Representations of a non-pointed Hopf algebra

: In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations


Introduction
There are a lot of works about classification of Hopf algebras or Nichols algebras of finite GKdimension.For example, the readers can refer to [1,2].In the paper [2], Liu tried to classify all prime Hopf algebras of GK-dimension one and constructed a series of new examples of non-pointed Hopf algebras D(m, d, γ).As a by-product, a series of finite-dimensional non-semisimple quotient Hopf algebras are obtained, which have no Chevalley property.To understand this new class of quotient Hopf algebras, we see that those quotient Hopf algebras under the conditions that n is odd and d = 2n, denoted by D(n), are just isomorphic to an extension of the generalized quaternion group algebra kQ 4n equipped with a nontrivial suitable coalgebraic structure.It is well known that the representations of the generalized quaternion group Q 4n have been known for a long time.In [3,4], all the irreducible representations of Q 4n are given.As applications, [5] determined the complex representation rings of Q 4n and gave the isomorphism class of the n-th augmentation quotient of the augmentation ideal.In [6] the group code over the generalized quaternion group Q 4n is studied, which is based on representations of Q 4n .It is important in cryptography.
The task of this paper is to classify all the indecomposable modules of D(n) explicitly.The decomposition formulas of the tensor products of them are established.Finally, we describe the representation ring of D(n) by generators and generating relations.There is much effort to put into understanding and classifying all indecomposable modules of algebras of finite representation type.The readers can refer to the books [7,8] for the representation theory of algebras and some newest results [9,10] for example.In [11], Yang determined the representation type of a class of pointed Hopf algebras and classified all indecomposable modules of simple-pointed Hopf algebra R(q, α).In the paper [12], the representations of the half of the small quantum group u q (sl 2 ) were constructed by the technique of the deformed preprojective algebras.Furthermore, a lot of papers investigated the representation rings of various Hopf algebras, the readers can refer to [13][14][15][16][17].By techniques of generators and generating relations, Su and Yang described representation rings of the weak generalized Taft Hopf algebras as well as some small quantum groups in [15,16].Sun et al. described the representation rings of Drinfeld doubles of Taft algebras in [17].Motivated by the above works, we shall establish the decomposition formulas of the tensor products of the indecomposable D(n)-modules and determine the representation ring of D(n).This can help us to understand the structure and representation theory of D(n) in a better way.
The paper is organized as follows.In Section 2, we review the definition of D(n) and show that D(n) is of finite representation type.In Section 3, we shall construct all the indecomposable D(n)modules and establish all the decomposition formulas of the tensor product of two indecomposable D(n)-modules.In Section 4, we characterize the representation ring of D(n) by three generators and some generating relations.

Preliminaries
Throughout this paper, we work over an algebraic closed field k of characteristic 0. Unless otherwise stated, all algebras, Hopf algebras, and modules are finite-dimensional over k, all maps are k-linear, dim and ⊗ stand for dim k and ⊗ k , respectively.In this paper, we describe the representations of a quotient of the Hopf algebra D(m, d, γ) for the case of m = 2.
Firstly, we review the definition of the Hopf algebra D(2, d, γ) in [2].Let 2|d.As an algebra, it is generated by a ±1 , b ±1 , c, u 0 , u 1 , subject to the following relations ) where ω ∈ k is a primitive 4-th root of unity.The comultiplication ∆, the counit ϵ and the antipode S of D(2, d, γ) are given by From now on, assume that n is odd and d = 2n.Let D(n) be the quotient Hopf algebra We claim that the Hopf algebra D(n) can be viewed as the Hopf algebra generated by x, y, z satisfying the following relations The comultiplication ∆, the counit ϵ and the antipode S are given by Indeed, we define the maps φ and ψ as respectively.It is straightforward to check that φ and ψ are Hopf algebra isomorphisms and ψ • φ = id.
Hence we have the claim.Now, we use the later generators and relations to define the Hopf algebra D(n).Actually, D(n) is a class of non-pointed Hopf algebras.
It is noted that the generalized quaternion group algebra kQ 4n of order 4n is defined as Obviously, it can be embedded into the algebra D(n) as an algebra, but not as a Hopf algebra.Firstly, we determine the representation type of D(n) as an algebra.
Lemma 2.1.The algebra D(n) is of finite representation type.
Proof.Let Q 4n be the generalized quaternion group and A = k⟨y|y 2 = 0⟩ a k-algebra.It is obvious that A is of finite representation type.Define be viewed as subgroup of Aut Alg (A).Therefore, we have the skew group algebra Q 4n * A, whose multiplication is given by It is easy to see that A can be viewed as the subalgebra of Q 4n * A and

