Projective class rings of the category of Yetter-Drinfeld modules over the $2$-rank Taft algebra

1.   Introduction

In 1998, Andruskiewitsch and Schneider (see ^[1]) introduced the $lifting\; method$ which was extensively used in the classification of finite dimensional pointed and copointed Hopf algebras. In this process, Yetter-Drinfeld modules over a Hopf algebra (see ^[2]) play important roles. See for example ^{[3,4,5,6,7,8,9,10,11,12]}. Therefore, constructing Yetter-Drinfeld modules over a particular interesting Hopf algebra is very important.

In 2003, Radford ^[13] introduced a general technique to construct simple Yetter-Drinfeld modules. By this idea, one can get all simple Yetter-Drinfeld modules over a finite dimensional Hopf algebra. For example, Zhu and Chen ^[14] described all simple Yetter-Drinfeld modules over the Hopf Ore extension of the dihedral group $\mathcal{D}_n$. Xiong ^[15] (see also Zhang ^[16] in somewhat different idea) constructed all simple (resp. indecomposable projective) Yetter-Drinfeld modules and then described their corresponding projective class rings over a non-semisimple and non-pointed Hopf algebra $H_{4n}$ which was firstly introduced in ^[17] and reconstructed via Hopf Ore extension of automorphism type in ^[18] by Yang and Zhang. It is remarked that all finite dimensional Hopf algebras over $H_{8}$ and $H_{12}$ were given in ^[7] and ^[8] respectively. In ^[19], all Yetter-Drinfeld modules and their corresponding Grothendieck rings for a $2n^2$-dimensional semisimple Hopf algebra $H_{2n^2}$ of Kac-Paljutkin type were constructed. In ^[20], all simple (resp. indecomposable projective) Yetter-Drinfeld modules and their corresponding projective class rings for a family of non-semisimple $8m$-dimensional Hopf algebra $H_{8m}$ of tame type with even number $m\geq2$.

Motivated by the above works, we will firstly construct and classify all simple (resp. indecomposable projective) Yetter-Drinfeld modules over $\mathcal{\bar{A}}$ by the Radford technique where $\mathcal{\bar{A}}$ is the $2$-rank Taft algebra which Green ring was constructed in ^[21]. It is remarked that $\mathcal{\bar{A}}$ is isomorphic to $H_2(0, -1)$ as Hopf algebras given by $f(x_1) = b, f(x_2) = c, f(y_1) = ab, f(y_2) = cd$ where $a, b, c, d$ are generators of $H_n(p, q)$. Here, $H_n(p, q)$ is firstly constructed in ^[22] which is a family of $n^4$-dimensional Hopf algebras where $n\geq 2$ is an integer, $p$ and $q$ are scalars in the ground field and $q$ is a primitive $n$-th root of unity. The second task of this paper is to describe the projective class rings of the category of the Yetter-Drinfeld modules over $\mathcal{\bar{A}}$. It is pointed that the projective class rings of $H_n(0, q)$ and $H_n(1, q)$ for $n > 2$ in ^[23] are desribed. Also, we note that the $n$-rank Taft algebra $\mathcal{\bar{A}}_q(n)$ based on the work in ^[24] was firstly introdcued in ^[25] in which its Drinfeld double $\mathcal{D}(\mathcal{\bar{A}}_q(n))$ as well as all simple modules of $\mathcal{D}(\mathcal{\bar{A}}_q(n))$ by the different method from the Radford technique, were constructed.

We organize the paper as follows. In Section $2$, we recall some definitions and results. In Section $3$, all simple (or indecomposable projective) Yetter-Drinfeld modules over $\mathcal{\bar{A}}$ are constructed. In Section $4$, the indecomposable projective Yetter-Drinfeld modules corresponding to the simple Yetter-Drinfeld modules of $\mathcal{\bar{A}}$ are constructed and the tensor products of arbitrary simple and indecomposable projective Yetter-Drinfeld modules are established. Furthermore, the projective class ring of the category of the Yetter-Drinfeld modules over $\mathcal{\bar{A}}$ is described.

We firstly fix some notations. Throughout the paper, $\mathbb{K}$ is assumed to be an algebraically closed field of characteristic zero and $\left[\kern-0.15em\left[{{a, b}}\right]\kern-0.15em\right] = \{a, a+1, \cdots, b\}$ for $a\leq b\in\mathbb{Z}$. The Sweedler notation

for a Hopf algebra is used and some other notations are referred to ^[26].

