Representations of a non-pointed Hopf algebra

1.   Introduction

There are a lot of works about classification of Hopf algebras or Nichols algebras of finite GK-dimension. 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 $D(\underline{m}, d, \gamma)$. 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 $\mathbb{k}Q_{4n}$ 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 $R(q, \alpha).$ In the paper ^[12], the representations of the half of the small quantum group $\frak{u}_q(\frak{sl}_2)$ 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.

For the theories of Hopf algebras and representation theory, we refer to ^[7,8,18,19].

2.   Preliminaries

Throughout this paper, we work over an algebraic closed field $\mathbb{k}$ of characteristic $0$. Unless otherwise stated, all algebras, Hopf algebras, and modules are finite-dimensional over $\mathbb{k},$ all maps are $\mathbb{k}$-linear, dim and $\otimes$ stand for $\rm{dim}_{ \mathbb{k}}$ and $\otimes_{ \mathbb{k}},$ respectively. In this paper, we describe the representations of a quotient of the Hopf algebra $D(\underline{m}, d, \gamma)$ for the case of $m = 2$.

Firstly, we review the definition of the Hopf algebra $D(\underline{2}, d, \gamma)$ in ^[2]. Let $2|d$. As an algebra, it is generated by $a^{\pm1}, \; b^{\pm1}, \; c, \; u_{0}, \; u_{1},$ subject to the following relations

where $\omega\in{ \mathbb{k}}$ is a primitive $4$-th root of unity. The comultiplication $\Delta$, the counit $\epsilon$ and the antipode $S$ of $D(\underline{2}, d, \gamma)$ 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 $\Delta$, the counit $\epsilon$ and the antipode $S$ are given by

Indeed, we define the maps $\varphi$ and $\psi$ as

and

respectively. It is straightforward to check that $\varphi$ and $\psi$ are Hopf algebra isomorphisms and $\psi\circ\varphi = id$. 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 $\mathbb{k}Q_{4n}$ 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 $A = \mathbb{k}\langle y|y^{2} = 0\rangle$ a $\mathbb{k}$-algebra. It is obvious that $A$ is of finite representation type. Define

Since $(y\cdot x)(y\cdot x) = 0, \; (y\cdot z)(y\cdot z) = 0,$ $Q_{4n}$ can be viewed as subgroup of $\rm{Aut}_{\rm Alg}(A)$. Therefore, we have the skew group algebra $Q_{4n}\ast A$, whose multiplication is given by

for any $h, k\in Q_{4n}, \; a, b\in A.$

It is easy to see that $A$ can be viewed as the subalgebra of $Q_{4n}\ast A$ and

and $D(n)\cong Q_{4n}\ast A$ as an algebra. Consequently, $D(n)$ is of finite representation type as an algebra by [20,Theorem 1.1,Theorem 1.3(a)].

3.   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 $G\ast B$ 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 $\xi\in{ \mathbb{k}}$ 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 $0\leq i\leq3$, 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 $\{v_{j}^{1}, v_{j}^{2}\}$ for $1\leq j\leq n-1$, the action of $D(n)$ is defined by

(c) There are $4$ pairwise non-isomorphic $2$-dimensional indecomposable projective $D(n)$-modules $P_{i}$ with basis $\{\mu_{i}^{1}, \mu_{i}^{2}\}$ for $0\leq i\leq 3$, the action of $D(n)$ is defined by

(d) There are $n-1$ pairwise non-isomorphic $4$-dimensional indecomposable projective $D(n)$-modules $T_{j}$ with basis $\{\vartheta_{j}^{1}, \vartheta_{j}^{2}, \vartheta_{j}^{3}, \vartheta_{j}^{4}\}$ for $1\leq j\leq n-1$, the action of $D(n)$ is defined by

Proof. The results of $(a)$, $(b)$ are showed in ^[5,6,21].

