Projective class rings of a kind of category of Yetter-Drinfeld modules

1.   Introduction

The category of Yetter-Drinfeld modules over a Hopf algebra $H$ was introduced firstly in ^[1], which provides a solution to the Yang-Baxter equation ^[2] when the antipode of $H$ is bijective. In 1998, Andruskiewitsch and Schneider ^[3] introduced the $lifting\; method$ which was extensively used in the classification of finite dimensional pointed and copointed Hopf algebras. It is remarked that the Yetter-Drinfeld modules play an important role in this process. More precisely, by determining the braiding in ${ }_H^H{\mathcal{YD}}$ and indecomposable objects in the category of Yetter-Drinfeld modules over a Hopf algebra $H$, one can construct all finite dimensional Nichols algebras in ${ }_H^H{\mathcal{YD}}$, and then all finite dimensional Hopf algebras over $H$ by the lifting method. There are a lot of works to classify finite dimensional Hopf algebras by lifting method, see for example ^{[4,5,6,7,8,9,10,11,12,13]}. Therefore, it is important to understand the structures of Yetter-Drinfeld modules for a finite dimensional Hopf algebra.

In 2003, Radford ^[14] gave an idea of constructing the simple Yetter-Drinfeld modules: any Yetter-Drinfeld $H$-module $M$ is the form $M = H\cdot N$ for some simple subcomodule $N$ of $H\otimes L$, where $L$ is a left $H$-module. In 2012, Zhu and Chen ^[15] gave the classification of all simple Yetter-Drinfeld modules over the Hopf-Ore extension $A(n, 0)$ of the dihedral group $\mathcal{D}_n$ for even or odd nummber $n$. In 2020, Yang and Zhang ^[16] classified all Hopf algebra structures on the quotient of Ore extensions $H_4[z; \sigma]$ of automorphism type for the Sweedler's 4-dimension Hopf algebra $H_4$, thereby obtaining a family of non-pointed and non-seimsimple Hopf algebras $H_{4n}$ of rank one. The classifications of finite dimensional Hopf algebras over $H_{8}$ and $H_{12}$ were given in ^[8] and ^[9] respectively. The Green ring of $H_{4n}$ was determined by Chen and Yang et al. ^[17]. Xiong ^[18] (see also Zhang ^[19] in somewhat different idea) classified all simple Yetter-Drinfeld modules over $H_{4n}$ and gave the structures of projective class rings of the category of the Yetter-Drinfeld modules of $H_{4n}$. In ^[20], we classified the Yetter-Drinfeld modules over $H_{2n^2}$ and gave the Grothendieck rings of the category of the Yetter-Drinfeld modules of $H_{2n^2}$, where $H_{2n^2}$ is a family of Kac-Paljutkin semisimple Hopf algebra of dimension $2n^2$.

Motivated by the above works, a family of non-pointed $8m$-dimension Hopf algebras $H_{8m}$ of tame type with rank two are studied in ^[21], where $m$ is even. It is pointed that $H_{8m}$ is a special biserial algebra, which is just one subclass of basic tame Hopf algebras with only one block in ^[22]. By the technique of special biserial algebras (see for example ^[23]), we construct and classify the isomorphism classes of all indecomposable modules of $H_{8m}$, and determine the components of Auslander-Reiten quivers. Furthermore, we establish the tensor product of arbitrary simple (or projective) modules and indecomposable modules, and characterize the projective class rings and Grothendieck rings of $H_{8m}$.

In this paper, we focus on the classification of simple (or indecomposable projective) Yetter-Drinfeld modules over $H_{8m}$ by Radford's method and the description of the projective class rings of the category of the Yetter-Drinfeld modules over $H_{8m}$. In our further works, we hope that the classification of finite dimensional Hopf algebras over $H_{8m}$ is established.

Throughout this paper, $\mathbb{K}$ is assumed to be an algebraically closed field of characteristic zero, $m\geq 2$ is even and $\omega$ the $2m$-th primitive root of unity. The Sweedler's notation

for a Hopf algebra is used, and some other notations see ^[24].

2.   Preliminaries

In this section, let us recall some basic definition and results for the Hopf algebra $H_{8m}$.

