On Hopf algebras over basic Hopf algebras of dimension 24

We determine finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, whose Hopf coradical is isomorphic to a non-pointed basic Hopf algebra of dimension $24$ and the infinitesimal braidings are indecomposable objects. In particular, we obtain families of new finite-dimensional Hopf algebras without the dual Chevalley property.


Introduction
Let k be an algebraically closed field of characteristic zero.It is a fundamental and difficult question in Hopf algebra theory to classify finite-dimensional ones.The research in this direction is very rich.Most of the classification results consist of Hopf algebras that are basic or have the dual Chevalley property (that is, its coradical is a subalgebra) .But there are very few results on finite-dimensional Hopf algebras without the dual Chevalley property in the literature, unless examples without pointed duals were constructed in [12,20,21,24,29] via the generalized lifting method [4].
As a generalization of the lifting method [8], the generalized lifting method gives a technical framework to classify the Hopf algebras without the dual Chevalley property.It consists of the following steps (see [4]): • Step 1. Classify all Hopf algebras L that are generated by a cosemisimple coalgebra.
• Step 2. Classify all connected graded Hopf algebras R in the category L L YD of left Yetter-Drinfeld modules over L. • Step 3. Given L and R as in previous items, classify all Hopf algebras A such that gr A ∼ = R♯L.
Here A is called a lifting of R over L.
It works because of the following facts.Suppose that A is a Hopf algebra over k and denote by A [0] the Hopf coradical of A (it is generated by the coradical A 0 of A).If S A (A [0] ) ⊆ A [0] , then the standard filtration {A [n] } n≥0 , defined recursively by , is a Hopf algebra filtration.Therefore the associated graded coalgebra gr = 0 is a Hopf algebra and so there is a connected graded braided Hopf algebra R = ⊕ n≥0 R(n) in Theorem 1.1.[Theorems 4. 7 & 5.11] Let A be a finite-dimensional Hopf algebra over K 24,1 whose infinitesimal braiding V is indecomposable in K24,1 K24,1 YD.Then V is isomorphic either to k χ i,j,k for (i, j, k) ∈ Λ 0 or to V i,j,k,ι for (i, j, k, ι) ∈ ∪ 6 i=1 Λ i , and A is isomorphic to one of the following objects: • k χ i,j,k ♯K 24,1 for (i, j, k) ∈ Λ 0 ; • B(V i,j,k,ι )♯K 24,1 for (i, j, k, ι) ∈ ∪ 6 i=1 Λ i − Λ 1 * ; • C i,j,k,ι (µ) for µ ∈ k and (i, j, k, ι) ∈ Λ 1 * .
The Nichols algebra B(V i,j,k,ι ) for (i, j, k, ι) ∈ ∪ 6 i=4 Λ i is isomorphic as an algebra to a quantum plane.They have appeared in [5] and were shown that the braidings are of non-diagonal type.The Nichols algebra B(V i,j,k,ι ) for (i, j, k, ι) ∈ ∪ 3 i=1 Λ i is an algebra of dimension 18 or 36 with no quadratic relations.They are examples of Nichols algebra of non-diagonal type, which are (up to isomorphism) arising from Nichols algebras of standard type B 2 by using the techniques in [2].
The paper is organized as follows: In section 2, we recall some basic knowledge and notations of Yetter-Drinfeld modules, Nichols algebras.In section 3, we introduce the structure of the Hopf algebra K 24,1 .In section 4, we determine all finite-dimensional Nichols algebras over simple objects in K24,1 K24,1 YD and present them by generators and relations.In section 5, we determine all finite-dimensional Hopf algebras over K 24,1 , whose infinitesimal braidings are simple objects in K24,1 K24,1 YD.

