Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classi ﬁ cation 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 classi ﬁ cation of Nichols algebras of diagonal type with a ﬁ nite 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 ﬁ nite dimensional Nichols algebras of diagonal type.

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 such that H 0 is a Hopf subalgebra of H and the associated graded coalgebra Then, grH is a graded Hopf algebra, since the coradical H 0 of H is a Hopf subalgebra. In addition, consider a projec-tion π : grH ⟶ H 0 ; 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 H 0 : grH ≃ R#H 0 . 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 H 0 -coinvariants of H 1 /H 0 , namely, Nichols algebra [9] B ðVÞ 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][12][13][14][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 BðVÞ 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][22][23]. Further, all finite Weyl groupoids were classified in [24,25]. Those constructions are very important theoretical tools for the classification of Nichols algebra BðVÞ.
The crucial step to classify pointed Hopf algebras is to determine that all Nichols algebras BðVÞ is finite dimensional. Andruskiewitsch stated the following question.
Question 1 (Andruskiewitsch [5]). Given a braiding matrix ðq ij Þ 1≤i,j≤θ whose entries are roots of 1, when BðVÞ is finite-dimensional, where V is a vector space with basis x 1 , ⋯, x θ and braiding cðx i ⊗ x j Þ = q ij ðx j ⊗ x i Þ? If so, compute dim k BðVÞ, 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][28][29][30], that is, when V is the direct sum of 1-dimensional Yetter-Drinfel'd modules over H 0 . 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][36][37].
In this paper, we illustrate the main properties of the finite Cartan graphs of rank 6 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][36][37] are followed, and several results from these papers will be used.

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].
Definition 2. Let V be a θ-dimensional vector space over k. The pair ðV, cÞ is called a braided vector space, if c ∈ AutðV ⊗ VÞ is a solution of the braid equation, that is, A braided vector space ðV, cÞ is termed of diagonal type if V admits a basis fx i | i ∈ Ig such that for all i, j ∈ I one has The matrix ðq ij Þ i,j∈I is termed of the braiding matrix of V. We say that the braiding matrix ðq ij Þ i,j∈I is indecomposable if for any i ≠ j there exists a sequence i 1 In this paper, we mainly concern the braided vector spaces of diagonal type with an indecomposable braiding matrix.
We say that V 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 for all V, W ∈ H H YD, v ∈ V, and w ∈ W. Actually, the category H H YD ia a braided monoidal category, where the monoidal structure is given by the tensor product over k. Then, any Yetter-Drinfel'd module V∈ H H YD over H admits a braiding c V,V , and hence, ðV, c V,V Þ 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).
If H = k'G, then we write G G YD for the category of Yetter-Drinfel'd modules over k'G and say that V ∈ H H YD is a Yetter-Drinfel'd module over G. Notice that if V ∈ G G YD is of diagonal type, then ðV, c V,V Þ 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 ðV, cÞ is a braided vector space of diagonal type with an indecomposable braiding matrix ðq ij Þ i,j∈I of a basis fx i | i ∈ Ig. Let G 0 be a free abelian group generated by elements fg i | i ∈ Ig. Define the left coaction and left action by

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Then, V = ⊕ i∈I kx i , and each kx i is one-dimensional Yetter-Drinfel'd modules over G 0 . Hence, V is a Yetter-Drinfel'd module of diagonal type over G 0 .