By definition, the Hopf algebra $H_{8m}$ as an algebra is generated by $x_i(i = 1, 2), z$ with the following relations

The co-multiplication, counit and antipode are given as follows:

for $i = 1, 2$. It is easy to see that

is a basis of $H_{8m},$ and

for $l\in\mathbb{Z}_{2m},$ where $a_l = \frac{1-\omega^{-2l}}{1-\omega^{-2}}.$ In particular,

In ^[21], all finite dimensional simple modules of $H_{8m}$ were constructed and classified. There exist exactly $2m$ pairwise non-isomorphic $1$-dimension simple $H_{8m}$-modules $S_i$ with the basis $\{\upsilon^i\}$ for $i\in\mathbb{Z}_{2m}$. The actions are given by

Let ${ }_H \mathcal{M}$ be the category of left $H$-modules. Recall that a left Yetter-Drinfeld $H$-module $M$ for a finite dimensional Hopf algebra $H$ is a left $H$-module $(M, \cdot)$ and a left $H$-comodule $(M, \rho)$ satisfying

where $S$ is the antipode of $H$ and $\rho(m) = \sum\limits_{(m)} m_{(-1)}\otimes m_{(0)}$. The category of left Yetter-Drinfeld $H$-modules is denoted by ${ }_H^H{\mathcal{YD}}$, whose morphisms are both $H$-linear and $H$-colinear maps (see ^[1]). Let $V\in{ }_H^H{\mathcal{YD}}$, the left dual $V^*$ is defined by

According to Radford's results in ^[14], we have the following results.

Proposition 2.1. If $V, W\in{ }_H^H{\mathcal{YD}},$ then $V\otimes W\in{ }_H^H{\mathcal{YD}}.$ The actions and coactions are as follows:

where $\upsilon\in V, \omega\in W, h\in H.$

Lemma 2.2. Let $L\in{ }_H \mathcal{M}$. Then, we have

(1) $H\otimes L\in { }_H^H{\mathcal{YD}}$; the module and comodule actions are given by

(2)If $M$ is a simple Yetter-Drinfeld $H$-module, then $M = H\cdot N$ for some simple subcomodule $N$ of $H\otimes L$ where $L$ is a simple left $H$-module.

By Proposition 2.1 and Lemma 2.2, we can construct all simple Yetter-Drinfeld modules over any Hopf algebra $H$.

3.   Simple and indecomposable projective Yetter-Drinfeld modules over $H_{8m}$

To classify the simple Yetter-Drinfeld modules over the Hopf algebra $H_{8m}$, we firstly give all simple subcomodules of $H_{8m}\otimes S_i$ for all $i\in\mathbb{Z}_{2m}$.

Lemma 3.1. Let $0\neq L$ be a subcomodule of $H_{8m}\otimes S_i$, then there exists $j\in\mathbb{Z}_{2m}$ such that $z^j\otimes\upsilon^i\in L$ or $(z^j+lx_2z^j)\otimes\upsilon^i\in L$, where $0\neq l\in\mathbb{K}$.

Proof. Let $0\neq L$ be a subcomodule of $H_{8m}\otimes S_i$ and $0\neq u = (\sum_{j = 0}^{2m-1}\sum_{s, t = 0}^{1}k_{s, t, i}x_1^sx_2^tz^i)\otimes\upsilon^i\in L$, where $k_{s, t, j}\in\mathbb{K}$. By (2.4), we have

where

Assume that $j\neq 0, m$ (the discussions when $j = 0, m$ are similar). Note that there exist some $j$ such that the vector $\theta_j = (k_{0, 0, j}, k_{1, 0, j}, k_{0, 1, j}, k_{1, 1, j})\neq 0$ and $a_j\neq 0$. Now, we complete the proof by discussing the cases $(i)$-$(vi)$.

(i) $k_{1, 1, j}\neq 0, k_{1, 0, j}\neq 0$.

(a) If $\alpha_{0, j}$ and $\alpha_{2, j}$ are linearly independent, we have $z^j\otimes\upsilon^i\in L$. Moreover, $x_2z^{m+j}\otimes\upsilon^i\in L$ since

(b) If $\alpha_{0, j}$ and $\alpha_{2, j}$ are linearly dependent, there exist a $0\neq l\in \mathbb{K}$ such that $(z^j+lx_2z^j)\otimes\upsilon^i\in L$. Moreover, $(a_jx_2z^{m+j}+lz^{m+j})\otimes\upsilon^i\in L$ since

(ii) $k_{1, 1, j}\neq 0, k_{1, 0, j} = 0$, we have $z^j\otimes\upsilon^i\in L$;

