Module algebra structures of nonstandard quantum group $X_{q}(A_{1})$ on the quantum plane

1.   Introduction

The nonstandard quantum groups were studied in ^[1], where Ge et al. ^[1] obtained new solutions of the Yang-Baxter equations. For these new solutions, they followed the Faddeev-Reshetikhin-Takhtajan ^[2] method to establish the related quantum group structure, which, in general, may not be the same as the standard ones. In ^[3] one class of nonstandard quantum deformation corresponding to simple Lie algebra $sl_{n}$ was given, which is denoted by $X_q(A_{n-1})$. For each vertex $i\; (i = 1, \cdots, n-1)$ of the Dynkin diagram, the parameter $q_i$ is equal to $q$ or $-q^{-1}$, and if $q_i = q$ for all $i$, then $X_q(A_{n-1})$ is just $U_q(sl_n)$. However, if $q_i\neq q_{i+1}$ for some $1\leq i\leq n-1$, it has the relation $E_{i}^{2} = F_{i}^{2} = 0$ in $X_q(A_{n-1})$, such that $X_q(A_{n-1})$ is different from $U_q(sl_n)$. For more results for nonstandard quantum groups, one can refer to ^[4,5,6].

The notion of Hopf algebra actions on algebras was introduced by Sweedler ^[7] in 1969. The Brauer groups of $H$-module and $H$-dimodule algebras were researched by Beattie ^[8]. A duality theorem for Hopf module algebras was studied by Blattner and Montgomery ^[9] in 1985. Moreover, the actions of Hopf algebras and their generalizations ^[10,11] play an important role in quantum group theory ^[12,13], and the actions of Hopf algebras have various applications in physics ^[14]. Duplij and Sinel'shchikov ^[15,16] used a general form of the automorphism of the quantum plane to render the notion of weight for $U_{q}(sl_{2})$-actions, and they completely classified $U_{q}(sl_{2})$-module algebra structures on the quantum plane, which consist of 6 non-isomorphic cases. Moreover, in ^[17] the authors used the method of weights ^[15,16] to study the module algebra structures of $U_{q}(sl_{m+1})$ on the coordinate algebra of quantum vector spaces. More relevant research can be found at ^[18,19]. However, the module algebras of nonstandard quantum groups have not yet achieved research results. Consequently, based on the above research results, we consider here the actions of the nonstandard quantum group $X_{q}(A_{1})$ on the quantum $n$-space $A_{q}(n)$. In particular, a complete list of $X_{q}(A_{1})$-module algebra structures on the quantum plane $A_{q}(2)$ is produced and the isomorphism classes of these structures are described.

This paper is organized as follows. In Section 1, we introduce some necessary notations and concepts, as well as prove a lemma about actions on generators and any elements of ${A_{q}(n)}$. In Section 2, using the method of weights ^[15,16,17], the 0-th homogeneous component and 1-st homogeneous component of the action matrix are given. We have $2n+1$ cases for the 0-th homogeneous component $(M_{EF})_{0}$, and $2n(n-1)+1$ cases for the 1-st homogeneous component $(M_{EF})_{1}$. In Section 3, we study the actions of ${X_{q}(A_{1})}$ on $A_q(2)$, and characterize all module algebra structures of ${X_{q}(A_{1})}$ on the quantum plane $A_q(2)$, which rely upon considering the 0-th and 1-st homogeneous components of an action.

2.   Preliminaries

Throughout, we work over the complex field $\mathbb{C}$ unless otherwise stated. All algebras, Hopf algebras, and modules are defined over $\mathbb{C}$; all maps are $\mathbb{C}$-linear.