Nichols
Algebras of Diagonal Type. Let ðV, cÞ be a θ -dimensional braided vector space of diagonal type. In this section, we recall a definition of the Nichols algebra BðVÞ generated by ðV, cÞ. In order to do that, we introduce one more notion in the category H H YD.
where I ðVÞ is the unique maximal coideal among all coideals of TðVÞ which are contained in ⊕ n≥2 V ⊗n . Nichols algebra BðVÞ 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 BðVÞ.
Note that if B ∈ H H YD and B is an algebra in H H YD, then B ⊗ B is an algebra in H H YD with the product given by for all a, b, c, d ∈ B, where · denotes the left action of H on B.
The tensor algebra TðVÞ admits a natural structure of a Yetter-Drinfel'd module and an algebra structure in H H YD. It is then a braided Hopf algebra in H H YD with coproduct δ ðvÞ = 1 ⊗ v + v ⊗ 1 ∈ TðVÞ ⊗ TðVÞ and counit εðvÞ = 0 for all v ∈ V such that δ and ε are the algebra morphisms. The antipode of TðVÞ exists (see [7], Section 2.1). Notice that the product defined by Equation (9) on TðVÞ is the unique algebra structure such that δðvÞ = 1 ⊗ v + v ⊗ 1 ∈ TðVÞ ⊗ TðVÞ for all v ∈ V. The coproduct can be extended from V to TðVÞ. For example, for all v, w ∈ V, one gets (we write the elements of TðVÞ without the tensor product sign for brevity) Let ðI i Þ i∈I be the family of all ideals of TðVÞ contained in ⊕ n≥2 V ⊗n , i.e., Then, the coideal I ðVÞ ≔ ∑ i∈I I i is the largest element in ðI i Þ i∈I . Hence, BðVÞ is a braided Hopf algebra in H H YD. As proved in [6] (Proposition 3.2.12), Nichols algebra BðVÞ is the unique ℕ 0 -graded braided Hopf algebra generated by V in H H YD with homogenous components BðVÞð0Þ = k, BðVÞð1Þ = V, and PðBðVÞÞ = V, where PðBðVÞÞ is the space of primitive elements of BðVÞ.

Definition 6. A generalized Cartan matrix is a matrix
is called a semi-Cartan graph if r 2 i = id X for all i ∈ I, and a X ij = a r i ðXÞ ij for all X ∈ X and i, j ∈ I. We say that a semi-Cartan For the sake of simplicity, the elements of the set fr i ðXÞ, i ∈ Ig 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 9. The exchange graph of C 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 r i ðXÞ = Y (and r i ðYÞ = 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 that C = CðI, X, r, ðA X Þ X∈X Þ is a connected semi-Cartan graph. Let ðα i Þ i∈I be the standard basis of ℤ I . For all X ∈ X, let 3 Advances in Mathematical Physics for all j ∈ I. Let DðX, IÞ be the category such that ObDðX, IÞ = X and morphisms HomðX, Let W ðCÞ be the smallest subcategory of DðX, IÞ, where the morphisms are generated by ðr i ðXÞ, s X i , XÞ, with i ∈ I, X ∈ X . From now on, we write s X i instead of ðr i ðXÞ, s X i , XÞ, if no confusion is possible. Notice that all generators s X i are reflections and hence are invertible. Then, W ðCÞ is a groupoid.
For any category D and any object X in D, let HomðD, XÞ = ∪ Y∈D HomðY, XÞ.
Definition 10. For all X ∈ X, the set is called the set of real roots of C at X. The elements of Δ X re + = Δ X re ∩ ℕ I 0 are called positive roots and those of Δ X re ∩ −ℕ I 0 negative roots, denoted by Δ X re − . If the set Δ X re is finite for all X ∈ X, then we say that C is finite.
Definition 11. We say that R = RðC, ðΔ X Þ X∈X Þ is a root system of type C if for all X ∈ X, the sets Δ X are the subsets of ℤ I such that We say that W ðRÞ ≔ W ðCÞ 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 a ij = 0 for all i ∈ I ′ , j ∈ I ′ ′ and In this case, we write R = Rj I 1 ⊕ Rj I 2 . If R ≠ Rj I 1 ⊕ Rj I 2 for all nonempty disjoint subsets I 1 , I 2 ⊂ I, then R is termed irreducible.
Let R = RðC, ðΔ X Þ X∈X Þ be a root system of type C. We recall some properties of R from [19,24].
Notice that the finiteness of R does not mean that W ðRÞ is also finite, since the set X might be infinite.