2.   Preliminaries

The $2$-rank Taft algebra $\mathcal{\bar{A}}$ was introduced in ^[21,22] which is generated by $x_i, y_j, i, j = 1, 2,$ subject to the relations

$\mathcal{\bar{A}}$ is a Hopf algebra with the coalgebra structure and the antipode given by

Note that ${\rm dim}\mathcal{\bar{A}} = 16$ and $\left\{x_1^kx_2^ly_1^sy_2^t|k, l, s, t = 0, 1\right\}$ forms a $\mathbb{K}$-basis for $\mathcal{\bar{A}}$.

For the definition of $n$-rank Taft algebra $\mathcal{\bar{A}}_q(n)$, the authors are referred to ^[25]. It is well-known that among $n$-rank Taft algebras, the only $2$-rank one is of tame type. From the viewpoint of representation theory, we are more interested in describing all simple (resp. indecomposable projective) modules in ${{_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}}$. Note that $\mathcal{\bar{A}} = \mathcal{\bar{A}}_{-1}(2)$.

Lemma 2.1. ^[21] There are $4$ non-isomorphic $1$-dimensional simple $\mathcal{\bar{A}}$-modules $S\left(s_1, s_2\right)$ with the basis $\left\{\upsilon^{s_1, s_2}\right\}$. The actions are given by

where $i = 1, 2$ and $s_1, s_2 = \pm1.$

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 ^[2]). Let $V\in{_{H}^{H}}{\mathcal{YD}}$, the left dual $V^*$ is defined by

According to Radford's results in ^[13], we have the following results.

Proposition 2.2. ^[13] 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 2.3. ^[13] 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.

Applying Proposition 2.2 and Lemma 2.3, we can construct all simple Yetter-Drinfeld modules over a specific Hopf algebra $H$, in particular, for $H = \mathcal{\bar{A}}$.

3.   Simple Yetter-Drinfeld modules over $\mathcal{\bar{A}}$

Let $U(k, l, s_1, s_2): = \mathbb{K}\left\{x_1^kx_2^l\otimes\upsilon^{s_1, s_2}\right\}$ where $s_1, s_2 = \pm1, k, l\in\mathbb{Z}_2.$ We have

Lemma 3.1. For $s_1, s_2 = \pm1$, the set

forms a complete set of all simple subcomodules of $\mathcal{\bar{A}}\otimes S(s_1, s_2)$.

Proof. Since $\dim\left(S(s_1, s_2)\right) = 1$, $\mathcal{\bar{A}}\otimes S(s_1, s_2)\cong \mathcal{\bar{A}}$ as (left) $\mathcal{\bar{A}}$-comodules. Since $\mathcal{\bar{A}}$ is pointed, any simple subcomodule of $\mathcal{\bar{A}}$ is equal to $\mathbb{K}g$ for some group-like element $g$ in $\mathcal{\bar{A}}$ by [26, Corollary 5.1.8]. Thus, the lemma follows.

Let

and

It's easy to see that $I = \bigcup\limits_{t = 0}^{3}I_t$.

Now, we consider the Yetter-Drinfeld module $\mathcal{\bar{A}}\cdot U(k, l, s_1, s_2)$ by the rules given in Lemma 2.3 where $(k, l, s_1, s_2)\in I$.

(a) Assume that $(k, l, s_1, s_2)\in I_0$. In this case

We get $1$-dimensional Yetter-Drinfeld modules $M_0(k, l): = \mathbb{K}\left\{\theta^{k, l}\right\}$ where $\theta^{k, l} = x_1^kx_2^l\otimes \upsilon^{(-1)^{k+l}, (-1)^{k+l}}, k, l\in\mathbb{Z}_2$. The actions and the coactions are given by

where $i = 1, 2.$

(b) Assume that $(k, l, s_1, s_2)\in I_1$. In this case

We get $2$-dimensional Yetter-Drinfeld modules $M_1(k, l): = \mathbb{K}\left\{\mu_0^{k, l}, \mu_1^{k, l}\right\}$ where $\mu_0^{k, l} = x_1^kx_2^l\otimes \upsilon^{s_1, s_2}, \mu_1^{k, l} = x_1^kx_2^ly_{2}\otimes \upsilon^{s_1, s_2}$, $s_1 = -s_2 = (-1)^{k+l}, k, l\in\mathbb{Z}_2$. The actions and the coactions are given by