(iii) $k_{1, 1, j} = 0, k_{1, 0, j}\neq 0$, we have $x_2z^j\in L$. Moreover, $z^{m+j}\otimes\upsilon^i\in L$ since

(iv) $k_{1, 1, j} = 0, k_{1, 0, j} = 0, k_{0, 0, j}\neq 0, k_{0, 1, j}\neq 0$, similar to $(i)$;

(v) $k_{1, 1, j} = 0, k_{1, 0, j} = 0, k_{0, 0, j}\neq 0, k_{0, 1, j} = 0$, similar to $(ii)$;

(vi) $k_{1, 1, j} = 0, k_{1, 0, j} = 0, k_{0, 0, j} = 0, k_{0, 1, j}\neq 0$, similar to $(iii)$.

Lemma 3.2. For $i\in\mathbb{Z}_{2m},$ all simple subcomodules of $H_{8m}\otimes S_i$ are as follows:

(1) $U(i, j): = \mathbb{K}\{z^{j}\otimes\upsilon^i\},$ where $j = 0, m;$

(2) $V(i, j): = \mathbb{K}\{z^j\otimes\upsilon^i, x_2z^{m+j}\otimes\upsilon^i\},$ where $j \in \mathbb{Z}_{2m} \;\mathit{\mbox{and}}\; j\neq 0, m;$

(3) $W(i, j, l): = \mathbb{K}\{(z^{j}+lx_2z^j)\otimes\upsilon^i, (a_jx_2z^{m+j}+lz^{m+j})\otimes\upsilon^i\},$ where $0\neq l\in\mathbb{K}, j \in \mathbb{Z}_{2m} \;\mathit{\mbox{and}}\; j\neq 0, m.$

Proof. It is easy to see that $U(i, j)$ is a simple subcomodule of $H_{8m}\otimes S_i$ for $j = 0, m$. One also see that $V(i, j)$ is a subcomodule of $H_{8m}\otimes S_i$ by (2.1). We show that $V(i, j)$ is a simple. Indeed, let $0\neq V$ be a subcomodule of $V(i, j)$ and $0\neq \upsilon = (k_{i, j}z^{j}+l_{i, j}x_2z^{m+j})\otimes\upsilon^i\in V$, where $k_{i, j}, l_{i, j}\in\mathbb{K}$ and $(k_{i, j}, l_{i, j})\neq 0$. By (2.4), we have

One can check that $k_{i, j}z^j+l_{i, j}x_2z^{m+j}, k_{i, j}a_jx_2z^{j}+l_{i, j}z^{m+j}$ are linearly independent. It follows that $z^{j}\otimes\upsilon^i, x_2z^{m+j}\otimes\upsilon^i\in V$ and $V = V(i, j)$. Hence, $V(i, j)$ is a simple subcomodule of $H_{8m}\otimes S_i$. Similarly, we have $W(i, j, l)$ is a simple subcomodule of $H_{8m}\otimes S_i$ for $0\neq l\in\mathbb{K}, j \in \mathbb{Z}_{2m} \mbox{ and } j\neq 0, m.$ By Lemma3.1, the vector spaces defined in $(1)$–$(3)$ are all simple subcomodules of $H_{8m}\otimes S_i$.

Lemma 3.3. As Yetter-Drinfeld modules, $H_{8m}\cdot V(i, j) = H_{8m}\cdot W(i, j, l)$, where $0\neq l\in\mathbb{K}, j \in \mathbb{Z}_{2m} \;\mathit{\mbox{and}}\; j\neq 0, m.$

Proof. By (2.3), we have

By (2.4), we have

Then $z^j\otimes\upsilon^i\in H_{8m}\cdot W(i, j, l)$. Hence

Therefore, as Yetter-Drinfeld modules, we have $H_{8m}\cdot V(i, j) = H_{8m}\cdot W(i, j, l)$, where $0\neq l\in\mathbb{K}, j \in \mathbb{Z}_{2m} \mbox{ and } j\neq 0, m.$

By Lemma 3.2 and Lemma 3.3, we just need to consider the following Yetter-Drinfeld modules:

$\bullet \; H_{8m}\cdot U(i, j),$ where $i\in\mathbb{Z}_{2m}, j = 0, m$;

$\bullet \; H_{8m}\cdot V(i, j)$, where $i, j\in\mathbb{Z}_{2m}, j\neq 0, m$.