Lemma 14.
Let C = CðI, X, r, ðA X Þ X∈X Þ be a connected semi-Cartan graph and R = RðC, ðΔ X Þ X∈X Þ be a root system of type C. Then, the following are equivalent.
(1) R is finite Recall that C is a connected semi-Cartan graph. Then, we get the following.
Proposition 15. Let R = RðC, ðΔ X Þ X∈X Þ be a root system of type C. Then, the following are equivalent.
(1) There exists X ∈ X such that A X is indecomposable If R is finite, then the semi-Cartan graph C is indecomposable if and only if the root system R is irreducible. Definition 16. We say C is a Cartan graph if the following hold: In this case, W ðCÞ is called the Weyl groupoid of C.
Let R re ≔ RðC, ðΔ X re Þ X∈X Þ. Then, C is a Cartan graph if and only if R re is a root system of type C. Indeed, we get that s X i ðΔ X re Þ = Δ r i ðXÞ re by Equation (14). For all X ∈ X, we obtain that Δ X re = Δ X re + ∪ Δ X re − , since ωs The following proposition implies that if R is a finite root system of type C, then R = R re ; namely, all roots are real and R is uniquely determined by C.
Proposition 17. Let R = RðC, ðΔ X Þ X∈X Þ be a root system of type C. Let X ∈ X, m ∈ ℕ 0 , and i 1 , ⋯, i m ∈ I such that

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and ℓðωÞ = m. Then, the elements are pairwise different, where n ∈ f1, 2, ⋯, mg (and β 1 = α i 1 ). Here, is the length of ω ∈ HomðW ðCÞ, XÞ. In particular, if R is finite and ω ∈ HomðW ðCÞÞ is the longest element, then Remark 18. If C is a finite Cartan graph, then R is finite, and hence, is the unique root system of type C by Proposition 17; that is, R is uniquely determined by C.

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. Let G be an abelian group. Let F G θ be the set of θ-tuples of finite-dimensional irreducible objects in G G YD and X G θ be the set of θ-tuples of isomorphism classes of finite-dimensional irreducible objects in G G YD. For any Assume that V M = ⊕ i∈I kx i ∈ G G YD is a Yetter-Drinfel'd module of diagonal type over G, where fx i | i ∈ Ig is a basis of V. Then, there exists a matrix ðq ij Þ i,j∈I such that δðx i Þ = g i ⊗ x i and g i · x j = q ij x j for all i, j ∈ I. We fix that M = ðkx 1 , kx 2 , ⋯, kx θ Þ ∈ F G θ is a tuple of one-dimensional Yetter-Drinfel'd over G and ½M ∈ X G θ . We say that the matrix ðq ij Þ i,j∈I is the braiding matrix of M. Recall that the matrix is independent of the basis fx i | i ∈ Ig up to permutation of I. We say BðV M Þ = Bð ⊕ n i=1 kx i Þ is the Nichols algebra of the tuple M, denoted by BðMÞ.
Recall that the adjoint representation ad of a Nichols algebra BðVÞ from [9] is the linear map ad c :V ⟶ End ðBðVÞÞ for all x ∈ V, y ∈ BðVÞ, where μ is the multiplication map of BðVÞ and c is defined by Equation (6). In particular, the braided commutator ad c of BðMÞ takes the form In order to construct a semi-Cartan graph to M, we recall some finiteness conditions from [7,22].
Lemma 20. For any i, j ∈ I with i ≠ j, s, the following are equivalent.
Here, ðnÞ q ≔ 1 + q + ⋯ + q n−1 , which is 0 if and only if q n = 1 for q ≠ 1 or p | n for q = 1. Notice that ð1Þ q ≠ 0 for any q ∈ ℕ.
Hence, we get the following from Lemma 20.
where y i ∈ ðkx i Þ * \ f0g. If M is not i-finite, then let R i ðMÞ = M. Then, R i ðMÞ is a θ-tuple of one-dimensional Yetter-Drinfel'd modules over G. Let Definition 22. We say that M admits all reflections if N is i -finite for all N ∈ F G θ ðMÞ.
Notice that the reflections depend only on the braiding matrix ðq ij Þ i,j∈I . We recall the notion of generalized Dynkin diagram for a braided vector space of diagonal type [40].
Definition 23. Let V be a θ-dimensional braided vector space of diagonal type with the braiding matrix ðq ij Þ i,j∈I . The generalized Dynkin diagram of V is a nondirected graph D with the following properties: 5 Advances in Mathematical Physics (i) There is a bijective map ϕ from I to the vertices of D (ii) For all i ∈ I, the vertex ϕðiÞ is labeled by q ii (iii) For all i, j ∈ I with i ≠ j, the number n ij of edges between ϕðiÞ and ϕðjÞ is either 0 or 1. If q ij q ji = 1, then n ij = 0; otherwise, n ij = 1 and the edge is labeled by q ij q ji We say that the generalized Dynkin diagram of M is the generalized Dynkin diagram of braided vector space ⊕ i∈I M i . 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 of R i ðMÞ = ðR i ðMÞ j Þ j∈I by the following lemma.