Firstly, we construct the 2-dimensional indecomposable non-simple $D(n)$-modules. Let $x\cdot\mu^{i} = \lambda_{i}\mu^{i}$ for $i = 1, 2, \; \lambda_{i}\in \mathbb{k}$ and $y\cdot\mu^{1}\neq0.$ Suppose that $y\cdot\mu^{1} = \bar{\mu}^{1},$ it is obvious that $\bar{\mu}^{1}$ and $\mu^{1}$ are linearly independent. We might let $y\cdot\mu^{1} = \bar{\mu}^{1} = :\mu^{2}$ as well, then $y\cdot\mu^{2} = 0.$ Since $x\cdot\mu^{2} = x\cdot(y\cdot\mu^{1}) = -y\cdot(x\cdot\mu^{1}) = -\lambda_{1}\mu^{2},$ there is $\lambda_{2} = -\lambda_{1}.$ Now consider $z\cdot\mu^{1}.$ If $z\cdot\mu^{1}$ and $\mu^{1}, \mu^{2}$ are linearly dependent, let $z\cdot\mu^{1} = p_{1}\mu^{1}+p_{2}\mu^{2}, \; p_{1}, p_{2}\in \mathbb{k},$ then $z\cdot\mu^{2} = z\cdot(y\cdot\mu^{1}) = -\omega y\cdot(z\cdot\mu^{1}) = -\omega p_{1}\mu^{2}.$ By $z^{2} = x^{n},$ it's easy to see $(1+\omega)p_{1}p_{2} = 0$ and $p_{1}^{2} = \lambda_{1}^{n},$ so $p_{2} = 0$ for $p_{1}\neq0.$ Since $\lambda_{1}^{2n} = 1, \; \lambda_{1}^{n} = p_{1}^{2} = \pm1.$ When $\lambda_{1}^{n} = 1$, $p_{1} = \pm1,$ there is $\lambda_{1} = \lambda_{1}^{-1}$ by $xz = zx^{-1}$, thus $\lambda_{1} = \pm1.$ But if $\lambda_{1} = -1$, then $\lambda_{1}^{n} = -1\neq1$, which is a contradiction. Thus $\lambda_{1} = 1.$ Similarly, if $\lambda_{1}^{n} = -1$, then $\lambda_{1} = -1$, it also gets a contradiction. Hence we get $(c).$

Next, let $x\cdot v_{1} = \xi^{i}v_{1}$ for some $i\in \mathbb{Z}_{2n}$, $z\cdot v_{1} = v_{2}$, $y\cdot v_{1} = v_{3}$, here $\xi$ is a primitive $2n$-th root of unity. Then

and

Obviously, $z\cdot v_{3}\neq0.$

If $z\cdot v_{3}$ and $v_{1}, v_{2}, v_{3}$ are linearly dependent, let $z\cdot v_{3} = av_{1}+bv_{2}+cv_{3}, \; \; a, b, c\in \mathbb{k}$. By $xz = zx^{-1},$ it's easy to get that $a = b = 0.$ Since $z^{4} = 1$, we can set $z\cdot v_{3} = \omega^{k}v_{3}$ for some $k\in \mathbb{Z}_{4}$, where $\omega$ is a primitive $4$-th root of unity, then $y\cdot v_{2} = \omega^{k+1} v_{3}$. At this time, the matrices of $x, \; y, \; z$ acting on $\{v_{1}, \; v_{2}, \; v_{3}\}$ are

respectively. It is directly checked that all the generating relations are satisfied only when $i = 0$ or $i = n.$ If $i = 0,$ then

If $i = n,$ then

Now we use a unified expression to describe such modules $V_{k}$ with a basis $\{\nu_{k}^{1}, \nu_{k}^{2}, \nu_{k}^{3}\}$ for $0\leq k\leq 3,$ 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 $\{\omega_{k}^{1}, \; \omega_{k}^{2}, \; \omega_{k}^{3}\}$ are

respectively. Thus we get that

Moreover, if $z\cdot v_{3}$ and $v_{1}, v_{2}, v_{3}$ are linearly independent, let $z\cdot v_{3} = v_{4}$, then

and the matrices of $x, \; y, \; z$ acting on $\{v_{1}, \; v_{2}, \; v_{3}, \; v_{4}\}$ are

respectively. We set $v_{4}' = \omega v_{4}$ to get the result (d). It is noted that when $n < i < 2n,$ let $\bar{v}_{1}: = v_{2}, \; \bar{v}_{2}: = (-1)^{i}v_{1}, \; \bar{v}_{3}: = v_{4}, \; \bar{v}_{4}: = (-1)^{i}v_{3},$ 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 $P_{0}\oplus P_{2}$. Similarly, for the case $i = n,$ the module is decomposable and isomorphic to $P_{1}\oplus P_{3}$. Indeed, we choose the basis

then the matrices of $x, \; y, \; z$ acting on this basis are

respectively. Therefore, we get the result $(d)$.