Let $(H, m, \eta, \Delta, \varepsilon, S)$ be a Hopf algebra, where $\Delta$, $\varepsilon$, and $S$ are the comultiplication, counit, and antipode of $H$, respectively. Let $A$ be a unital algebra with unit $\bf{1}$. Sweedler's notations ^[7] are used in the sequel. For example, for $h\in H$, we denote

Definition 2.1. By a structure of an $H$-module algebra on $A$, we mean a homomorphism $\pi: H\rightarrow \mathrm{End}_{\mathbb{C}}A$ such that:

1) for all $h\in H, a, b\in A$, $\pi(h)(ab) = \sum\limits_{(h)}\pi(h_{(1)})(a)\pi(h_{(2)})(b);$

2) for all $h\in H, \pi(h)(1) = \varepsilon(h)1$.

Let $\pi_{1}$ and $\pi_{2}$ be two $H$-module algebras on $A$, and the structures $\pi_{1}, \pi_{2}$ are said to be isomorphic, if there exists an automorphism $\Psi$ of the algebra $A$, such that $\Psi\pi_{1}(h)\Psi^{-1} = \pi_{2}(h)$ for all $h\in H.$

Throughout the paper we assume that $q\in {\mathbb{C}}^{*} = {\mathbb{C}}\setminus\{0\}$ is not a root of the unit ($q^{n}\neq 1$ for all non-zero integers $n$). A class of the nonstandard quantum group $X_q(A_1)$ was studied by the authors of ^[3,4]. Now, we recall the definition of $X_q(A_1)$.

Definition 2.2. The nonstandard quantum group $X_q(A_1)$ is a unital associative ${\mathbb{C}}$-algebra generated by $E, F, K_1, K_2, K_{1}^{-1}, K_{2}^{-1}$ subject to the relations:

The algebra ${X_{q}(A_{1})}$ is also a Hopf algebra, and the comultiplication $\Delta$, counit $\varepsilon$, and antipode $S$ are given as the following:

Let us review the definition of the quantum $n$-space (see ^[20,21]).

Definition 2.3. The quantum $n$-space $A_{q}(n)$ is a unital algebra, generated by $n$ generators $x_{i}$ for $i\in \{1, 2, \cdots, n\}$, and for any $i > j$ it satisfies the relation:

The quantum $n$-space $A_{q}(n)$ is also called a coordinate algebra of quantum $n$-dimensional vector space. If $n = 2$, $A_{q}(2)$ is called a quantum plane.

For all $n \geq 2$ and $n\neq3$, by ^[22,23,24], one has a description of automorphisms of the algebra ${A_{q}(n)}$, as follows. Let $\Psi$ be an automorphism of ${A_{q}(n)}$, and then there exist nonzero constants $\alpha_{i}$ for $i\in \{1, 2, 4, \cdots, n\}$, such that

All such automorphisms form the automorphism group of ${A_{q}(n)}$, which we denote by $\mathrm{Aut}({A_{q}(n)})$, and in addition, one can get

It should be pointed out that there are more automorphisms of $A_q(3)$. Let $\sigma$ be an automorphism of $A_q(3)$, and then there exist nonzero constants $\alpha, \beta, \gamma\in \mathbb{C}^{*}$ and $t\in \mathbb{C}$, such that

and $\mathrm{Aut}(A_q(3))\cong \mathbb{C}\rtimes\left(\mathbb{C}^{*}\right)^{3}$. Obviously, the automorphism group of $A_q(3)$ is more complex, and therefore, we separately discussed the module algebra structures of nonstandard quantum group $X_{q}(A_{1})$ on $A_q(3)$, as detailed in ^[25].

Unless otherwise specified, in the following text, we fix the integers $n \geq 2$ and $n\neq3$.

Next, we give a lemma which will be useful for checking the module algebra structures of $X_{q}(A_{1})$ on ${A_{q}(n)}$.

Lemma 2.4. Given the module algebra actions of the generators $E, F, K_1, K_2$ of $X_{q}(A_{1})$ on ${A_{q}(n)}$, if an element in the ideal generated by the relations (2.1)–(2.7) of $X_{q}(A_{1})$, which acting on the generators $x_i$ of ${A_{q}(n)}$ produces zero for all $i = 1, 2, 4, \cdots, n$, then this element acting on any $v\in {A_{q}(n)}$ produces zero.