Now, we consider $M(i, j): = H_{8m}\cdot U(i, j)$, where $i\in\mathbb{Z}_{2m}, j = 0, m$. There are two cases:

Case 1: $M_i: = M(i, j)$ for $i\in\mathbb{Z}_{2m}$, where $j = 0$ if $i$ is even and $j = m$ if $i$ is odd. $M_i$ are $1$-dimension simple Yetter-Drinfeld modules. The actions on the basis $\{\nu^i = z^{im}\otimes \upsilon^i\}$ and the coactions are given by

Case 2: $M(i, j)$ for $i\in\mathbb{Z}_{2m}$, where $j = 0$ if $i$ is odd and $j = m$ if $i$ is even. $M(i, j)$ is a $4$-dimension simple Yetter-Drinfeld module. The action on the basis $\{\nu_0^{i, j} = z^j\otimes\upsilon^i, \nu_1^{i, j} = x_1z^{m+j}\otimes\upsilon^i, \nu_2^{i, j} = x_2z^{m+j}\otimes\upsilon^i, \nu_3^{i, j} = x_1x_2z^{j}\otimes\upsilon^i\}$ and the coaction are given by

For $N(i, j): = H_{8m}\cdot V(i, j)$, $i, j\in\mathbb{Z}_{2m}, j\neq 0, m$. $N(i, j)$ are $4$-dimension simple Yetter-Drinfeld modules. The action on the basis $\{\nu_0^{i, j} = z^j\otimes\upsilon^i, \nu_1^{i, j} = x_1z^{m+j}\otimes\upsilon^i, \nu_2^{i, j} = x_2z^{m+j}\otimes\upsilon^i, \nu_3^{i, j} = x_1x_2z^{j}\otimes\upsilon^i\}$ and the coaction are given by

Here $a_{i, j} = (-1)^{i}+(-1)^{j+1}\omega^{-j}, b_{i, j} = (-1)^{i}-\omega^{-j}, a_j = \frac{1-\omega^{-2j}}{1-\omega^{-2}}.$

Let $\Lambda = \{(i, j)|i, j\in\mathbb{Z}_{2m} \text{ and } j\not\equiv im\pmod{2m}\}$ and

where $(i, j)\in\Lambda.$ Note that

In the sequel the operations of subscripts are in $\mathbb{Z}_{2m}$.

We get the first result of this paper.

Theorem 3.4. The set

forms a complete list of non-isomorphic simple Yetter-Drinfeld modules over $H_{8m}$.

Proof. Firstly, we show that

Let $f: M_{i}\rightarrow M_{i'}, \nu^{i}\mapsto a\nu^{i'}$ be a Yetter-Drinfeld module isomorphism, where $0\neq a\in\mathbb{K}.$ Then, we have

Hence $i = i'.$

Now we show that each $N_{i, j}$ for $(i, j)\in\Lambda$ is simple Yetter-Drinfeld module.

Let $0\neq V$ be a Yetter-Drinfeld submodule of $N(i, j)$ and $0\neq \upsilon = \sum\limits_{k = 0}^3\alpha_k^{i, j}\nu_k^{i, j}\in V$, where $\alpha_k^{i, j}\in\mathbb{K}$ and $(\alpha_0^{i, j}, \alpha_1^{i, j}, \alpha_2^{i, j}, \alpha_3^{i, j})\neq 0$. By (2.4), we have

If $(\alpha_1^{i, j}, \alpha_3^{i, j}) = 0$, it is easy to see that $\nu_0^{i, j}\in V$. If $(\alpha_1^{i, j}, \alpha_3^{i, j})\neq 0$, $\alpha_0^ {i, j}z^j+\alpha_1^{i, j}x_1z^{m+j}+\alpha_2^{i, j}x_2z^{m+j}+\alpha_3^{i, j}x_1x_2z^{j}$ and $\alpha_1^{i, j}a_jx_2z^{m+j}+\alpha_3^{i, j}z^{j}$ are linearly independent. We also get that $\nu_0^{i, j}\in V$. It follows that

Hence, $V = N(i, j)$ and $N(i, j)$ is a simple Yetter-Drinfeld module.