Preliminaries
Conventions.In the paper, the base field k is algebraically closed of characteristic zero and ξ is a primitive 6th root of unity.Let Z n := Z/nZ and I k,n := {k, k + 1, . . ., n} for n ≥ k ≥ 0.
Let H be a Hopf algebra over k.Denote by G(H) the set of group-like elements of H.For any In particular, c := c V,V is a linear isomorphism satisfying the braid equation (c If H is finite-dimensional, then by [6, Proposition 2.2.1.],H H YD ∼ = H * H * YD as braided monoidal categories via the functor (F, η) defined as follows : F (V ) = V as a vector space, where ⊗ r (2) for the comultiplication.By the Radford biproduct or bosonization of R by H ([27]), written as R♯H, means a usual Hopf algebra, as a vector space, R♯H = R ⊗ H, whose multiplication and comultiplication are provided by the smash product and smash coproduct, respectively: 2.2.Nichols algebras and skew-derivations.Let H be a Hopf algebra with bijective antipode and The Nichols algebra B(V ) is isomorphic to T (V )/I(V ), where I(V ) ⊂ T (V ) is the largest N-graded ideal and coideal in H H YD such that I(V ) ∩ V = 0.Moreover, the ideal I(V ) is the kernel of the quantum symmetrizer associated to the braiding c and B(V ) as a coalgebra and an algebra depends only on (V, c).[14]).
Remark 2.2.The Nichols algebra B(V ) is of diagonal type if there is a linear basis {x i , i ∈ I 1,n } such that c(x i ⊗ x j ) = q ij x j ⊗ x i for some q ij ∈ k.The matrix q = (q ij ) i,j∈I1,n is called the matrix of the braiding.The generalized Dynkin diagram of the matrix q is a graph with n vertices, the vertex i labeled with q ii , and an arrow between the vertices i and j only if q ij q ji = 1, labelled with q ij q ji .Finitedimensional Nichols algebras of diagonal type were classified by Heckenberger [18], with the help of the Weyl groupoid and generalized root systems.Their defining relations were given by Angiono [9,10].See [1] for a survey on Nichols algebras of diagonal type.
Let C be a coalgebra, D a subcoalgebra of C and W ∈ C M. Denote the largest D-subcomodule of W by Now we recall the standard tool, the so called skew-derivation, for working with Nichols algebras.Let (V, c) be a (rigid) braided vector space of dimension n and ∆ i,m−i : Let {v i } 1≤i≤n and {v i } 1≤i≤n be the dual bases of V and V * .We write ∂ i := ∂ v i for simplicity.This is useful for seeking the relations of B(V ) due to: Furthermore, ∂ f can be defined on B(V ) and ∩ f ∈V * ker ∂ f = k.For details, see [7,2].
3. The Hopf algebra K 24,1 and The category We introduce the structures of the Hopf algebra K 24,1 and the category K 24,1 admits a Hopf algebra structure, where the coalgebra structure and antipode are given as follows: The set {a i , da i , ba i , ca i , i ∈ I 0,5 } is a linear basis of K 24,1 .
(1) The set {g j , g j h, g j x, g j hx, j ∈ I 0,5 } is a linear basis of A 24,1 .We have ( gr A24,1 YD as braided monoidal categories via the tensor functor (G, γ) defined as follows: G(V ) = V as vector spaces and coactions, transforming the action • to Now we give explicitly the structure of the Drinfeld double D(K cop 24,1 ) of K cop 24,1 .From now on, we set D := D(K cop 24,1 ) := A cop op 24,1 ⊗ K cop 24,1 for convenience.Recall that D(H cop ) = H * op cop ⊗ H cop is a Hopf algebra with the tensor product coalgebra structure and the algebra structure given by (p 24,1 ⊗ K cop 24,1 is isomorphic to the algebra generated by the elements g, h, x, a, b, c, d, subject to the relations in K cop 24,1 , the relations in A cop op