Then we claim that $P_{i}(0\leq i\leq 3)$ and $T_{j}(1\leq j\leq n-1)$ are indecomposable projective modules.

In fact, we know that the primitive idempotents of $D(n)$ are listed in ^[6] as

Since

and

then $\mathbb{k}\{e_{i}, \; ye_{i}|0\leq i \leq 3\}$ consist of four indecomposable modules of $D(n)$ and are isomorphic to $P_{i},$ respectively. Thus $D(n)e_{i}\cong P_{i}$ is an indecomposable projective module.

For $0\leq j\leq 2n-1,$ set

then $\{\theta_{0}, \; \theta_{1}, \; \cdots\; ,\theta_{2n-1}\}$ is a set of orthogonal idempotents of $D(n)$. Since $x\theta_{j} = \xi^{j}\theta_{j},$ the matrices of $x, \; y, \; z$ act on $\{\theta_{j}, \; z\theta_{j}, \; y\theta_{j}, \; yz\theta_{j}\},$ are

respectively. By the result of (d), we know that $\mathbb{k}\{\theta_{j}, \; z\theta_{j}, \; y\theta_{j}, \; yz\theta_{j}\}\cong T_{j}$ for $1\leq j\leq n-1$, so $D(n)\theta_{j}\cong T_{j}$ is an indecomposable projective module.

The straightforward verification shows that $P_{i}(0\leq i\leq 3)$ and $T_{j}(1\leq j\leq n-1)$ are uniserial, that is $0\subset S_{i-1(mod\; 4)}\subset P_{i}$ and $0\subset M_{n-j}\subset T_{j}$ 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 $0\leq i\leq3, \; P_{i}$ is the projective cover of $S_{i}$.

(2) For all $1\leq j\leq n-1, \; T_{j}$ 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 $M\otimes _{ \mathbb{k}}N$ is a left $H$-module defined by

for all $h\in H, \; m\in M$ and $n\in N,$ where $\Delta(h) = \sum_{(h)}h_{(1)}\otimes h_{(2)}.$

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 $0\leq i, j\leq3,$

(b) For $1\leq j\leq n-1,$

(c) For $0\leq i, j\leq 3,$

(d) For $1\leq j\leq n-1,$

(2) (a) For $1\leq i, j\leq n-1,$

(b) For $1\leq i\leq n-1, \; 0\leq j\leq 3,$

(c) For $1\leq i, j\leq n-1,$

(3) (a) For $0\leq i, j \leq 3,$

(b) For $1\leq j \leq n-1, 0\leq i \leq 3,$

(4) For $1\leq i, j\leq n-1,$

Proof. For $S_{i}\otimes S_{j},$ let $v_{ij}: = v_{i}\otimes v_{j},$ then

thus $S_{i}\otimes S_{j}\cong S_{i+j(mod\; 4)}\cong S_{j}\otimes S_{i}.$

For $S_{i}\otimes M_{j},$ let $v_{ij}^{1}: = v_{i}\otimes v_{j}^{1}, \; v_{ij}^{2}: = v_{i}\otimes v_{j}^{2}, \;$ then

thus $S_{i}\otimes M_{j}\cong M_{j}$ when $i = 0, 2$ and $S_{i}\otimes M_{j}\cong M_{n-j}$ when $i = 1, 3.$ Similarly, we can get the same results of $M_{j}\otimes S_{i}.$

For $S_{i}\otimes P_{j},$ let $\mu_{ij}^{1}: = v_{i}\otimes \mu_{j}^{1}, \; \mu_{ij}^{2}: = v_{i}\otimes \mu_{j}^{2},$ then

thus $S_{i}\otimes P_{j}\cong P_{i+j(mod\; 4)}.$ The same results of $P_{j}\otimes S_{i}$ can be obtained in the same way.