Finally we show that

Let $g: N_{i, j}\rightarrow N_{i', j'}, \nu_{k}^{i, j}\mapsto \sum\limits_{l = 0}^{3}\alpha_{k, l}\nu_{l}^{i', j'},$ where $k\in\{0, 1, 2, 3\}, \; \alpha_{k, l}\in\mathbb{K}$ and there exists $l\in\{0, 1, 2, 3\}$ such that $\alpha_{k, l}\neq 0$ for each $k\in\{0, 1, 2, 3\}.$ Then we have

Hence $i = i'$ and $\alpha_{k, l} = 0$ for $k\neq l.$ Also, we have

Hence $j = j'$ and $\alpha_{0, 0} = \alpha_{1, 1} = \alpha_{2, 2} = \alpha_{3, 3}.$ Therefore,

The proof is finished.

Corollary 3.5. $N_{i, j}^*\cong N_{m+2-i, -j}$ for $(i, j)\in\Lambda.$

Proof. Let $\{\mu_k^{i, j}|k\in\{0, 1, 2, 3\}\}$ be the dual basis of $\{\nu_k^{i, j}|k\in\{0, 1, 2, 3\}\}$ such that

By (2.2), we have

and

Hence $N_{i, j}^*\cong N_{m+2-i, -j}$.

4.   Projective class rings of ${ }_{H_{8 m}}^{H_{8 m}}{\mathcal{YD}}$

Assume that $H$ is a finite dimensional Hopf algebra. Let $F(H)$ be the free abelian group generated by the isomorphic classes $[M]$ of $H$-modules $M$. Then the abelian group $F(H)$ becomes a ring equipped with a multiplication given by the tensor product $[M][N] = [M\otimes N]$. The Green ring $r(H)$ is defined to be the quotient ring of $F(H)$ module the relations $[M\oplus N] = [M]+[N]$. The projective class ring of $H$ is the subring of $r(H)$ generated by their projective and simple representations of $H$. As is known, ${ }_H^H{\mathcal{YD}}\cong _{\mathcal{D}\left(H^{c o p}\right)} \mathcal{M}.$ where $_{\mathcal{D}\left(H^{c o p}\right)} \mathcal{M}$ is the category of the left modules of Drinfeld double $\mathcal{D}(H^{\rm cop})$. The projective class ring of $\mathcal{D}(H^{\rm cop})$, or equivalently, the projective class ring of ${ }_H^H{\mathcal{YD}}$ is denoted by $r_{P}(\mathcal{D}(H^{\rm cop})).$

In this section, we denote $\mathcal{D}: = \mathcal{D}(H_{8m}^{\rm cop}).$ Let $\mathcal{P}(V)$ be the projective cover of a simple $\mathcal{D}$-module $V$, or equivalently, a simple Yetter-Drinfeld module $V\in { }_{H_{8 m}}^{H_{8 m}}{\mathcal{YD}}$. It is well-known that $\mathcal{P}(V)$ is unique (up to isomorphism) indecomposable projective $\mathcal{D}$-module which maps onto $V$. Let ${\rm Irr}(\mathcal{D})$ be the set of isomorphism classes of simple $\mathcal{D}$-modules. One sees that

and $\mathcal{D}$ is unimodular and quasi-triangular (see ^[24,25]).

Let us determine the projective class ring $r_{P}(\mathcal{D})$. For this purpose, we denote $\mathcal{P}_i: = \mathcal{P}(M_{i}), i\in\mathbb{Z}_{2m}$ and firstly introduce some lemmas.

Lemma 4.1. (1) $M_i\otimes M_j\cong M_{i+j}\cong M_j\otimes M_i$ and $M_i\otimes N_{k, l}\cong N_{i+k, im+l}\cong N_{k, l}\otimes M_i$ for $i, j\in\mathbb{Z}_{2m}, (k, l)\in\Lambda;$

(2) $\mathcal{P}(N_{i, j})\cong N_{i, j},$ $(i, j)\in\Lambda;$

(3) $M_i\otimes \mathcal{P}_j\cong \mathcal{P}_{i+j}\cong \mathcal{P}_j \otimes M_i$ and $\dim \mathcal{P}_i = 16$ for $i, j\in \mathbb{Z}_{2m};$

(4) For $(i, j), (k, l)\in\Lambda, h\in\mathbb{Z}_{2m}, {\rm Hom}(N_{i, j}\otimes N_{k, l}, M_h)\neq 0$ if and only if $h\equiv i+m+k-2\pmod{2m}$ and $j\equiv (i+k)m-l\pmod{2m}$.