Proof. Here, we only prove that, if

where $x_{i}, x_{j}$ are arbitrary generators of ${A_{q}(n)}$, then

The other relationships can be proven similarly. Indeed, by (2.9) and (2.10), we have

and by Definition (2.1), then,

Thus, $\left[(EF-FE)-\frac{K_{2}K_{1}^{-1}-K_{2}^{-1}K_{1}}{q-q^{-1}}\right](x_ix_j) = 0$, and the lemma holds. □

Therefore, by Lemma 2.4, in checking whether the relations of $X_{q}(A_{1})$, acting on any $v\in {A_{q}(n)}$, produces zero, we only need to check whether they produce zero when they act on the generators $x_1, x_2, \cdots, x_n$.

3.   Properties of $X_{q}(A_{1})$-module algebras on ${A_{q}(n)}$

In this section, we will study the module algebra structures of $X_{q}(A_{1})$ on ${A_{q}(n)}$, where $K_1, K_2\in \mathrm{Aut}({A_{q}(n)})$, $n\geq2$, and $n\neq3$.

The $s$-th homogeneous component of ${A_{q}(n)}$ is denoted by ${A_{q}(n)}_{s}$, which is linear spanned by the monomials $x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}$ with $m_{1}+m_{2}+\cdots+m_{n} = s$. Also, given a polynomial $p\in{A_{q}(n)}$, the $s$-th homogeneous component of $p$ is denote by $(p)_{s}$, which is the projection of $p$ onto ${A_{q}(n)}_{s}$ parallel to the direct sum of all other homogeneous components of ${A_{q}(n)}$.

By the definition of module algebra, it is easy to see that any action of ${X_{q}(A_{1})}$ on ${A_{q}(n)}$ is determined by the following $4\times n$ matrix with entries from ${A_{q}(n)}$:

which is called the full action matrix, see^[22]. Given a ${X_{q}(A_{1})}$-module algebra structure on ${A_{q}(n)}$, obviously, the action of $K_{1}$ or $K_{2}$ determines an automorphism of ${A_{q}(n)}$. Therefore, by the assumption $K_{1}, K_{2}\in \mathrm{Aut}(A_{q}(n)$, we can set

where $\alpha_{i}, \beta_{i}\in \mathbb{C}^{*}$ for $i\in\{1, 2, \cdots, n\}$.

It is easy to see that every monomial $x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}\in{A_{q}(n)}$ is an eigenvector of $K_{1}$ and $K_{2}$, and the associated eigenvalues $\alpha_{1}^{m_{1}}\alpha_{2}^{m_{2}}\cdots \alpha_{n}^{m_{n}}$ and $\beta_{1}^{m_{1}}\beta_{2}^{m_{2}}\cdots \beta_{n}^{m_{n}}$ are called the $K_{1}$-weight and $K_{2}$-weight of this monomial, respectively, which will be written as

We will also need another matrix $M_{EF}$ as follows:

and we call $M_{K_{1}K_{2}}$ and $M_{EF}$ the action $K_{1}K_{2}$-matrix and $EF$-matrix, respectively. It follows from relations (2.2)–(2.5) that all entries of $M$ are weight vectors for $K_{1}$ and $K_{2}$, and we have

where the relation $\left(a_{st}\right)\bowtie \left(b_{st}\right)$ means that for every pair of indices $s, t$ such that both $a_{st}$ and $b_{st}$ are nonzero, one has $a_{st} = b_{st}$.

