BGG category for the quantum Schr\"odinger algebra

TL;DR

This paper investigates the structure of the BGG category for the quantum Schrödinger algebra, establishing equivalences with known categories and demonstrating the wild representation type of finite-dimensional modules.

Contribution

It provides a classification of the BGG category for the quantum Schrödinger algebra and constructs an explicit equivalence with quiver representations, revealing the wild nature of finite-dimensional modules.

Findings

Equivalence between certain subcategories and quantum group categories

Construction of a functor to quiver representations

Finite-dimensional modules category is wild

Abstract

In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{\"o}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module $B_{\dot z}$ over the quantum Weyl algebra $H_q$, we show that there is an equivalence between the full subcategory $\mathcal{O}[\dot z]$ consisting of modules with the central charge $\dot z$ and the BGG category $\mathcal{O}^{(\mathfrak{sl}_2)}$ for the quantum group $U_q(\mathfrak{sl}_2)$. In the case that $\dot z=0$, we study the subcategory $\mathcal{A}$ consisting of finite dimensional $U_q(\mathfrak{s})$-modules of type $1$ with zero action of $Z$. Motivated by the ideas in \cite{DLMZ, Mak}, we directly construct an equivalent functor from $\mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\mathfrak{s})$-modules is wild.

Equations

$$\begin{array}[]{ll}[E,F]=\frac{K-K^{-1}}{q-q^{-1}},&KXK^{-1}=qX,\\ KEK^{-1}=q^{2}E,&KFK^{-1}=q^{-2}F,\\ KYK^{-1}=q^{-1}Y,&qYX-XY=Z,\\ EX=qXE,&EY=X+q^{-1}YE,\\ FX=YK^{-1}+XF,&FY=YF,\end{array}$$

$$U_q(\mathfrak{s})=U_q({\mathfrak{n}}^{-})\otimes U_q(\mathfrak{h})\otimes U_q({\mathfrak{n}}^{+}).$$

$$V={\oplus }_{\lambda \in {\mathbb{C}}^{\ast }}{V}_{\lambda },$$

$$\mathrm{supp}(V)=\{\lambda \in {\mathbb{C}}^{\ast }|V_{\lambda }\ne 0\}.$$

$$0=M_0\subset M_1\subset \cdots \subset M_n=M$$

$$B_{\dot{z}}:=H_q/(H_q(Z-\dot{z})+H_{q}X)$$

$$X{v}_i=-\frac{\dot{z}(q^i-1)}{q-1}{v}_{i-1},i\in {\mathbb{Z}}_{+}.$$

$$K{v}_i$$

$$F{v}_i$$

$$E{v}_i$$

$$(EF-FE)v_i$$

$$EY{v}_i$$

$$FX{v}_i$$

$$\Delta :U_q(\mathfrak{s})\to U_q(\mathfrak{s})\otimes U_q(\mathfrak{s})$$

$$\Delta (E)$$

$$\Delta (K)$$

$$\Delta (X)$$

$$\Delta (Z)$$

$$U_q(\mathfrak{s})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$u(n\otimes b)={\Sigma }_i{u}_{i}n\otimes w_{i}b,n\in N,b\in {\tilde{B}}_{\dot{z}}.$$

$$0\to {\Delta }_{{\mathfrak{sl}}_2}(q^{-d-2})\to {\Delta }_{{\mathfrak{sl}}_2}(q^d)\to L_{{\mathfrak{sl}}_2}(q^d)\to 0,$$

$$0\to {\Delta }_{{\mathfrak{sl}}_2}(-q^{-d-2})\to {\Delta }_{{\mathfrak{sl}}_2}(-q^d)\to L_{{\mathfrak{sl}}_2}(-q^d)\to 0.$$

$$0\to \Delta (q^{-d-\frac{5}{2}},\dot{z})\to \Delta (q^{d-\frac{1}{2}},\dot{z})\to L(q^{d-\frac{1}{2}},\dot{z})\to 0,$$

$$0\to \Delta (-q^{-d-\frac{5}{2}},\dot{z})\to \Delta (-q^{d-\frac{1}{2}},\dot{z})\to L(-q^{d-\frac{1}{2}},\dot{z})\to 0.$$