where $i = 1, 2.$

Similarly, assume that $(k, l, s_1, s_2)\in I_2$ and we get $2$-dimensional Yetter-Drinfeld modules $M_2(k, l): = \mathbb{K}\left\{\nu_0^{k, l}, \nu_1^{k, l}\right\}$ where $\nu_0^{k, l} = x_1^kx_2^l\otimes \upsilon^{s_1, s_2}, \nu_1^{k, l} = x_1^kx_2^ly_{1}\otimes \upsilon^{s_1, s_2}$, $s_1 = -s_2 = (-1)^{k+l+1}, k, l\in\mathbb{Z}_2.$ The actions and the coactions are given by

where $i = 1, 2.$

(c) Assume that $(k, l, s_1, s_2)\in I_3$. In this case

We get $4$-dimensional Yetter-Drinfeld modules $M_3(k, l): = \mathbb{K}\left\{\eta_0^{k, l}, \eta_1^{k, l}, \eta_2^{k, l}, \eta_3^{k, l}\right\}$ where $\eta_0^{k, l} = x_1^kx_2^l\otimes \upsilon^{s_1, s_1}, \eta_1^{k, l} = x_1^kx_2^ly_{1}\otimes \upsilon^{s_1, s_1}, \eta_2^{k, l} = x_1^kx_2^ly_{2}\otimes \upsilon^{s_1, s_1}, \eta_3^{k, l} = x_1^kx_2^ly_{1}y_2\otimes \upsilon^{s_1, s_1}$, $s_1 = (-1)^{k+l+1}, k, l\in\mathbb{Z}_2$. The actions and the coactions are given by

where $i = 1, 2$.

In summary, we have

Theorem 3.2. The set

forms a complete list of non-isomorphic simple Yetter-Drinfeld modules over $\mathcal{\bar{A}}$.

Proof. (1) Let $f_0: M_0(k, l)\rightarrow M_0(k', l'), \theta^{k, l}\mapsto a\theta^{k', l'}$ be a Yetter-Drinfeld module isomorphism where $k, l, k', l'\in\mathbb{Z}_2$ and $0\neq a\in\mathbb{K}.$ Then, for $i = 1, 2$ we have

Hence, $k = k', l = l'.$ Therefore, $M_0(k, l)\cong M_0(k', l')$ if and only if $k = k', l = l'$.

(2) Let $0\neq L$ be a simple Yetter-Drinfeld submodule of $M_1(k, l)$ and $0\neq v = a_0\mu_0^{k, l}+a_1\mu_1^{k, l}\in L$ where $k, l\in\mathbb{Z}_2, a_0, a_1\in\mathbb{K}$. Then, we have

Since $a_0x_1^kx_2^l+a_1x_1^kx_2^ly_{2}$ and $x_1^{k}x_2^{l+1}$ are linearly independent, $\mu_0^{k, l}\in L$. Moreover,

Thus, $L = M_1(k, l)$. Therefore, $M_1(k, l)$ is a simple Yetter-Drinfeld module. Similarly, $M_2(k, l)$ is also a simple Yetter-Drinfeld module.

Let

be a Yetter-Drinfeld module isomorphism where $k, l, k', l' = 0, 1$ and $b_0, b_1, b_2,$ $b_3\in\mathbb{K}.$ Then, we have

Hence, $b_1 = b_2 = 0, b_0, b_3\neq 0, k = k', l = l'$. Therefore, $M_1(k, l)\cong M_1(k', l')$ if and only if $k = k', l = l'$. Similarly, we have $M_2(k, l)\cong M_2(k', l')$ if and only if $k = k', l = l'$.

Let

be a Yetter-Drinfeld module isomorphism where $k, l, k', l' = 0, 1$ and $c_0, c_1, c_2,$ $c_3\in\mathbb{K}.$ Then, for $i = 1, 2$ we have

Hence, $c_1 = c_2 = 0$ and

Then, $c_0 = c_3 = 0$. Therefore, $M_1(k, l)\ncong M_2(k', l')$ for $k, l, k', l'\in\mathbb{Z}_2$.