In the following, we denote the $j$-th homogeneous component of $M$, whose elements are just the $j$-th homogeneous components of the corresponding entries of $M$, by $\left(M\right)_{j}$. Set

where $a_{i}, b_{i}\in {\mathbb{C}}$ for all $i\in\{1, 2, \cdots, n\}$. Then, we obtain

Therefore, the relations (3.6) and (3.7) imply that $a_{i}$ and $b_{i}$ are at most one nonzero for any $i\in\{1, 2, \cdots, n\}$, and

An application of $E$ and $F$ to the relation (2.13) and by Eq (3.2), one has the following equalities:

After projecting (3.10) and (3.11) to ${A_{q}(n)}_{1}$, we obtain

which certainly implies

For any $i, j\in\{1, 2, \cdots, n\}$ and $i > j$, we will determine the weight constants $\alpha_{i}$ and $\beta_{i}$ as follows:

Lemma 3.1. For any $i, j, s, t\in\{1, 2, \cdots, n\}$, $a_{i}$, $a_{j}$, $b_{s}$, and $b_{t}$ are at most one nonzero.

Proof. For any $i, j\in\{1, 2, \cdots, n\}$, and $i > j$, we only prove that $a_{i}$ and $a_{j}$ are at most one nonzero. Assume $a_{i}\neq0$ and $a_{j}\neq0$, and then

by Eqs (3.8), (3.12), and (3.13). However

which are impossible, since it is contradictory to $q$ not being a root of the unit. Therefore at least one of $a_{i}$ and $a_{j}$ is zero for $i, j\in {1, 2, \cdots, n}$.

The remaining statements can be proven in a similar way. □

In summary, we have obtained the following results for the $0$-th homogeneous component $\left(M_{EF}\right)_{0}$ of $M_{EF}$.

Theorem 3.2. There are $2n+1$ cases for the $0$-th homogeneous component $\left(M_{EF}\right)_{0}$ of $M_{EF}$, as follows:

1) $a_i\neq 0, a_j = 0$ for $i\neq j$ and all $b_s = 0$ for any $i, j, s\in\{1, 2, \cdots, n\}$, i.e.,

and we have

2) $b_i\neq 0, b_j = 0$ for $i\neq j$ and all $a_s = 0$ for any $i, j, s\in\{1, 2, \cdots, n\}$, i.e.,

and we have

3) all $a_i = b_i = 0$ for any $i\in\{1, 2, \cdots, n\}$, i.e.,

Therefore, it does not determine the weight constants at all.

Next, for the $1$-st homogeneous component $\left(M_{EF}\right)_{1}$, due to $q$ not being a root of the unit, one has

which implies

for some $c_{is}\in {\mathbb{C}}$. In a similar way, we have

for some $d_{is}\in {\mathbb{C}}$. Hence

where $c_{is}, d_{is} \in {\mathbb{C}}$.

Now project (3.10) and (3.11) to ${A_{q}(n)}_{2}$, and we can obtain

for any $i, j\in\{1, 2, \cdots, n\}$ and $i > j$, Where

As a consequence, for any $i, j\in\{1, 2, \cdots, n\}$ and $i > j$, we have

Lemma 3.3. For any $i\in\{1, 2, \cdots, n\}$, every $1$-st homogeneous component $(E(x_i))_1$ and $(F(x_i))_1$, if nonzero, reduces to a monomial.

Proof. We assume that

and $c_{is}\neq 0, \; c_{is'}\neq 0 \; (s\neq s')$ for some $s, s'\in \{1, 2, \cdots, i-1, i+1, \cdots, n\}$. Without loss of generality, we stipulate that $s < s'$.

If $s, s' < i$, then

However, $s$ must be one of the $\{1, 2, \cdots, s'-1\}$, and one gets $q^{-1} = 1$, which is impossible. Hence, $c_{is}$ and $c_{is'}$ are at most one nonzero, and $(E(x_i))_1$ is equal to zero or a monomial. The remaining situations can be proven in a similar way.

Similarly, $(F(x_i))_1$ is equal to zero or a monomial. □

Additionally, since

We obtain the following result.

Lemma 3.4. For any $i, j, s, t\in\{1, 2, \cdots, n\}$, $(E(x_i))_1$, $(E(x_j))_1$, $(F(x_s))_1$, $(F(x_t))_1$ are at most one nonzero.

Proof. Here, we only prove that $(E(x_i))_1$ and $(E(x_j))_1$ are at most one nonzero. The other statements can be proven similarly.