For $S_{i}\otimes T_{j},$ let $\vartheta_{ij}^{1}: = v_{i}\otimes \vartheta_{j}^{1}, \; \vartheta_{ij}^{2}: = v_{i}\otimes \vartheta_{j}^{2}, \; \vartheta_{ij}^{3}: = v_{i}\otimes \vartheta_{j}^{3}, \; \vartheta_{ij}^{4}: = v_{i}\otimes \vartheta_{j}^{4}, \;$ then

thus $S_{i}\otimes T_{j}\cong T_{j}$ when $i = 0, 2$ and $S_{i}\otimes T_{j}\cong T_{n-j}$ when $i = 1, 3.$ Similarly, we get the same results of $T_{j}\otimes S_{i}.$

For $M_{i}\otimes M_{j},$ let $\omega_{ij}^{1}: = v_{i}^{1}\otimes v_{j}^{1}, \; \omega_{ij}^{2}: = v_{i}^{1}\otimes v_{j}^{2}, \; \omega_{ij}^{3}: = v_{i}^{2}\otimes v_{j}^{1}, \; \omega_{ij}^{4}: = v_{i}^{2}\otimes v_{j}^{2}, \;$ then

Obviously, when $0 < i+j, \; i-j < n, \; M_{i}\otimes M_{j}\cong M_{i+j}\otimes M_{i-j}.$ Besides, we need to note that when $n < i+j < 2n,$ let $\bar{\omega}_{ij}^{1}: = \omega_{ij}^{4}, \; \bar{\omega}_{ij}^{2}: = (-1)^{i+j}\omega_{ij}^{1},$ then

Therefore, $\mathbb{k}\{\bar{\omega}_{ij}^{1}, \; \bar{\omega}_{ij}^{2}\}\cong M_{i+j}.$ When $-n < i-j < 0, \; n < i-j+2n < 2n,$ using the previous conclusion, we directly get that $\mathbb{k}\{\bar{\omega}_{ij}^{3}: = \omega_{ij}^{3}, \; \bar{\omega}_{ij}^{4}: = (-1)^{i}\omega_{ij}^{2}\}\cong M_{j-i}.$

In particular, when $i+j = n,$ the matrices of $x, \; y, \; z$ acting on the basis $\{\bar{\omega}_{ij}^{1}, \; \bar{\omega}_{ij}^{2}\}$ are simultaneously diagonalizable. Thus the modules are isomorphic to $S_{1}\oplus S_{3}$; when $i-j = 0,$ the matrices of $x, \; y, \; z$ acting on the basis $\{\bar{\omega}_{ij}^{3}, \; \bar{\omega}_{ij}^{4}\}$ are also simultaneously diagonalizable and isomorphic to $S_{0}\oplus S_{2}.$ In fact we might assume that $i\geq j$ since the result of $M_{j}\otimes M_{i}$ are the same as $M_{i}\otimes M_{j}.$

For $M_{i}\otimes P_{j},$ let $\bar{\nu}_{ij}^{1}: = v_{i}^{1}\otimes \mu_{j}^{1}, \; \bar{\nu}_{ij}^{2}: = \omega^{j}v_{i}^{2}\otimes \mu_{j}^{1}, \; \bar{\nu}_{ij}^{3}: = v_{i}^{1}\otimes \mu_{j}^{2}, \; \bar{\nu}_{ij}^{4}: = \omega^{j}v_{i}^{2}\otimes \mu_{j}^{2},$ then

Thus when $j = 0, 2, \; M_{i}\otimes P_{j}\cong P_{j}\otimes M_{i}\cong T_{i};$ when $j = 1, 3, \; M_{i}\otimes P_{j}\cong P_{j}\otimes M_{i}\cong T_{n-i},$

For $M_{i}\otimes T_{j},$ let $\{\bar{\vartheta}_{ij}^{1}: = v_{i}^{1}\otimes \vartheta_{j}^{1}, \; \bar{\vartheta}_{ij}^{2}: = v_{i}^{2}\otimes \vartheta_{j}^{2}, \; \bar{\vartheta}_{ij}^{3}: = v_{i}^{1}\otimes \vartheta_{j}^{3}, \; \bar{\vartheta}_{ij}^{4}: = v_{i}^{2}\otimes \vartheta_{j}^{4}, \; \bar{\vartheta}_{ij}^{5}: = v_{i}^{1}\otimes \vartheta_{j}^{2}, \; \bar{\vartheta}_{ij}^{6}: = (-1)^{j}v_{i}^{2}\otimes \vartheta_{j}^{1}, \; \bar{\vartheta}_{ij}^{7}: = v_{i}^{1}\otimes \vartheta_{j}^{4}, \; \bar{\vartheta}_{ij}^{8}: = (-1)^{j}v_{i}^{2}\otimes \vartheta_{j}^{3}\},$ then