(3) Let $0\neq U$ be a simple Yetter-Drinfeld submodule of $M_3(k, l)$ and $0\neq u = \sum\limits_{t = 0}^{3}d_t\eta_t^{k, l}\in U$ where $d_0, d_1, d_2, d_3\in\mathbb{K}, k, l = 0, 1$. Then,

Since $u\neq 0,$ the vector $(d_0, d_1, d_2, d_3)\neq 0,$ we have $\eta_0^{k, l}\in U$. Hence for $i = 1, 2,$ $y_i\cdot\eta_0^{k, l} = 2(-1)^{k+l}\eta_i^{k, l}\in U$ and $y_1\cdot\eta_2^{k, l} = 2(-1)^{k+l}\eta_3^{k, l}\in U$. Therefore, $U = M_3(k, l)$ and $M_3(k, l)$ is a simple Yetter-Drinfeld module.

Let $f_3: M_3(k, l)\rightarrow M_3(k', l'), \eta_s^{k, l}\mapsto\sum\limits_{t = 0}^3c_{s, t}\eta_t^{k', l'}$ be a Yetter-Drinfeld module isomorphism where $s\in\left[\kern-0.15em\left[{{0, 3}}\right]\kern-0.15em\right]$, $k, l, k', l'\in\mathbb{Z}_2$ and $c_{s, t}\in\mathbb{K}$. Hence, for $i = 1, 2$ we have

By (3.1) we have

Then,

By (3.2) we have

Then,

By (3.3) we have

Then, $c_{0, 1} = c_{0, 2} = c_{0, 3} = c_{1, 3} = c_{2, 3} = 0, c_{0, 0} = c_{1, 1} = c_{2, 2} = c_{3, 3}\neq 0, k = k', l = l'.$ Therefore, $M_3(k, l)\cong M_3(k', l')$ if and only if $k = k', l = l'$.

By (1)–(3), the set

forms a complete list of non-isomorphic simple Yetter-Drinfeld modules over $\mathcal{\bar{A}}$.

Corollary 3.3. (1) $M_0(k, l)^\ast = M_0(k, l), \; M_i(k, l)^\ast = M_i(k+i-1, l+i)$ for $k, l\in\mathbb{Z}_2, i = 1, 2.$

(2) $M_3(k, l)^\ast = M_3(k+1, l+1)$ for $k, l\in\mathbb{Z}_2.$

Proof. We only give the proof of (2), the other cases are similar.

Let $\left\{\varphi_t^{k, l}|t\in\left[\kern-0.15em\left[{{0, 3}}\right]\kern-0.15em\right]\right\}$ be the dual basis of $\left\{\eta_t^{k, l}|t\in\left[\kern-0.15em\left[{{0, 3}}\right]\kern-0.15em\right]\right\}$ such that

By (2.1) we have

where $i = 1, 2.$ Hence $M_3(k, l)^\ast = M_3(k+1, l+1)$.

4.   Projective class rings of ${_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}$

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 its simple and projective modules. As is known, ${_{H}^{H}}{\mathcal{YD}}\cong {_{\mathcal{D}(H^{cop})}}{\mathcal{M}}.$ where ${_{\mathcal{D}(H^{cop})}}{\mathcal{M}}$ is the category of the left modules of Drinfeld double $\mathcal{D}(H^{\rm cop})$. For this reason, we have the projective class ring of ${_{H}^{H}}{\mathcal{YD}}$ which is denoted by $r_{P}({_{H}^{H}}{\mathcal{YD}}).$

In this section, we briefly set $\mathcal{D}: = \mathcal{D}(\mathcal{\bar{A}}^{\rm cop}).$ Let $\mathcal{P}(V)$ be the projective cover of a simple $\mathcal{D}$-module $V$, or equivalently, a simple Yetter-Drinfeld module $V\in {_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}$. 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 ^[26,27]).

For convenience, we let

where $i\in\left[\kern-0.15em\left[{{1, 2}}\right]\kern-0.15em\right]$.