By Lemma 3.3, we get that if $(E(x_i))_1$ and $(E(x_j))_1$ are nonzero, then they are a monomial for any $i, j\in\{1, 2, \cdots, n\}$. Assume

Without loss of generality, we stipulate that $i > j$. According to the Eqs (3.20) and (3.21), we have

In addition,

So, $\alpha_{i} = q\alpha_{s}, \; \; \beta_{i} = -q\beta_{s}, \; \; \alpha_{j} = q\alpha_{s'}, \; \; \beta_{j} = -q\beta_{s'}$.

On the other hand, since $c_{is}\neq 0$ and $c_{js'}\neq 0$, it follows that

by (3.18). Then $q^{-1} = -q^{-2}$ or $q^{-1} = -1$, and $q = -q^{2}$ or $q = -1$, which are impossible. Hence, $(E(x_i))_1$ and $(E(x_j))_1$ are at most one nonzero. □

From the above discussion, we have the following result for the $1$-st homogeneous component $\left(M_{EF}\right)_{1}$ of $M_{EF}$.

Theorem 3.5. There are $2n(n-1)+1$ cases for the $1$-st homogeneous component $\left(M_{EF}\right)_{1}$ of $M_{EF}$, as follows:

1) $c_{is}\neq 0\; (i\neq s)$, and otherwise $c_{i's'} = 0$ and all $d_{jt} = 0$ for any $i, s, j, t, i', s'\in\{1, 2, \cdots, n\}$, i.e.,

and we have $\alpha_{i} = q\alpha_{s}, \beta_{i} = -q\beta_{s}$, and

2) $d_{is}\neq 0\; (i\neq s)$, and otherwise $d_{i's'} = 0$ and all $c_{jt} = 0$ for any $i, s, j, t, i', s'\in\{1, 2, \cdots, n\}$, i.e.,

and we have $\alpha_{i} = q^{-1}\alpha_{s}, \beta_{i} = -q^{-1}\beta_{s}$, and

3) all $c_{is} = 0$ and $d_{i's'} = 0$, for any $i, s, i', s'\in\{1, 2, \cdots, n\}$, i.e.,

Therefore, it does not determine the weight constants at all.

4.   The structures of $X_{q}(A_{1})$-module algebra on $A_q(2)$

In this section, our aim is to describe the concrete $X_{q}(A_{1})$-module algebra structures on the quantum plane $A_q(2)$, where $K_{1}, K_{2}\in \mathrm{Aut}( A_q(2) )\cong \left(\mathbb{C}^{*}\right)^{2}$.

By Theorems 3.2 and 3.5, it follows that if both the $0$-th homogeneous component and the $1$-st homogeneous component of $M_{EF}$ are nonzero, it is easy to see that these series are empty, so we only need to consider 9 possibilities.

where $a_i\neq 0, b_i\neq 0$ for $i = 1, 2$ and $c_{12}, c_{21},$$d_{12}, d_{21}$ are not zero.

Lemma 4.1. If the $0$-th homogeneous component of $M_{EF}$ is zero and the $1$-st homogeneous component of $M_{EF}$ is nonzero, then these series are empty.

Proof. Now we show that the $\left[\left(\begin{array}{ccccccccc} 0 & 0\\ 0 & 0 \end{array} \right)_{0}, \left(\begin{array}{ccccccccc} c_{12}x_{2} & 0\\ 0 & 0 \end{array} \right)_{1}\right]$-series is empty. If we suppose the contrary, then it follows from

that within this series, one can have

By $c_{12}\neq0$, one can get $\alpha_{1} = q\alpha_{2}, \beta_{1} = -q\beta_{2},$ and $\beta_{2}\alpha_{2}^{-1} = q$. Hence, $\beta_{1}\alpha_{1}^{-1} = -q$, and

On the other hand, projecting $(EF-FE)(x_{1})$ to $A_q(2) _{1}$, we obtain

However, $0\neq -x_{i}$. We get the contradiction, and prove our claim.