Next, we describe two-dimensional simple objects in
Clearly, |Λ| = 120.Lemma 3.7.For any (i, j, k, ι) ∈ Λ, there is a 2-dimensional simple object V i,j,k,ι ∈ D M, whose matrices defining the D-action on a fixed basis are given by Any two-dimensional simple object in D M is isomorphic to V i,j,k,ι for some (i, j, k, ι) ∈ Λ.Furthermore, V i,j,k,ι ∼ = V p,q,r,κ if and only if (i, j, k, ι) = (p, q, r, κ).
Proof.Let V be a simple D-module of dimension 2. As the generators g, h, a, d commute with each other and g 6 = h 2 = a 6 = d 6 = 1, we may assume that the matrix defining the action on V are of the form where The relations xh = −hx, bh = −hb and ch = −hc imply that are zero matrices and hence V is not a simple D-module.Now we claim that g 1 = g 2 .Indeed, if g 1 = g 2 , then the relations gx = xg, bg = gb and cg = gc yield (g are zero matrices and hence V is not simple.
From the relations b 2 = 0 = c 2 and bc = 0 = cb, we have that By permuting the elements of the basis, we may assume that b We claim that b 2 = 0 or c 2 = 0. Suppose that b 2 = 0 = c 2 .Then x 2 x 3 = 0 and by equations ( 10), (11), we have Hence a 1 − ξ −1 a 2 = 0 and a 2 − ξ −1 a 1 = 0, which implies that a 1 = 0 = a 2 , a contradiction.Thus the claim follows.We may also assume that c 2 = 1.

The category
gr A24,1 YD, we transport the information from the category Proof.The K 24,1 -action is given by the restriction of the character of D. The coaction is of the form Proof.Since K 24,1 ∼ = A * 24,1 , by [6, Proposition 2.2.1.],we have the equivalence A24,1 YD via the functor (F, η) defined by (2).More precisely, by the formula (2), Lemma 3.9 and Remark 3.4 (1), we have Then by Remark 3.4 (2), we have gr A24,1 YD via the functor (G, γ) defined by the formulae ( 8)-( 9), and then GF (k χ i,j,k ) = k χ i,j,k ∈ gr A24,1 gr A24,1 YD with the module structure given by K24,1 YD with the module structure given by and the comodule structure given by (1) for k = 0, (2) for k = 1, where Proof.Let {h i } i∈I1,24 and {h i } i∈I1,24 be the dual bases of K 24,1 and K * 24,1 .The K 24,1 -action is given by the restriction of the D-action and the K 24,1 -comodule structure is given by δ . By Lemma 3.3 and Remark 3.4, we have Then the lemma follows by a direct computation.