By a complex computation, we can obtain the following three classes of Yetter-Drinfeld modules for $\mathcal{\bar{A}}.$

(1) $\mathcal{P}_0(k, l), k, l\in\mathbb{Z}_2:$ it is of a basis $\{u_{0}, u_{1}, \cdots, u_{15}\}$, the matrices of the action and coaction of $\mathcal{\bar{A}}$ on $\mathcal{P}_0(k, l)$ are given as follows.

where $W = (u_{0}, u_{1}, \cdots, u_{15})^T.$

(2) $\mathcal{P}(k, l, j), k, l\in\mathbb{Z}_2, j\in\left[\kern-0.15em\left[{{1, 2}}\right]\kern-0.15em\right]:$ it is of a basis $\{p_{0}, p_{1}, \cdots, p_{7}\}$, the matrices of the action and coaction of $\mathcal{\bar{A}}$ on $\mathcal{P}(k, l, j)$ are given as follows.

where $P = (p_{0}, p_{1}, \cdots, p_{7})^T.$

(3) $T_j(k, l), k, l\in\mathbb{Z}_2, j\in\left[\kern-0.15em\left[{{1, 2}}\right]\kern-0.15em\right]:$ it is of a basis $\{v_{0}^j, v_{1}^j, v_{2}^j, v_{3}^j\}$, the matrices of the action and coaction of $\mathcal{\bar{A}}$ on $T_j(k, l)$ are given as follows.

where $U = (v_{0}^j, v_{1}^j, v_{2}^j, v_{3}^j)^T.$

Lemma 4.1. For $k, l\in\mathbb{Z}_2, j\in\left[\kern-0.15em\left[{{1, 2}}\right]\kern-0.15em\right],$ $\mathcal{P}_0(k, l),$ $\mathcal{P}(k, l, j)$ and $T_j(k, l)$ are indecomposable Yetter-Drinfeld modules over $\mathcal{\bar{A}}.$

Proof. Suppose that $\mathcal{P}(k, l, j)$ is not indecomposable. Then, there exist two non-trivial submodules $M$ and $N$ such that $\mathcal{P}(k, l, j) = M\oplus N$. We claim that $p_7\notin M$ and $p_7\notin N$. If $p_7\in M$ then $p_4, p_5, p_6\in M$ since

which implies that

Then, $M = \mathcal{P}(k, l, j).$ It's a contradiction. Similarly, if $p_7\in N$ then $N = \mathcal{P}(k, l, j)$, a contradiction too.

Therefore, we may assume $p = \sum\limits_{i = 0}^{6}\alpha_{i}p_i+p_{7}\in M$ for some $\alpha_{i}\in\mathbb{K}$, $i \in\left[\kern-0.15em\left[{{0, 6}}\right]\kern-0.15em\right]$. Then,

Hence, $p_5, \alpha_1p_1+p_7 \in M$ which implies that

It is a contradiction. Therefore, $\mathcal{P}(k, l, j)$ is an indecomposable Yetter-Drinfeld module over $\mathcal{\bar{A}}.$ Similarly, one can check that $\mathcal{P}_0(k, l)$ and $T_j(k, l)$ are also indecomposable Yetter-Drinfeld modules over $\mathcal{\bar{A}}$.

The results are followed.

It is easy to see $\mathcal{P}(k, l, j),$ $\mathcal{P}_0(k, l)$ and $T_j(k, l),$ $k, l\in\mathbb{Z}_2, j = 1, 2$ are non-isomorphic to each other. Denote $\mathcal{P}_i(k, l) = \mathcal{P}(k+i, l+i-1, i)$ for $i\in\left[\kern-0.15em\left[{{1, 2}}\right]\kern-0.15em\right], k, l\in\mathbb{Z}_2.$

Lemma 4.2. For $k, l\in\mathbb{Z}_2, i\in\left[\kern-0.15em\left[{{0, 2}}\right]\kern-0.15em\right],$ $\mathcal{P}(M_i(k, l))\cong \mathcal{P}_i(k, l)$ and $\mathcal{P}(M_3(k, l))\cong M_3(k, l).$

Proof. It is well known that $\mathcal{D}$ is a symmetric algebra ^[28] and every projective module is injective. Then, $\mathcal{P}(M_0(k, l)) = E(M_t(k, l))$ for some $t\in\mathbb{Z}_{4}$ and the socle and top of $\mathcal{P}(M_0(k, l))$ coincides. Therefore, $\mathcal{P}(M_0(k, l))\cong E(M_0(k, l))$. By Lemma 4.1, $\mathcal{P}_0(k, l)$ is an indecomposable module with $\mathrm{Soc}\mathcal{P}_0(k, l)\cong M_0(k, l).$ Thus, $\mathcal{P}_0(k, l)$ embeds in $E(M_0(k, l))$ which implies that $\dim\mathcal{P}(M_0(k, l))\geq \dim\mathcal{P}_0(k, l) = 16.$

