details.


## 3.2. Some Remarks

Remark 29.
Before starting the algorithms, we give several remarks on Algorithm28.
(1)
In step 1, we restrict the input root systems to Appendix 5. In details, since Cuntz and Heckenberger illustrate all the finite Weyl groupoids in [25], the finite root system of Nichols algebras of diagonal type must be contained in [25] (Cor. 3.13 Appendix B). The classification of Nichols algebras of diagonal type with a finite root system which appears in [25] (Cor. 3.13) is not difficult to achieve(2)
From steps 1 to 3, we obtain all the possible reflected root systems of the given root systemrs. By Lemma 14, the Weyl groupoid is finite. Hence, in step 3, the implementation of the reflections terminates(3)
From Table1 of Appendix A in [25], for each root system rs, there is an object such that the generalized Cartan matrix is of type A6. Hence, for simplification, we collect all objects with generalized Cartan matrixes of type A6 in step 4 of the algorithmAlgorithm 28:GoodNeighborhoods(rs).
Check the neighborhoods of root systems which have Cartan matrix of typeA6.Input: all root systems in Appendix 5.Output: All the A6 neighborhoods1. Copy the data in Appendix 5 to setrs and allrs.2. CallSortrs,cmpvect.3. CallReflectionrs,i, for i∈1,⋯,6 and sort all the reflected root systems. Check if the reflections yield new root systems. If yes, then sort them and add to the set allrs. Repeat above to all new found root systems and update the set allrs until the reflections yield no new root systems.4. Callcmallrsi, for i=1,⋯,Lengthallrs. Check if its Cartan matrix is of type A6 by calling IsTypeA6(cm(allrs[i])). Collect root systems which have Cartan matrix of finite type of A6 to set “A6roots”.5. Reflect all rank 6 root systems in set “A6roots” in 6 directions and remember all the positions of the reflected root systems in allrs.6. Return part of the Exchange graph including all objects with Cartan matrix of typeA6 and the Cartan matrixes of the reflected root systems.Table 1
Finite dimensional Nichols algebra of diagonal type overk (p≠3).
XGener. Dynkin diagramsFixed parametersX1ζ∈G′3X2X3X4X5X6X7


## 3.3. The Algorithms

The fundamental function in Algorithm28 is the function “Sort”:By Algorithm30, every root system could be sorted under the same given partial order “cmpvect.” Then, comparing and determining whether two root systems are equal are possible.Algorithm 30:Sort(rs,cmpvect).
Sort the input root system with the given partial order.Input: Every rank 6 finite root system in Appendix 5 and function “cmpvect”.Output: An ordered root system, where the order is given by function “cmpvect”.1. The function “cmpvect”: given two different rootsβi,βj in rs, the order of the two roots depends on the natural order of the two integers in the first place k where two roots differ (counting from the beginning of the roots): βi<βj if and only if βik<βjk. Return True, if βi<βj. Else, return False.2. For roots inrs, sort them with the partial order given by “cmpvect” in Step 1. Update rs and return.As a result, we could append new root systems to a container with an order, which is essential in the following function.Remark 32.
The reflection in step 2 of Algorithm31 is given in Equation (13).Algorithm 31:Reflection(rs,i).
Reflect a root system.Input: A root system and a direction.Output: Reflected root systems.1. Input a root systemrs in Appendix 5 and direction i, where i∈1,⋯,6.2. Calculate the matrix ofi-th reflection with respect to given standard basis αk,k∈1,⋯6, and denote it as si.3. For every positive rootα in the given root system rs, calculate si·α.4. Construct a setN consisting of αi and si·α, for all α in the given root system rs.5. ReturnN.Algorithm 33:cm(rs).
Calculate the generalized Cartan matrix of a root system.Input: A root system.Output: The generalized Cartan matrix of the root system.1. Input a root systemrs in Appendix 5.2. Create a matrixcm with entries aij, where aii=2, aij=−maxk∣kαi+αj∈rs, for i,j∈1,⋯,6 and i≠j.3. Returncm.Remark 34.
The entries of the generalized Cartan matrix of the given root system are defined in Lemma13.Finally, we still need a function to check which of the root system is indeed type ofA6.Algorithm 35:IsTypeA6(cm).
Check whether a matrix is type ofA6.Input: The generalized Cartan matrix of a root system.Output: True or False.1. Construct a matrixA6=2−10000−12−10000−12−10000−12−10000−12−10000−12.2. If there is a permutation of1,⋯,6 satisfying the input matrix cm is of type A6, then return True. Otherwise, return False.