Proof. (1) It follows from a direct computation.

(2) Suppose that $\mathcal{P}(N_{i, j})\ncong N_{i, j}$ for some $(i, j)\in\Lambda$. Since $\mathcal{D}$ is unimodular, ${\rm Soc}\mathcal{P}(N_{i, j})\cong N_{i, j},$ and $\dim\mathcal{P}(N_{i, j})\geq 2\dim N_{i, j} = 8.$ Since $\mathcal{P}(N_{i-k, j-mk})\otimes M_k$ is projective and

we have $\mathcal{P}(N_{i, j})\cong \mathcal{P}(N_{i-k, j-mk}\otimes M_k)\subset \mathcal{P}(N_{i-k, j-mk})\otimes M_k,$ which implies that

Let $J = \{(s, t)\in\Lambda|(s, t) = (i-k, j-mk) \text{\; for }\; k\in\mathbb{Z}_{2m}\}.$ It is obvious that

Then

It is a contradiction. Hence $\mathcal{P}(N_{i, j})\cong N_{i, j}$ for $(i, j)\in\Lambda.$

(3) Since $\mathcal{P}(N_{i, j})\cong N_{i, j}$ for any fixed $(i, j)\in\Lambda,$ it follows that $2m\dim \mathcal{P}_i = \dim\mathcal{D}-4|\Lambda|\dim N_{i, j} = 32m$ and hence $\dim \mathcal{P}_i = 16.$ Similar to (2), we have $\mathcal{P}_i\subseteq \mathcal{P}_0\otimes M_i$. Then, $\mathcal{P}_i\cong\mathcal{P}_0 \otimes M_i$. Hence, $\mathcal{P}_j\otimes M_i\cong \mathcal{P}_0\otimes M_j\otimes M_i\cong \mathcal{P}_0 \otimes M_{i+j}\cong \mathcal{P}_{i+j}.$

(4) Since $\mathcal{D}$ is quasi-triangular, $M\otimes_{\mathbb{K}} N\cong N\otimes_{\mathbb{K}} M$ for $M, N\in{ }_{\mathcal{D}}{\mathcal{M}}$,

Then by Schur's lemma, ${\rm Hom}_{\mathcal{D}}(N_{i, j}\otimes N_{k, l}, M_h)\neq 0$ if and only if $i\equiv h+m+2-k\pmod{2m}$ and $j\equiv hm-l\pmod{2m}$, if and only if $h\equiv i+m+k-2\pmod{2m}$ and $j\equiv (i+k)m-l\pmod{2m}$.

Corollary 4.2. We have

Now we describe the projective cover $\mathcal{P}_i$ of the simple module $M_i$ for $i\in\mathbb{Z}_{2m}.$ For convenience, we let

Let $\mathcal{P}$ be a vector space with a basis $\{p_{0}, p_{1}, \cdots, p_{15}\}$ and the action and coaction of $H_{8m}$ on $\mathcal{P}$ be the form:

where $P = (p_{0}, p_{1}, \cdots, p_{15})^T.$ By a complex computation, we know that $\mathcal{P}$ is a left Yetter-Drinfeld module.

Lemma 4.3. $\mathcal{P}$ is an indecomposable Yetter-Drinfeld module over $H_{8m}.$

Proof. Suppose that $\mathcal{P}$ is not indecomposable. Then there exist two non-trivial submodules $M$ and $N$ such that $\mathcal{P} = M\oplus N$. Let

and $K_{-1} = 0, K_l = \sum_{i = 0}^{l}\mathbb{K}\kappa_{i}$ for $l\in\{0, \cdots, 15\}$. One can check that

is a Yetter-Drinfeld submodules chain of $\mathcal{P}$ such that $K_l/K_{l-1}$ is a one dimensional Yetter-Drinfeld module. And

We claim that $\kappa_{15}\notin M$ and $\kappa_{15}\notin N$. If $\kappa_{15}\in M$, then $\kappa_{11}-\kappa_{2}, \kappa_{4}-\kappa_{12}, \kappa_{8}, \kappa_{13}, \kappa_{14}\in M$ since

which implies that

Then $M = \mathcal{P}.$ It's a contradiction. Similarly, if $\kappa_{15}\in N$, then $N = \mathcal{P}$ and also a contradiction. Hence the claim follows. Therefore, there exist some $\alpha_{i}\in\mathbb{K}$, for $i \in\{0, 1, \cdots, 14\},$ such that