Similarly, $\dim\mathcal{P}(M_i(k, l))\geq \dim\mathcal{P}_i(k, l) = 8$ since $\mathrm{Soc}\mathcal{P}_i(k, l)\cong M_i(k, l)$ for $i = 1, 2.$

It follows that

Hence, we only have $\dim\mathcal{P}(M_0(k, l)) = 16,$ $\dim\mathcal{P}(M_1(k, l)) = \dim\mathcal{P}(M_2(k, l)) = 8,$ $\dim\mathcal{P}(M_3(k, l)) = 4.$

Therefore, $\mathcal{P}(M_i(k, l))\cong \mathcal{P}_i(k, l)$ and $\mathcal{P}(M_3(k, l))\cong M_3(k, l)$ for $i\in\left[\kern-0.15em\left[{{0, 2}}\right]\kern-0.15em\right].$

Corollary 4.3. We have

By Corollary 4.3, it is obvious that $T_i(k, l)$ is not projective for $i = 1, 2.$

Note that the tensor products of Yetter-Drinfeld modules are commutative since $\mathcal{D}$ is quasi-triangular.

Lemma 4.4. For $k, k', l, l'\in\mathbb{Z}_2,$ we have

(1) $M_0(k, l)\otimes M_t(k', l')\cong M_t(k+k', l+l')$ where $t\in\left[\kern-0.15em\left[{{0, 3}}\right]\kern-0.15em\right].$

(2) $M_0(k, l)\otimes \mathcal{P}_i(k', l')\cong\mathcal{P}_i(k+k', l+l')$ where $i\in\left[\kern-0.15em\left[{{0, 2}}\right]\kern-0.15em\right].$

Proof. (1) The results of (1) can be obtained by a direct computation.

(2) By (1) we have

By Lemma 4.2 we have $\dim\mathcal{P}_0(k+k', l+l') = \dim\left(M_0(k, l)\otimes \mathcal{P}_0(k', l')\right)$ and $M_0(k, l)\otimes \mathcal{P}_0(k', l')$ is projective. Hence, $M_0(k, l)\otimes \mathcal{P}_0(k', l')\cong\mathcal{P}(k+k', l+l').$ Similarly, we have $M_0(k, l)\otimes\mathcal{P}_1(k', l')\cong\mathcal{P}_1(k+k', l+l')$ and $M_0(k, l)\otimes\mathcal{P}_2(k', l')\cong\mathcal{P}_2(k+k', l+l').$

Lemma 4.5. For $k, k', k'', l, l', l''\in\mathbb{Z}_2, i = 1, 2,$ we have

(1) $M_1(k, l)\otimes M_2(k', l')\cong M_3(k+k', l+l').$

(2) $M_i(k, l)\otimes M_i(k', l')\cong T_i(k+k', l+l').$

(3) $M_i(k, l)\otimes M_i(k', l')\otimes M_i(k'', l'')$

  $\cong M_i(k+k'+k''+i-1, l+l'+l''+i)^{\oplus2}\oplus M_i(k+k'+k'', l+l'+l'')^{\oplus2}.$

(4) $M_i(k, l)\otimes M_3(k', l')\cong \mathcal{P}_{3-i}(k+k'+i-1, l+l'+i).$

(5) $M_3(k, l)\otimes M_3(k', l')\cong \mathcal{P}_0(k+k'+1, l+l'+1).$

Proof. The statements (1), (2) can be obtained by a direct computation.

On the other hand, by Lemma 4.4 (1) it suffices to prove the cases for $M_i(0, 0)^{\otimes3}, M_i(0, 0)\otimes M_3(0, 0)$ and $M_3(0, 0)\otimes M_3(0, 0)$.