Lemma 24.
Let i ∈ I. Assume that M is i-finite and let a ij ≔ a M ij for all j ∈ I. Let ðq ′ jk Þ j,k∈I be the braiding matrix of R i ðMÞ with respect to ðy j Þ j∈I . Then, Here, G n′ denotes the set of primitive nth roots of unity in k, that is, G n ′ = fq ∈ k * | q n = 1, q k ≠ 1 for all 1 ≤ k < ng for n ∈ ℕ.
If M admits all reflections, then we are able to construct a semi-Cartan graph CðMÞ of M by ( [36], Proposition 1.5).

Theorem 25.
Assume that M admits all reflections. For all X ∈ X G θ ðMÞ, let ½X θ = fY ∈ X G θ ðMÞjY and X have the same generalized Dynkin diagramg: is a connected semi-Cartan graph. We say that CðMÞ is the semi-Cartan graph attached to M. Furthermore, one can attach a groupoid W ðMÞ ≔ W ðC ðMÞÞ to M if M admits all reflections.
Notice that Nichols algebra BðMÞ is ℕ θ 0 -graded with deg M i = α i for all i ∈ I. Following the terminology in [22], we say that the Nichols algebra BðMÞ is decomposable if there exists a totally ordered index set ðL, ≤Þ and a sequence For each decomposition (28), we define the set of positive roots Δ ½M + ⊂ ℤ I and the set of roots Δ ½M ⊂ ℤ I of ½M by

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 Theorem 39.
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.

Some Remarks
Remark 29. Before starting the algorithms, we give several remarks on Algorithm 28.
(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 system rs. By Lemma 14, the Weyl groupoid is finite. Hence, in step 3, the implementation of the reflections terminates (3) From Table 1 of Appendix A in [25], for each root system rs, there is an object such that the generalized Cartan matrix is of type A 6 . Hence, for simplification, we collect all objects with generalized Cartan matrixes of type A 6 in step 4 of the algorithm 3.3. The Algorithms. The fundamental function in Algorithm 28 is the function "Sort": By Algorithm 30, 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.
As a result, we could append new root systems to a container with an order, which is essential in the following function.
Check the neighborhoods of root systems which have Cartan matrix of type A 6 . Input: all root systems in Appendix 5. Output: All the A 6 neighborhoods 1. Copy the data in Appendix 5 to set rs and allrs.

Call Sortðrs, cmpvectÞ.
3. Call Reflectionðrs, iÞ, for i ∈ f1,⋯,6g 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. Call cmðallrs½iÞ, for i = 1, ⋯, LengthðallrsÞ. Check if its Cartan matrix is of type A 6 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 type A 6 and the Cartan matrixes of the reflected root systems.

X
Gener. Dynkin diagrams Fixed parameters

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Remark 32. The reflection in step 2 of Algorithm 31 is given in Equation (13).
Remark 34. The entries of the generalized Cartan matrix of the given root system are defined in Lemma 13.
Finally, we still need a function to check which of the root system is indeed type of A 6 .

Classification Theorem of Finite Cartan Graphs
Let M ≔ ðkx 1 , ⋯, kx 6 Þ be a tuple of one-dimensional Yetter-Drinfel'd modules over k G. Assume that M admits all reflections and CðMÞ = CðI, X, r, ðA X Þ X∈X Þ be the attached indecomposable semi-Cartan graph to M. Assume that Δ ½M is the set of roots of BðMÞ. If Δ ½M is finite, then CðMÞ is a finite Cartan graph by Theorem 27.