In a similar way, one can prove that all other series where the $0$-st homogeneous component of $M_{EF}$ is zero and the $1$-st homogeneous component of $M_{EF}$ is nonzero are empty. □

Lemma 4.2. If the $0$-th homogeneous component of $M_{EF}$ is nonzero and the $1$-st homogeneous component of $M_{EF}$ is zero, then these series are empty.

Proof. We only show that the $\left[\left(\begin{array}{ccccccccc} a_1 & 0\\ 0 & 0 \end{array} \right)_{0}, \left(\begin{array}{ccccccccc} 0 & 0\\ 0 & 0 \end{array} \right)_{1}\right]$-series is empty, and in a similar way, one can prove that all other series are empty.

Consider this series and we obtain that

and suppose that it is not empty. We set

where $\alpha_{2}, \beta_{2}\in \mathbb{C}^{\ast}$, and $\rho_{m_{1}m_{2}}, \; \theta_{l_{1}l_{2}}, \; \sigma_{t_{1}t_{2}}, \; \tau_{h_{1}h_{2}}\in {\mathbb{C}}$.

Then we apply the relations (2.1)–(2.7) to the generators of $A_q(2)$. It is easy to see that the application of relation (2.1) to the generators of $A_q(2)$ produces zero. So, we consider the residue, as follows.

and then $\rho_{m_{1}m_{2}} = 0$ for all $m_{1}, m_{2}\in \mathbb{N}$ with $m_{1}+m_{2}\geq2$, or $\alpha_{2}^{m_{2}} = q^{-m_{1}}$ for some $m_{1}, m_{2}\in \mathbb{N}$ with $m_{1}+m_{2}\geq2$.

and then $\rho_{m_{1}m_{2}} = 0$ for all $m_{1}, m_{2}\in \mathbb{N}$ with $m_{1}+m_{2}\geq2$, or $\beta_{2}^{m_{2}} = (-q)^{-m_{1}}$ for some $m_{1}, m_{2}\in \mathbb{N}$ with $m_{1}+m_{2}\geq2$.

If some $\rho_{m_{1}m_{2}}\neq 0$, and it meets the conditions, i.e.,

and $\beta_{2}\alpha_{2}^{-1} = q$, one can get $q^{m_{2}} = (-1)^{m_{1}},$ since $q$ is not a unit root, which is impossible. Therefore, we have $E(x_{1}) = a_{1}$.

Similar to the discussion above, we can obtain that

where $\sigma_{20}, \tau_{11}\in{\mathbb{C}}.$

From $EF-FE = \frac{K_{2}K_{1}^{-1}-K_{2}^{-1}K_{1}}{q-q^{-1}}$, we have

If $F(x_2) = 0$, then

if $F(x_{2}) = \tau_{11}x_{1}x_{2}\neq 0$, then

Hence, we have $\tau_{11} = \frac{1}{a_{1}}$ and $F(x_{2}) = \frac{1}{a_{1}}x_{1}x_{2}$.

By $F^{2} = 0$, one has that

If $F(x_{1}) = 0$, then

if $F(x_{1}) = \sigma_{20}x_{1}^{2}$, then

So $\sigma_{20} = -\frac{q}{a_{1}}$ and $F(x_1) = -\frac{q}{a_{1}}x_{1}^{2}.$

With an application of $F$ to $x_{2}x_{1} = qx_{1}x_{2}$, we have

In summary, this series is empty.

In a similar way, one can prove that all other series where the $0$-th homogeneous component of $M_{EF}$ is nonzero and the $1$-st homogeneous component of $M_{EF}$ is zero are empty. □

Theorem 4.3. The $\left[\left(\begin{array}{ccccccccc} 0 & 0\\ 0 & 0 \end{array} \right)_{0}, \left(\begin{array}{ccccccccc} 0 & 0\\ 0 & 0 \end{array} \right)_{1}\right]$ -series has $X_{q}(A_1)$-module algebra structures on the quantum plane $A_q(2)$ given by

where $\lambda_1, \lambda_2\in {\mathbb{C}}^{*}$, and therefore, they are pairwise nonisomorphic.