For the statement (3), note that $T_j(0, 0) = M_{j}(0, 0)\otimes M_{j}(0, 0), j = 1, 2.$ Let $\{u_0^{j}, u_1^{j}\}$ be a basis of $M_{j}(0, 0).$ The actions on the basis $\{u_0^{j}, u_1^{j}\}$ and the coactions are given by

where $i = 1, 2.$ Let

and $L_t = \mathbb{K}\lambda_{2t}^j+\mathbb{K}\lambda_{2t+1}^j$ where $t\in\left[\kern-0.15em\left[{{0, 3}}\right]\kern-0.15em\right].$ By a direct computation, we have $L_0\cong L_3 \cong M_j(j-1, j), L_1\cong L_2\cong M_j(0, 0).$ Therefore,

For the statement (4), we have

where $i = 1, 2.$ By Schur's lemma, ${\rm Hom}_{\mathcal{D}}(M_i(0, 0)\otimes M_{3}(0, 0), M_{3-i}(k, l))\neq0$ if and only if $k = i-1, l = i.$ Since $M_i(0, 0)\otimes M_{3}(0, 0)$ is projective and $\dim\left(M_i(0, 0)\otimes M_{3}(0, 0)\right) = \dim\mathcal{P}_{3-i}(0, 1) = 8,$ we get that $M_i(0, 0)\otimes M_{3}(0, 0)\cong\mathcal{P}_{3-i}(i-1, i).$

Similarly for the statement (5) we have

Hence, $M_3(0, 0)\otimes M_3(0, 0)\cong \mathcal{P}_0(1, 1).$

Lemma 4.6. For $k, k', k'', l, l', l''\in\mathbb{Z}_2, i = 1, 2$ we have

(1) $M_i(k, l)\otimes \mathcal{P}_i(k', l')\cong \mathcal{P}_0(k+k'+i-1, l+l'+i).$

(2) $M_i(k, l)\otimes \mathcal{P}_{3-i}(k', l')\cong M_3(k+k', l+l')^{\oplus2}\oplus M_3(k+k'+i-1, l+l'+i)^{\oplus2}.$

Proof. It suffices to prove the lemma for $M_i(0, 0)\otimes \mathcal{P}_i(0, 0)$ and $M_i(0, 0)\otimes \mathcal{P}_{3-i}(0, 0)$ by Lemma 4.4 (1) (2), where $i = 1, 2.$

(1) We have

Hence, $M_i(0, 0)\otimes \mathcal{P}_i(0, 0)\cong \mathcal{P}_0(i-1, i).$

(2) Let $N_{-1} = 0, N_t = \sum\limits_{s = 0}^{t}\mathbb{K}v_s^i$ for $t\in\left[\kern-0.15em\left[{{0, 3}}\right]\kern-0.15em\right],$ where $i\in\left[\kern-0.15em\left[{{1, 2}}\right]\kern-0.15em\right].$ One can check that

is a Yetter-Drinfeld submodules chain of $T_i(k, l)$ such that $N_t/N_{t-1}$ is a one dimensional Yetter-Drinfeld module and

Hence, by Lemma 4.5 (4) we have

Lemma 4.7. For $k, k', k'', l, l', l''\in\mathbb{Z}_2, i = 1, 2$ we have

(1) $M_3(k, l)\otimes \mathcal{P}_i(k', l')\cong M_{3-i}(k, l)\otimes \mathcal{P}_0(k'+1, l'+1)$

$\quad\quad\quad\quad\quad\quad\quad \cong \mathcal{P}_{3-i}(k+k'+1, l+l'+1)^{\oplus2}\oplus \mathcal{P}_{3-i}(k+k'+i-1, l+l'+i)^{\oplus2}.$

(2) $\mathcal{P}_i(k, l)\otimes \mathcal{P}_i(k', l')\cong \mathcal{P}_0(k+k'+1$, $l+l'+1)^{\oplus2}\oplus \mathcal{P}_0(k+k'+i-1, l+l'+i)^{\oplus2}.$

(3) $\mathcal{P}_1(k, l)\otimes \mathcal{P}_2(k', l')\cong M_3(k, l)\otimes \mathcal{P}_0(k', l')\cong \bigoplus\limits_{s, t = 0}^1M_3(s, t)^{\oplus4}.$

(4) $\mathcal{P}_i(k, l)\otimes \mathcal{P}_0(k', l')\cong \bigoplus\limits_{s, t = 0}^1\mathcal{P}_i(s, t)^{\oplus4}.$

(5) $\mathcal{P}_0(k, l)\otimes \mathcal{P}_0(k', l')\cong \bigoplus\limits_{s, t = 0}^{1}\mathcal{P}_0(s, t)^{\oplus4}.$

Proof. It suffices to prove the lemma for $M_3(0, 0)\otimes \mathcal{P}_i(0, 0),$ $M_3(0, 0)\otimes \mathcal{P}_0(0, 0),$ $\mathcal{P}_i(0, 0)\otimes \mathcal{P}_i(0, 0),$ $\mathcal{P}_1(0, 0)\otimes \mathcal{P}_2(0, 0),$ $\mathcal{P}_i(0, 0)\otimes \mathcal{P}_0(0, 0)$ and $\mathcal{P}_0(0, 0)\otimes \mathcal{P}_0(0, 0)$ by Lemma 4.4 (1) (2).