The indecomposable modules of D(n) and their tensor products
In this section, we mainly construct all the indecomposable D(n)-modules and establish their tensor products.It is remarked that the simple modules of the algebra G * B are completely understood, and coincide with those of the group G for which B acts as zero (see [21]).
Firstly we classify all the indecomposable modules of D(n).Let ξ ∈ k be a primitive 2n-th root of unity.Theorem 3.1.(a) There are 4 pairwise non-isomorphic 1-dimensional D(n)-modules S i with basis {v i } for 0 ≤ i ≤ 3, the action of D(n) is defined by (b) There are n − 1 pairwise non-isomorphic 2-dimensional simple D(n)-modules M j with basis (c) There are 4 pairwise non-isomorphic 2-dimensional indecomposable projective D(n)-modules P i with basis (d) There are n − 1 pairwise non-isomorphic 4-dimensional indecomposable projective D(n)-modules T j with basis {ϑ Proof.The results of (a),(b) are showed in [5,6,21].Firstly, we construct the 2-dimensional indecomposable non-simple , respectively.It is directly checked that all the generating relations are satisfied only when i = 0 or i = n.
. Now we use a unified expression to describe such modules V k with a basis {ν 1 k , ν 2 k , ν 3 k } for 0 ≤ k ≤ 3, and the matrices of x, y, z acting on this basis are , respectively.In fact, these modules are decomposable.For, let be another basis, then the matrices of x, y, z acting on , respectively.Thus we get that and the matrices of x, y, z acting on , then the matrices of x, y, z acting on this basis are , respectively.Therefore when n < i < 2n, the modules are isomorphic to the case of 2n−i.Furthermore, when i = 0, we choose the basis then the matrices of x, y, z acting on this basis are , respectively.Hence it is decomposable and isomorphic to P 0 ⊕ P 2 .Similarly, for the case i = n, the module is decomposable and isomorphic to P 1 ⊕ P 3 .Indeed, we choose the basis then the matrices of x, y, z acting on this basis are , respectively.Therefore, we get the result (d).
In fact, we know that the primitive idempotents of D(n) are listed in [6] as and ze 0 = e 0 , ze 1 = ωe 1 , ze 2 = −e 2 , ze 3 = −ωe 3 , then k{e i , ye i |0 ≤ i ≤ 3} consist of four indecomposable modules of D(n) and are isomorphic to P i , respectively.Thus D(n)e i P i is an indecomposable projective module.For 0 ≤ j ≤ 2n − 1, set Since xθ j = ξ j θ j , the matrices of x, y, z act on {θ j , zθ j , yθ j , yzθ j }, are , respectively.By the result of (d), we know that k{θ j , zθ j , yθ j , yzθ j } T j for 1 ≤ j ≤ n − 1, so D(n)θ j T j is an indecomposable projective module.
The straightforward verification shows that P i (0 ≤ i ≤ 3) and T j (1 ≤ j ≤ n − 1) are uniserial, that is 0 ⊂ S i−1(mod 4) ⊂ P i and 0 ⊂ M n− j ⊂ T j are the unique composition series of P i and T j , respectively.Since D(n) is a Frobenius algebra and thus is self-injective, we get that all the indecomposable projective modules are indecomposable injective modules.Therefore D(n) is a Nakayama algebra.By [8, Theorem V.3.5], the modules listed above are all the indecomposable modules of D(n).
The proof is completed.□ Corollary 3.2.(1) For all 0 ≤ i ≤ 3, P i is the projective cover of S i .
(2) For all 1 ≤ j ≤ n − 1, T j is the projective cover of M j .
Proof.The results is directly obtained by [8,Lemma 5.6].□ Let H be a Hopf algebra, M and N be left H-modules.It has been known that The remaining of this section is devoted to establishing all the decomposition formulas of the tensor products of two indecomposable D(n)-modules.Theorem 3.3.(1) (a) For 0 ≤ i, j ≤ 3, S i ⊗ S j S j ⊗ S i S i+ j(mod 4) .
thus S i ⊗ S j S i+ j(mod 4) S j ⊗ S i .
thus S i ⊗ M j M j when i = 0, 2 and S i ⊗ M j M n− j when i = 1, 3. Similarly, we can get the same results of M j ⊗ S i .
thus S i ⊗ P j P i+ j(mod 4) .The same results of P j ⊗ S i can be obtained in the same way.
thus S i ⊗ T j T j when i = 0, 2 and S i ⊗ T j T n− j when i = 1, 3. Similarly, we get the same results of Obviously, when 0 < i + j, i − j < n, M i ⊗ M j M i+ j ⊗ M i− j .Besides, we need to note that when n < i + j < 2n, let ω1 i j := ω 4 i j , ω2 i j := (−1) i+ j ω 1 i j , then , using the previous conclusion, we directly get that k{ ω3 i j := ω 3 i j , ω4 i j := (−1) i ω 2 i j } M j−i .In particular, when i + j = n, the matrices of x, y, z acting on the basis { ω1 i j , ω2 i j } are simultaneously diagonalizable.Thus the modules are isomorphic to S 1 ⊕ S 3 ; when i − j = 0, the matrices of x, y, z acting on the basis { ω3 i j , ω4 i j } are also simultaneously diagonalizable and isomorphic to S 0 ⊕ S 2 .In fact we might assume that i ≥ j since the result of M j ⊗ M i are the same as M i ⊗ M j .
For M i ⊗ P j , let ν1 It has been shown that in Theorem 3.1 when n < i + j < 2n, these modules k{ θ1 i j , θ2 i j , θ3 i j , θ4 i j } are isomorphic to the cases of 2n − (i + j).When −n < i − j < 0, n < i − j + 2n < 2n, it's clear to get that the modules are isomorphic to the case of 2n − (i − j + 2n) = j − i.
For P i ⊗ P j , we have known that 0 ⊂ S i−1(mod 4) ⊂ P i is the unique composition series of P i for 0 ≤ i ≤ 3. Thus there is an exact sequence 0 −→ S i−1(mod 4) −→ P i −→ S i −→ 0. By Corollary 3.2, P i = P(S i ), so there is a split exact sequence 0 −→ S i−1(mod 4) ⊗ P j −→ P i ⊗ P j −→ S i ⊗ P j −→ 0.
For P i ⊗ T j , similarly, we have the split exact sequence 0 −→ S i−1(mod 4) ⊗ T j −→ P i ⊗ T j −→ S i ⊗ T j −→ 0.
Hence P i ⊗ T j S i−1(mod 4) ⊗ T j ⊕ S i ⊗ T j T j ⊕ T n− j .
For T i ⊗ T j , i > j, the unique composition series of T i is 0 ⊂ M n−i ⊂ T i for 1 ≤ i ≤ n − 1, and there is an exact sequence 0 for T j = P(M j ).Hence Then applying the result of (2)(c), we get the result (4).For i < j, the result is similar.The proof is finished.□ .It follows that the representation ring r(H) is an associative ring with identity given by [k ε ], the trivial 1-dimensional H-module.Note that r(H) has a Z-basis consisting of isomorphic classes of finite dimensional indecomposable H-modules.In this section we will describe the representation ring r(D(n)) of the Hopf algebra D(n) explicitly by the generators and the generating relations.
Proof.The results of (1), ( 2) are easy to get from Theorem 3.3(1)(a)-(c) and (3)(a).We prove (3) by induction.By Theorem 3.3(2)(a), there is Suppose that (4.1) holds for j − 1 being odd and j being even, then for j + 1 we have Similarly, suppose that (4.1) holds for j − 1 being even and j being odd, then for j + 1 we directly get that By Theorem 3.3(2)(b), we know that M j ⊗ P 0 T j , thus the Eq (4.2) is obvious to be obtained.□ Corollary 4.3.Keep the notations above.