Proof. It is easy to check that (4.1)–(4.3) determine a well-defined $X_{q}(A_1)$-action consistent with the multiplication in $X_{q}(A_1)$ and in the quantum plane $A_q(2)$, as well as with comultiplication in $X_{q}(A_1)$. We prove that there are no other $X_{q}(A_1)$-actions here. Note that an application of (2.6) to $x_{1}$ or $x_{2}$ has zero projection to $A_q(2) _{1}$, i.e., $(EF-FE)(x_i) = 0, (i = 1, 2)$, because in this series $E$ and $F$ send any monomial to a sum of the monomials of higher degree. Therefore,

and we have

which leads to $\beta_{1}^{2} = \alpha_{1}^{2}$ and $\beta_{2}^{2} = \alpha_{2}^{2}$. Let $\alpha_{1} = \lambda_{1}$ and $\alpha_{2} = \lambda_{2}$, and we have $\beta_{1} = \pm\lambda_{1}$ and $\beta_{2} = \pm\lambda_{2}$. To prove (4.3), note that if $E(x_i)\neq 0$ or $F(x_i)\neq 0$, for $i = 1, 2$, then they are a sum of the monomials with degrees greater than 1. Similar to the proof of Lemma 4.2, we get that this is impossible, because they cannot satisfy the conditions of $X_{q}(A_1)$-module algebra on $A_q(2)$.

To see that the $X_{q}(A_1)$-module algebra structures are pairwise nonisomorphic, observe that all the automorphisms of the quantum plane commute with the actions of $K_{1}$ and $K_2$. □

Next, our immediate intention is to describe the composition series for these representations.

Proposition 4.4. The representations corresponding to the $\left[\left(\begin{array}{cccccc} 0 & 0\\ 0 & 0\\ \end{array} \right)_{0}, \left(\begin{array}{cccccc} 0 & 0\\ 0 & 0\\ \end{array} \right)_{1}\right]$-series described in (4.1)–(4.3) split into the direct sum $A_q(2) = \oplus_{m = 0}^{\infty}\oplus_{n = 0}^{\infty}{\mathbb{C}} x_{1}^{m}x_{2}^{n}$ of one-dimensional subrepresentations. These subrepresentations may belong to two isomorphism classes, depending on the weights of a specific monomial $x_{1}^{m}x_{2}^{n}$ which can be $K_1(x_{1}^{m}x_{2}^{n}) = \lambda^{m}_{1}\lambda^{n}_{2}x_{1}^{m}x_{2}^{n}$ and $K_2(x_{1}^{m}x_{2}^{n}) = (\pm1)^{m+n}\lambda^{m}_{1}\lambda^{n}_{2}x_{1}^{m}x_{2}^{n}$.

Proof. Since $E$ and $F$ are represented by zero operators and the monomials $x_{1}^{m}x_{2}^{n}$ are eigenvectors for $K_1$ and $K_2$, then every direct summand is $X_{q}(A_1)$-invariant. □

5.   Conclusions

In this paper, we discuss the module algebra structures of $X_{q}(A_{1})$ on the quantum $n$-space ${A_{q}(n)}$ for $n\geq2$ and $n\neq3$. However, we have presented only a complete list of $X_{q}(A_{1})$-module algebra structures on the quantum plane $A_{q}(2)$, and described the isomorphism classes of these structures. For all $n\geq 4$, it is complicated to give the solutions of (3.7) and (3.8). We will continue to classify the module algebra structures of $X_{q}(A_{1})$ on the quantum $n$-space ${A_{q}(n)}$ for $n\geq4$ in the future.

Use of AI tools declaration

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 12201187), Natural Science Foundation of Henan Province (Grant No. 222300420156), and Key Research Project Plan of Henan Province Higher Education Institutions (Grant No. 23B110006).

Conflict of interest

The authors declare there is no conflicts of interest.