(1) Let $V_{-1} = 0, V_t = \sum_{s = 0}^{t}\left(\mathbb{K}p_{2s}+\mathbb{K}p_{2s+1}\right)$ for $t\in\left[\kern-0.15em\left[{{0, 3}}\right]\kern-0.15em\right]$. One can check that

is a Yetter-Drinfeld submodules chain of $\mathcal{P}_i(k+i, l+i-1)$ such that $V_t/V_{t-1}$ is a two dimensional Yetter-Drinfeld module and

Hence,

Therefore,

since

(2) By Lemma 4.6 (1), we have

(3) By Lemma 4.5 (1) (4) (5) and Lemma 4.6 (2), we have

(4) By (1), we have

(5) Let

and $K_{-1} = 0, K_t = \sum_{s = 0}^{t}\mathbb{K}\kappa_{s}$ for $t\in\left[\kern-0.15em\left[{{0, 15}}\right]\kern-0.15em\right]$. One sees that

is a Yetter-Drinfeld submodules chain of $\mathcal{P}_0(k, l)$ such that $K_t/K_{t-1}$ is a one dimensional Yetter-Drinfeld module and

Hence,

The proof is finished.

Now, we can describe the projective class rings of ${{_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}}$.

Let $a_1 = [M_0(1, 0)], a_2 = [M_0(0, 1)], b_1 = [M_1(0, 0)], b_2 = [M_2(0, 0)].$

Lemma 4.8. The following statements hold in $r_{P}({{_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}})$.

(1) For $k, l\in\mathbb{Z}_2, i = 1, 2,$

(2) For $i = 1, 2,$

Proof. The results are easy to get from Lemma 4.4 and Lemma 4.5.

Corollary 4.9. The following set is a $\mathbb{Z}$-basis of $r_{P}({{_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}}):$

Proof. By Lemma 4.8(2), $a_i^{2} = 1$ and $b_i^3 = 2(1+a_1^{i-1}a_2^{i})b_i$ for all $i\in\left[\kern-0.15em\left[{{1, 2}}\right]\kern-0.15em\right]$. By Lemma 4.5(2) we have $b_i^3 = 2(1+a_1^{i-1}a_2^{i})b_i$ for $i = 1, 2.$ Hence, the highest degree of $b_i$ is $2$. It is easy to check that the set

is an independent set since $\#\{a_1^ka_2^lb_1^sb_2^t|k, l, s, t\in\mathbb{Z}_2\} = 36$, the number of $\mathbb{Z}$-basis of $r_{P}({{_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}})$. Therefore,

is a $\mathbb{Z}$-basis of $r_{P}({{_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}})$.

The results of this section is as follows.

Theorem 4.10. The projective class ring $r_{P}({{_{\mathcal{\bar{A}}}^{\mathcal{\bar{A}}}}{\mathcal{YD}}})$ is isomorphic to the quotient ring of the ring $\mathbb{Z}[x_1, x_2, y_1, y_2]$ module the ideal $I$ generated by the following elements

Proof. By Corollary 4.9, there is a unique ring epimorphism

such that $\Phi(x_i) = a_i, \Phi(y_i) = b_i$ for $i = 1, 2.$ By Lemma 4.8(2) we have

It follows that $\Phi(I) = 0$ and hence $\Phi$ induces a natural ring epimorphism

such that $\overline{\Phi}(\overline{\nu}) = \Phi(\nu)$ for all $\nu\in\mathbb{Z}[x_1, x_2, y_1, y_{2}],$ where $\overline{\nu} = \nu+I$. It is straightforward to check that the ring $\mathbb{Z}[x_1, x_2, y_1, y_{2}]/I$ is $\mathbb{Z}$-spanned by

This means the $\mathbb{Z}$-rank of $\mathbb{Z}[x_1, x_2, y_1, y_{2}]/I$ is $36$. Hence, we get the ring isomorphism $\overline{\Phi}.$

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

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

Conflict of interest

The authors declare there is no conflicts of interest.