Conclusions
We have constructed all the indecomposable modules of the non-pointed Hopf algebra D(n) and established the decomposition formulas of the tensor product of any two indecomposable modules.The representation ring r(D(n)) has been characterized by generators and relations.In the further work, we hope to construct all the simple Yetter-Drinfeld modules of D(n) and classify all the finite-dimensional Nichols algebras and finite-dimensional Hopf algebras over D(n).

Corollary 3 . 4 . 4 .
The tensor product of any two D(n)-modules is commutative.The representation ring of D(n) Let H be a finite dimensional Hopf algebra and F(H) the free abelian group generated by the isomorphic classes [M] of finite dimensional H-modules M. The abelian group F(H) becomes a ring if we endow F(H) with a multiplication given by the tensor product [M][N] = [M ⊗ N].The representation ring (or Green ring) r(H) of the Hopf algebra H is defined to be the quotient ring of F(H) modulo the relations [M ⊕ N] = [M] + [N]

Lemma 4 . 1 .
[13, Lemma 3.11]  Let Z[y, z] be the polynomial algebra over Z in two variables y and z.

2 ]Corollary 4 . 4 .
by Theorem 3.3(1)(b), and using the equations of (4.1) in Lemma 4.2, we can easily get the results.□Keep notations as above, then the sets