Obviously, when $0 < i+j, \; i-j < n, \; M_{i}\otimes T_{j}\cong T_{i+j}\oplus T_{i-j}.$ It has been shown that in Theorem 3.1 when $n < i+j < 2n,$ these modules $\mathbb{k}\{\bar{\vartheta}_{ij}^{1}, \; \bar{\vartheta}_{ij}^{2}, \; \bar{\vartheta}_{ij}^{3}, \; \bar{\vartheta}_{ij}^{4}\}$ 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 $P_{i}\otimes P_{j},$ we have known that $0\subset S_{i-1(mod\; 4)}\subset P_{i}$ is the unique composition series of $P_{i}$ for $0\leq i\leq 3.$ Thus there is an exact sequence

By Corollary 3.2, $P_{i} = P(S_{i}),$ so there is a split exact sequence

Hence

For $P_{i}\otimes T_{j},$ similarly, we have the split exact sequence

Hence

For $T_{i}\otimes T_{j}, \; i > j,$ the unique composition series of $T_{i}$ is $0\subset M_{n-i}\subset T_{i}$ for $1\leq i\leq n-1,$ and there is an exact sequence

and the split exact sequence

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.

Corollary 3.4. The tensor product of any two $D(n)$-modules is commutative.

4.   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 $[M][N] = [M\otimes N]$. 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 $[M \oplus N ] = [M] + [N ]$. It follows that the representation ring $r(H)$ is an associative ring with identity given by $[ \mathbb{k}_\varepsilon]$, the trivial 1-dimensional $H$-module. Note that $r(H)$ has a $\mathbb{Z}$-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.

Let $F_{q}(y, z)$ be the generalized Fibonacci polynomials defined by

for $q \geq 1$, while $F_{0}(y, z) = 0, \; F_{1}(y, z) = 1, \; F_{2}(y, z) = z$. These generalized Fibonacci polynomials appeared in ^[13,14].

Lemma 4.1. [13,Lemma 3.11] Let $\mathbb{Z}[y, z]$ be the polynomial algebra over $\mathbb{Z}$ in two variables $y$ and $z$. Then for any $q\geq1$, we have

where $\left[\frac{q-1}{2}\right]$ denotes the biggest integer which is not bigger than $\frac{q-1}{2}$.

Let $[S_{1}] = \alpha$, $[M_{1}] = \beta$ and $[P_{0}] = \gamma$. In the following the sum $\sum_{i = 0}^{m}$ disappears if $m < 0.$

Lemma 4.2. The following statements hold in $r(D(n))$.

1. $\alpha^4 = 1, \; \alpha^{2}\beta = \beta, \; \alpha\gamma = \gamma(\gamma-1)$;

2. $[S_{i}] = \alpha^{i}$, $[P_{i}] = \alpha^{i}\gamma\; (0\leq i\leq 3)$;

3. For $1\leq j\leq n-1,$

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

and

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 $M_{j}\otimes P_{0}\cong T_{j}$, thus the Eq (4.2) is obvious to be obtained.

Corollary 4.3. Keep the notations above.

1. If $\frac{n-1}{2}$ is odd, then

2. If $\frac{n-1}{2}$ is even, then

Proof. Since $[S_{1}][M_{\frac{n-1}{2}}] = [M_{\frac{n+1}{2}}]$ by Theorem 3.3(1)(b), and using the equations of (4.1) in Lemma 4.2, we can easily get the results.

Corollary 4.4. Keep notations as above, then the sets

form a $\mathbb{Z}$-basis of $r(D(n))$.