Nichols algebras in
We determine all finite-dimensional Nichols algebras over simple objects in We shall show that finite-dimensional Nichols algebras over one-dimensional objects in others.
Proof.It follows directly by Lemma 3.14 and Remark 2.1.
K24,1 YD for (i, j, k, ι) ∈ Λ.Then using the equivalence YD with the structure given by Corollary 3.13.By Proposition 2.3 (see also [2, Theorem 1.1]), B(V i,j,k,ι )♯B(X) ∼ = B(X ⊕ X i,j,k,ι ) in Γ Γ YD which is the identity on X ⊕ X i,j,k,ι , where It is clear that B(X ⊕ X i,j,k,ι ) is of diagonal type with the generalized Dynkin diagram given by −1 Now we show that infinite-dimensional Nichols algebras over two-dimensional simple objects in K24,1 K24,1 YD are parametrized by the following subsets: and present finite-dimensional ones by generators and relations.
. These diagrams do not appear in [18, Table 1], that is, they have infinite root systems.Therefore, dim B(X ⊕ X i,j,k,ι ) = ∞.
Proof.By Lemma 3.15, the braiding of V i,j,k,ι is given by , it follows by the formulae ( 3) and ( 4) that Similarly, we obtain that It is easy to verify that ∂ 1 (r) = 0 = ∂ 2 (r) for any relation r given in ( 12)-( 14).Then by ( 5), the quotient B of T (V i,j,k,ι ) by the relations ( 12)-( 14) projects onto B(V i,j,k,ι ).We claim that -( 14) and (v Proof.The braiding of V i,j,k,ι is given by c Then a direct computation shows that the relations ( 15) and ( 16) are zero in B(V i,j,k,ι ) being annihilated by ∂ 1 , ∂ 2 and hence the quotient B of T (V i,j,k,ι ) by the relations ( 15) and ( 16) projects onto B(V i,j,k,ι ).We claim that , ( 16) and (v , it is easy to show that v 1 I, v 2 I ⊂ I. Hence I linearly generates B since 1 ∈ I. • .Since j / ∈ {0, 3}, it is of standard type B 2 and by [9,10], dim B(X ⊕ X i,j,k,ι ) = 72.The claim follows since dim B(V i,j,k,ι ) ≥ 1 2 dim B(X ⊕ X i,j,k,ι ).Consequently, B ∼ = B(V i,j,k,ι ).
The Nichols algebra B(V i,j,k,ι ) for (i, j, k, ι) ∈ Λ 3 is generated by v 1 , v 2 , subject to the relations Proof.The braiding of V i,j,k,ι is given by c( v Then a direct computation shows that the relations (17) are zero in B(V i,j,k,ι ) being annihilated by ∂ 1 , ∂ 2 and hence the quotient B of T (V i,j,k,ι ) by the relations (17) projects onto B(V i,j,k,ι ).We claim that I = k{v i 1 (v 2 v 1 ) j v k 2 , i ∈ I 0,5 , j ∈ I 0,1 , k ∈ I 0,2 } is a left ideal.Indeed, from (17) and , it is easy to show that v 1 I, v 2 I ⊂ I. Hence I linearly generates B since clearly 1 ∈ I.
We claim that dim Proposition 4.6.
Proof.Assume that (i, j, k, ι) ∈ Λ 4 .The braiding of V i,j,k,ι = {v 1 , v 2 } is given by Then a direct computation shows that Therefore, the quotient B of T (V i,j,k,ι ) by the relations (18) projects onto B(V i,j,k,ι ).From (17), it is easy to show that I = k{v i 1 v j 2 , i, j ∈ I 0,1 } is a left ideal and linearly generates B. We claim that dim The proof follows for (i, j, k, ι) ∈ Λ 5 or Λ 6 the same lines as for (i, j, k, ι) ∈ Λ 4 .In these cases, the generalized Dynkin diagram of X ⊕ X i,j,k,ι is given by They are of standard A 2 type.Theorem 4.7.Let V be a simple object in The Nichols algebras B(V i,j,k,ι ) with (i, j, k, ι) ∈ Λ 4 ∪ Λ 5 ∪ Λ 6 have already appeared in [5].They are isomorphic to quantum planes as algebras.They can be recovered, up to isomorphism, by using the techniques in [2].Indeed, from the proofs of Proposition 4.6, they are arising from Nichols algebras of Cartan type A 2 or standard type A 2 .
Proof.We prove the assertion for C 2,j,1,0 (µ), being the proof for C 2,j,0,0 (µ) completely analogous.We write v 12 := v 1 v 2 for short.By Diamond Lemma, it suffices to show that all overlaps ambiguities are resolvable, that is, the ambiguities can be reduced to the same expression by different substitution rules with the order v 2 < v 1 v 2 < v 1 < d < c < b < a.Here we verify that the overlapping pair (f v 2 )v 2 2 = f (v 3 2 ) for f ∈ {a, b, c, d} is resolvable: The theorem follows by Propositions 5.7-5.10.The Hopf algebras from different families are pairwise non-isomorphic since the diagrams are not isomorphic as Yetter-Drinfeld modules over K 24,1 .

Proposition 2 . 3 .
[19, Proposition 8.8]  Let H be a Hopf algebra with bijective antipode, N ∈ H H YD and W ∈ B(N )♯H B(N )♯H YD.Assume that W is a semisimple object in the category of Z-graded left Yetter-Drinfeld modules over B(N )♯H.Let K = B(W ) in B(N )♯H B(N )♯H YD, and define M = W (H). Then there is a unique isomorphism K♯B(N ) ∼ = B(M ⊕ N ) of braided Hopf algebras in H H YD which is the identity on M ⊕ N .