Then

It is observe that $\hat{\kappa}_1 = \alpha_{0}\kappa_{0}+\alpha_{3}\kappa_{3}+\alpha_{5}\kappa_{5}+\alpha_{7}\kappa_{7}+\alpha_{8}\kappa_{8} +\alpha_{9}\kappa_{9}+\alpha_{10}\kappa_{10}+\kappa_{15}\in M.$ Hence

and

Then $\kappa_0, \kappa_1, \kappa_2, \kappa_3$ and $\alpha_{8}\kappa_4+\kappa_{14}, \alpha_{8}\kappa_6+\frac{\kappa_{11}}{2}\in M.$ Note that

Then $\kappa_4, \kappa_5, \kappa_6, \kappa_7, \kappa_9, \kappa_{10}, \kappa_{11}, \kappa_{14} \in M.$ Hence $\hat{\kappa}_2 = \alpha_{8}\kappa_{8}+\alpha_{12}\kappa_{12}+\alpha_{13}\kappa_{13}+\kappa_{15}\in M,$ and

Thus $\kappa_{15}\in M.$ It's a contradiction. Consequently, $\mathcal{P}$ is indecomposable.

Lemma 4.4. $\mathcal{P}\cong \mathcal{P}_0$ as $\mathcal{D}$-modules.

Proof. It is well known that $\mathcal{D}$ is a symmetric algebra ^[26] and every projective module is injective. In particular, $\mathcal{P}_0 = E(M_i)$ for some $i\in\mathbb{Z}_{2m}$ and the socle and top of $\mathcal{P}_0$ coincide. Therefore $\mathcal{P}_0\cong E(M_0)$. On the other hand, by Lemma 4.3, we know that $\mathcal{P}$ is an indecomposable module with $\mathrm{Soc}\mathcal{P}\cong M_0.$ Thus, $\mathcal{P}$ embeds in $E(M_0)$, which implies that $\mathcal{P}\cong E(M_0)$, since they have the same dimension. Hence, $\mathcal{P}\cong \mathcal{P}_0.$

Now we calculate the tensor decompositions of the simple and indecomposable projective Yetter-Drinfeld modules. Denote $\bigoplus_{\pm}V_{a\pm b}: = V_{a+b}\oplus V_{a-b},$ for $a, b\in\mathbb{Z}_{2m}, V\in{ }_{H_{8 m}}^{H_{8 m}}{\mathcal{YD}}.$

Lemma 4.5. (1) For $i, j\in\mathbb{Z}_{2m},$

(2) For $i, j\in\Lambda, k\in\mathbb{Z}_{2m},$

Proof. It suffices to prove the lemma for $\mathcal{P}_0\otimes \mathcal{P}_0$ and $\mathcal{P}_0\otimes N_{i, j}$ by Lemma 4.1 (1)(3).

By the proof of Lemma 4.3,

in the Grothendieck ring of the category of the Yetter-Drifeld modules over $H_{8m}.$ It follows that

and

The proof is finished.

Lemma 4.6. For $(i, j), (k, l)\in\Lambda,$ we have

Proof. If $j\equiv (i+k)m-l\pmod{2m},$ then by Lemma 4.1(4), ${\rm Hom}(N_{i, j}\otimes N_{k, l}, M_h)\neq 0$ if and only if $h\equiv i+m+k-2\pmod{2m}$. Since $N_{i, j}\otimes N_{k, l}$ is projective and $\mathrm{dim}\mathcal{P}_{i+m+k-2} = \mathrm{dim}(N_{i, j}\otimes N_{k, l}) = 16,$ we get that $\mathcal{P}_{i+m+k-2}\cong N_{i, j}\otimes N_{k, l}.$

If $j\not\equiv (i+k)m-l\pmod{2m},$ then by Lemma 4.1(4), ${\rm Hom}(N_{i, j}\otimes N_{k, l}, M_h) = 0$ for any $h\in\mathbb{Z}_{2m},$ which implies that $\mathcal{P}_h$ can not the direct summand of $N_{i, j}\otimes N_{k, l}$. Hence $N_{i, j}\otimes N_{k, l}$ has to be the direct sum of four 4-dimensional simple projective modules.