## 4. Classification Theorem of Finite Cartan Graphs

LetM≔kx1,⋯,kx6 be a tuple of one-dimensional Yetter-Drinfel’d modules over kG. Assume that M admits all reflections and CM=CI,X,r,AXX∈X be the attached indecomposable semi-Cartan graph to M. Assume that ΔM is the set of roots of BM. If ΔM is finite, then CM is a finite Cartan graph by Theorem 27.Theorem 36.
AssumeΔM is finite, then there exists a point X∈X satisfying that the set Δ+Xre appears in [25] (Corollary 3.13) or in one of the sets listed in Appendix.Proof.
IfCM is equivalent to a Cartan graph in [15] (Corollary 3.13), then the claim is true. Otherwise, CM has a unique finite root system by Theorem 27, say RM. Assume that R=RC,ΔXreX∈X is the unique root system, where ΔXre is the real roots of X. Moreover, the root system RM is irreducible by Proposition 15. For any X∈X, let Δ+Xre be the positive roots of X. By [25] (Theorem 40), there exists a point X∈X satisfying that the set Δ+Xre is in the list of Appendix or in [25] (Corollary 3.13) up to a permutation of I. There are precisely 4 such possible sets of real roots for the rank 6 case.☐The aim of this section is to classify all the finite Cartan graphsCM. The output of Algorithm 28 is crucial for the classification of Theorem 39.In general, the points of a finite Cartan graphC could have many different neighborhoods. Depending on the output, we construct the following “good A6 neighborhoods” in order to cover all the finite connected indecomposable Cartan graphs in such way that at least one point of C has one of the good neighborhoods.Definition 37.
We say thatX has a good A6 neighborhood if there exists a permutation of I and an integer a∈ℕ such that
(30)AX=Ar1X=A6,AX=Ar2X=A6,AX=Ar3X=A6,Ar4X=2−10000−12−10000−12−1−1000−12−1000−1−12−10000−12,Ar5X=2−10000−12−10000−12−10000−12−1a000−12−1000a−12,
andAX=Ar6X=A6, where a satisfies one of the following.
(1)
a=0, a36r5r4X=0,a32r5r4X=−1, a25r3r4X=0, a43r5r6X=−1(2)
a=−1, a36r5r4X=0, a25r3r4X=0Remark 38.
The conditions listed in Definition37 are possibly not enough to do the classification. In the classification of finite dimensional Nichols algebras of diagonal type over fields of positive characteristic, we need to add more restrictions about the points of the Cartan graph CM. For more explanation about how it works, we can check a demonstration in Section 5.Theorem 39.
Assume there exists a pointX∈X satisfying that the set Δ+Xre appears in Appendix 5 (except E6 type), then up to equivalence, there exists a point Y∈X such that Y has one of the good A6 neighborhoods.Proof.
IfΔM appears in Appendix A (except E6 type), then there are precisely 3 possible sets of real roots for the rank 6 case. We analyze each set of the real roots in the list. Since the reflection siX maps Δ+Xre\αi bijectively to Δ+riXre\αi for any i∈I, the Cartan matrices of all neighbors of X can be obtained from Δ+Xre by Lemma 13. If there exists a point Y of the Cartan graph CM which has a good A6 neighborhood, then the claim is true. Otherwise, repeat the previous step to the neighbors of X. Since X is finite, this algorithm terminates. The elementary calculations are done by GAP Algorithm 1.☐


## 5. Application

In this section, we present a case of finite dimensional rank 6 Nichols algebras of diagonal type over fields of positive characteristics. We illustrate them with the corresponding generalized Dynkin diagrams and give the exchange graph of the Cartan graph attached to the Nichols algebras.Theorem 40.
Letk be a field of characteristic p>0. Let I=1,2,3,4,5,6. Let V,c be a braided vector space of diagonal type over k with basis xk∣k∈I satisfying
(31)cxi⊗xj=qijxj⊗xiforsomeqij∈k∗.
Assume thatqiji,j∈I is an indecomposable braiding matrix. Let M≔kxii∈I. Assume that the Nichols algebra BV generated by V,c has a finite set of roots ΔM listed in Appendix 5 (except E6 type). Then the generalized Dynkin diagrams D of V partially appear in Table 1. In this case, the row of Table 1 containing D consists precisely of the generalized Dynkin diagrams of all the points of CM. Figure 1 is the exchange graph of the corresponding Cartan graph CM.Remark 41.
In the part of the proof, we give the following statement to avoid confusion.(1)
Label the vertices of the generalized Dynkin diagrams from left to right and then top to bottom by1,⋯,6 (for example: label D6 as in Figure 2)(2)
For a generalized Dynkin diagramD and i,j,k,l,m,n∈I, write τijklmnD for the graph D where the vertices of D change to i, j, k, l, m, n, respectivelyProof.
Assume thatBV has a finite set of roots ΔM. Let X=M6s, I=1,2,3,4,5,6, and AX≔aiji,j∈I be the Cartan matrix of X. Let qi,ji,j∈I be the braiding matrix of X and qi,jriXi,j∈I be the braiding matrix of riX. To simplify the labels, we write qij′≔qijqji for 1≤i,j≤6. Since BV has a finite set of roots ΔM, we obtain that CM is a finite Cartan graph by Theorem 27; we are free to assume that there exists a point X such that AX has a good A6 neighborhood by Theorem 39.☐Figure 1
Exchange graph ofCM.Figure 2
Generalized Dynkin diagram of typeD6.Consider the case thatq44=−1 and qiiqi,i+1′−1=qjjqj−1,j′−1=0 for all i∈1,2,3,5 and j∈2,3,5,6. Let a≔a46r5X. From Lemma 24, we get Ar5X=A6 and then a=0. Let q≔q′23 and r≔q′34. Then the condition where q=r=−1 does not hold.SinceX has a good A6 neighborhood, we obtain qr≠1. Then, we get 4th direction reflection
(32)According toa36r5r4X=0 by Definition 37 (6), then qr2=1. Hence, we get that
(33)The conditiona32r5r4X=−1 implies that rq=−1 or q2r=1 by Lemma 24. If rq=−1 then r=−1,q=1, which is a contradiction. If q2r=1, then q=r and q3=1. Hence,
(34)whereζ∈G′3. We denote it as X1 in Table 1. Doing all the reflections of X1, we obtain the generalized Dynkin diagrams X2, X3, X4, X5, X6, and X7 in Table 1. And the corresponding exchange graph of CM is Figure 1.



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