Theorem 36.
Assume Δ ½M 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. If CðMÞ is equivalent to a Cartan graph in [15] (Corollary 3.13), then the claim is true. Otherwise, CðMÞ has a unique finite root system by Theorem 27, say RðMÞ. Assume that R = RðC, ðΔ X re Þ X∈X Þ is the unique root system, where Δ X re is the real roots of X. Moreover, the root system RðM Þ is irreducible by Proposition 15. For any X ∈ X, let Δ X re + be the positive roots of X. By [25] (Theorem 40), there exists a point X ∈ X satisfying that the set Δ X re + 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 graphs CðMÞ. The output of Algorithm 28 is crucial for the classification of Theorem 39.
In general, the points of a finite Cartan graph C could have many different neighborhoods. Depending on the output, we construct the following "good A 6 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 that X has a good A 6 neighborhood if there exists a permutation of I and an integer a ∈ ℕ such that and A X = A r 6 ðXÞ = A 6 , where a satisfies one of the following.
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 roots β i , β j 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): β i < β j if and only if β i ½k < β j ½k. Return True, if β i < β j . Else, return False.
2. For roots in rs, sort them with the partial order given by "cmpvect" in Step 1. Update rs and return. Theorem 39. Assume there exists a point X ∈ X satisfying that the set Δ X re + appears in Appendix 5 (except E 6 type), then up to equivalence, there exists a point Y ∈ X such that Y has one of the good A 6 neighborhoods.
Proof. If Δ ½M appears in Appendix A (except E 6 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 s X i maps Δ X re + \ fα i g bijectively to Δ r i ðXÞ re + \ fα i g for any i ∈ I, the Cartan matrices of all neighbors of X can be obtained from Δ X re + by Lemma 13. If there exists a point Y of the Cartan graph CðMÞ which has a good A 6 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. ☐

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. Let k be a field of characteristic p > 0. Let I = f1, 2, 3, 4, 5, 6g. Let ðV, cÞ be a braided vector space of diagonal type over k with basis fx k | k ∈ Ig satisfying Check whether a matrix is type of A 6 . Input: The generalized Cartan matrix of a root system. Output: True or False.

Construct a matrix
2. If there is a permutation of f1,⋯,6g satisfying the input matrix cm is of type A 6 , then return True. Otherwise, return False.
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 system rs in Appendix 5.

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Assume that ðq ij Þ i,j∈I is an indecomposable braiding matrix. Let M ≔ ðkx i Þ i∈I . Assume that the Nichols algebra BðVÞ generated by ðV, cÞ has a finite set of roots Δ ½M listed in Appendix 5 (except E 6 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 CðMÞ. Figure 1 is the exchange graph of the corresponding Cartan graph CðMÞ.
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 by 1, ⋯, 6 (for example: label D 6 as in Figure 2) (2) For a generalized Dynkin diagram D and i, j, k, l, m , n ∈ I, write τ ijklmn D for the graph D where the vertices of D change to i, j, k, l, m, n, respectively Proof. Assume that BðVÞ has a finite set of roots Δ ½M . Let X = ½M s 6 , I = f1, 2, 3, 4, 5, 6g, and A X ≔ ða ij Þ i,j∈I be the Cartan matrix of X. Let ðq i,j Þ i,j∈I be the braiding matrix of X and ðq r i ðXÞ i,j Þ i,j∈I be the braiding matrix of r i ðXÞ. To simplify the labels, we write q ij ′ ≔ q ij q ji for 1 ≤ i, j ≤ 6. Since BðVÞ has a finite set of roots Δ ½M , we obtain that CðMÞ is a finite Cartan graph by Theorem 27; we are free to assume that there exists a point X such that A X has a good A 6 neighborhood by Theorem 39.

Data Availability
The data used to support the findings of this study is available inside the paper.

Disclosure
Since we are dealing with the finite dimensional Nichols algebras and related structures, the backgrounds in Introduction and the terminology in Section 2 are similar to that of the corresponding author's paper [37] (Introduction and Section 1) submitted as preprint in arXiv.org with the link http://arxiv.org/abs/1911.03555 and published in the Journal of Algebra [37]. 10 Advances in Mathematical Physics