Let

for $r, s\in\{0, 1, 2, 3\}$ and

A direct computation shows that

Hence,

By Lemma 4.1, Lemma 4.5, Lemma 4.6, the projective class ring $r_{P}(\mathcal{D})$ is a commutative ring. Let $\dot{y}_0 = [M_1], \dot{y}_i = [N_{0, i}], i\in\{1, \cdots, 2m-1\}$

Lemma 4.7. The following statements hold in $r_{P}(\mathcal{D})$.

(1) $[M_i] = \dot{y}_0^{i}$ for $i\in\mathbb{Z}_{2m};$

(2) $[N_{i, j}] = \dot{y}_0^i\dot{y}_{j+im}$ for $(i, j)\in\Lambda;$

(3) $[P_i] = \dot{y}_0^{i+m+2}\dot{y}_m^2$ for $i\in\mathbb{Z}_{2m};$

(4) For $i, j, k, j+k\in\{1, \cdots, 2m-1\},$

Proof. The results are easy to get from Lemma 4.1, Lemma 4.5, Lemma 4.6.

Corollary 4.8. The following set is a $\mathbb{Z}$-basis of $r_{P}(\mathcal{D})$:

Proof. By Lemma 4.7(4), $\dot{y}_0^{2m} = 1$, and for $k, l, k+l\in\{1, \cdots, 2m-1\},$ $\dot{y}_k\dot{y}_m^2, \dot{y}_k\dot{y}_l$ can be expressed as a linear combination of $\{\dot{y}_0^{i}, \dot{y}_0^{i}\dot{y}_j, \dot{y}_0^{i}\dot{y}_m^2|i\in\mathbb{Z}_{2m}, j\in\{1, 2, \cdots 2m-1\}\}.$ It is easy to check that the set

is a independent set since $\#\{\dot{y}_0^{i}, \dot{y}_0^{i}\dot{y}_j, \dot{y}_0^{i}\dot{y}_m^2|i\in\mathbb{Z}_{2m}, j\in\{1, 2, \cdots 2m-1\}\} = 4m^2+2m$, the number of $\mathbb{Z}$-basis of $r_{P}(\mathcal{D})$.

Hence, $\{\dot{y}_0^{i}, \dot{y}_0^{i}\dot{y}_j, \dot{y}_0^{i}\dot{y}_m^2|i\in\mathbb{Z}_{2m}, j\in\{1, 2, \cdots 2m-1\}\}$ is a $\mathbb{Z}$-basis of $r_{P}(\mathcal{D})$.

The results of this section is as follows.

Theorem 4.9. The projective class ring $r_{P}(\mathcal{D})$ is isomorphic to the quotient ring of the ring $\mathbb{Z}[y_0, y_1, \cdots, y_{2m-1}]$ modulo the ideal $I$ generated by the following elements

where $i, j, k, j+k\in\{1, \cdots, 2m-1\}.$

Proof. By Corollary 4.8, there is a unique ring epimorphism

such that $\Phi(y_i) = \dot{y}_i$ for $i\in\mathbb{Z}_{2m}.$ By Lemma 4.7(4), we have

Let $I$ is the ideal generated by the elements (4.1). It follows that, $\Phi(I) = 0$ and $\Phi$ induces a natural ring epimorphism

such that $\overline{\Phi}(\overline{\nu}) = \Phi(\nu)$ for all $\nu\in\mathbb{Z}[y_0, y_1, \cdots, y_{2m-1}],$ where $\overline{\nu} = \nu+I$.

It is straightforward to check that the ring $\mathbb{Z}[y_0, y_1, \cdots, y_{2m-1}]/I$ is $\mathbb{Z}$-spanned by

This means the $\mathbb{Z}$-rank of $\mathbb{Z}[y_0, y_1, \cdots, y_{2m-1}]/I$ is $4m^2+2m$. Hence we get the ring isomorphism $\overline{\Phi}.$

5.   Conclusions

It's a challenging work to classify all indecomposable modules or indecomposable Yetter-Drinfeld modules over the finite dimensional Hopf algebras over an algebraically closed field of characteristic $p > 0$, and to classify all finite dimensional (Nichols) Hopf algebras over $H_{8m}$ over any field. This is our future attempt.

Acknowledgments

The work is supported by National Natural Science Foundation of China (Grant No. 11671024). The authors are grateful to thank the referee for careful reading and helpful suggestions.

Conflict of interest

The authors declare that they have no competing interests.