Proof. By Lemma 4.1, $\alpha^4 = 1$, and there is a one-to-one correspondence between the set $\{\alpha^{i}, \alpha^{i}\gamma\mid 0 \leq i \leq 3\}$ and the set of $D(n)$-modules $\{[S_{i}], [P_{i}]\mid 0 \leq i \leq 3\}$. By the Eq $(4.1),$ we know that $[M_{j}]$ is a $\mathbb{Z}$-polynomial with $\alpha$ and $\beta$, and when $j\geq\frac{n+1}{2}, \; [M_{j}] = [S_{1}][M_{n-j}].$ By Corollary 4.3, we know that the highest degree of $\beta$ in this polynomial is $\frac{n-1}{2},$ and $\{[M_{j}]\mid1\leq j\leq n-1\}$ is a $\mathbb{Z}$-linear combination of $\{\alpha^{i}\mid0\leq i\leq 3\},$ $\{\alpha\beta^{j}\mid 1\leq j\leq \frac{n-1}{2}\}$ and $\{\beta^{j}\mid 1\leq j\leq \frac{n-1}{2}\}.$ Consequently, $[T_{j}]$ is a $\mathbb{Z}$-polynomial with $\alpha, \; \beta, \; \gamma$ and the highest degree of $\gamma$ in this polynomial is $1$ since $\gamma^{2} = \alpha\gamma+\gamma$. Therefore $[T_{j}]$ is a $\mathbb{Z}$-linear combination of $\{\alpha^{i}\mid0\leq i\leq 3\}$, $\{\alpha\beta^{j}\gamma\mid 1\leq j\leq \frac{n-1}{2}\}$ and $\{\beta^{j}\gamma\mid 1\leq j\leq \frac{n-1}{2}\}.$

The result is obtained.

Theorem 4.5. The representation ring $r(D(n))$ is a commutative ring generated by $a, \; b, \; c,$ subject to the following relations

and

if $\frac{n-1}{2}$ is odd, or

if $\frac{n-1}{2}$ is even.

Proof. By Corollary 3.4, we know that the ring $r(D(n))$ is a commutative ring generated by $\alpha$, $\beta$ and $\gamma$, there is a unique ring epimorphism

from $\mathbb{Z}[a, b, c]$ to $r(D(n))$ such that

By Lemma 4.1, there is

thus we have

Note that by Corollary 4.3,

when $\frac{n-1}{2}$ is odd, or

when $\frac{n-1}{2}$ is even. Thus $\Phi$ maps

or

to 0. It follows that $\Phi (I) = 0,$ and $\Phi$ induces a ring epimorphism

such that $\overline{\Phi}(\overline{\nu}) = \Phi (\nu)$ for all $\nu\in \mathbb{Z}[a, b, c]$, where $\overline{\nu} = \pi(\nu)$ and $\pi$ is the natural epimorphism $\mathbb{Z}[a, b, c]\rightarrow \mathbb{Z}[a, b, c]/I$. Note that the ring $r(D(n))$ is a free $\mathbb{Z}$-module of rank $2n+6$ with the $\mathbb{Z}$-basis $\{\alpha^{i}\gamma^{k}\mid\; 0 \leq i\leq 3, \; 0\leq k\leq 1\}\cup\{\alpha^{i}\beta^{j}\gamma^{k}\mid\; 0 \leq i\leq 1, \; 1\leq j\leq \frac{n-1}{2}, \; 0 \leq k \leq 1\}$, so we can define a $\mathbb{Z}$-module homomorphism

On the other hand, as a free $\mathbb{Z}$-module, $\mathbb{Z}[a, b, c]/I$ is generated by elements $a^{i}c^{k}\; (0\leq i\leq3, \; 0\leq k\leq 1$) and $a^{i}b^{j}c^{k}\; (0 \leq i\leq 1, \; 1\leq j\leq \frac{n-1}{2}, \; 0 \leq k \leq 1)$, we have

Hence $\Psi\overline{\Phi} = id$, and $\overline{\Phi}$ is injective. Thus, $\overline{\Phi}$ is a ring isomorphism.

The proof is finished.

Example 4.6. We have the following examples.

● The representation ring $r(D(3))$ is a commutative ring generated by $a, \; b, \; c,$ subject to the following relations

● The representation ring $r(D(5))$ is a commutative ring generated by $a, \; b, \; c,$ subject to the following relations

5.   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 $D(n)$.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11671024).

Conflict of interest

The authors declare that they have no competing interests.