$$M\cong \widetilde{N_1}\otimes {\tilde{B}}_{\dot{z}},V/M\cong \widetilde{N_2}\otimes {\tilde{B}}_{\dot{z}},$$

$$X^{k}E(w\otimes v_0)=w_k\otimes X^k{v}_k=q^{-k}(w\otimes X^k{v}_0)=0.$$

$$E{v}_i=\frac{-\dot{z}(q^i-1)(q^{i-1}-1)}{q^{i-2}(q^2-1)(q-1)}{v}_{i-2}$$

$$E{Y}^i=q^{-i}{Y}^{i}E+[i]Y^{i-1}X-\frac{(q^i-1)(q^{i-1}-1)}{q^{i-2}(q^2-1)(q-1)}Z{Y}^{i-2},$$

$$E(w\otimes v_i)=E{Y}^i(w\otimes v_0)$$

$$=$$

$$F(w\otimes v_0)=w_F\otimes v_0+K^{-1}w\otimes F{v}_0+\sum _{i=1}^k{w}_i^{\prime }\otimes v_i,$$

$$0=$$

$$=$$

Full text

Keywords: BGG category, highest weight module, quiver, wild.

1. Introduction

In this paper, we denote by $\mathbb{Z}$, $\mathbb{Z}_{+}$, $\mathbb{N}$, $\mathbb{C}$ and $\mathbb{C}^{*}$ the sets of all integers, nonnegative integers, positive integers, complex numbers, and nonzero complex numbers, respectively. Let $q$ be a nonzero complex number which is not a root of unity. For $n\in\mathbb{Z}$, denote $[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}$.

The BGG category $\mathcal{O}$ for complex semisimple Lie algebras was introduced by Joseph Bernstein, Israel Gelfand and Sergei Gelfand in the early 1970s, see [BGG], which includes all highest weight modules such as Verma modules and finite dimensional simple modules. This category is influential in many areas of representation theory. About the knowledge of $\mathcal{O}$, one can see the recent monograph [Hu] for details.

The Schrödinger Lie algebra $\mathfrak{s}$ is the semidirect product of $\mathfrak{sl}_{2}$ and the three-dimensional Heisenberg Lie algebra. This algebra can describe symmetries of the free particle Schrödinger equation, see [DDM1, Pe]. The representation theory of the Schrödinger algebra has been studied by many authors. A classification of the simple highest weight representations of the Schrödinger algebra were given in [DDM1]. All simple weight modules with finite dimensional weight spaces were classified in [D], see also [LMZ]. All simple weight modules of the Schrödinger algebra were classified in [BL2, BL3]. The BGG category $\mathcal{O}$ of $\mathfrak{s}$ was studied in [DLMZ].

In 1996, in order to research the $q$-deformed heat equations, a $q$-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in [DDM2]. This algebra is called the quantum Schrödinger algebra.

The quantum Schrödinger algebra $U_{q}(\mathfrak{s})$ over $\mathbb{C}$ is generated by the elements $E$, $F$, $K$, $K^{-1}$, $X$, $Y$, $Z$ with defining relations:

[TABLE]

where $Z$ is central in $U_{q}(\mathfrak{s})$. This definition is somewhat different from that of [DDM2] in form. This algebra is a kind of quantized symplectic oscillator algebra of rank one, see [GK]. The paper [GK] gave the PBW Theorem and showed that the category $\mathcal{O}$ is a highest weight category. In the present paper, we will give specific characterizations for each block of $\mathcal{O}$ for $U_{q}(\mathfrak{s})$ using some quivers.

The subalgebra generated by $E,F,K,K^{-1}$ is the quantum group $U_{q}(\mathfrak{sl}_{2})$. The subalgebra generated by $X,Y,Z$ is called the quantum Weyl algebra $H_{q}$, see [B]. All simple weight modules over $U_{q}(\mathfrak{s})$ with zero action of $Z$ were classified in [BL1].

The paper is organized as follows. In Section 2 we recall some basic facts about the category $\mathcal{O}$ for $U_{q}(\mathfrak{s})$ . For a $\dot{z}\in\mathbb{C}$, we denote by $\mathcal{O}[\dot{z}]$ the full subcategory of $\mathcal{O}$ consisting of all modules which are annihilated by some power of the maximal ideal $\langle Z-\dot{z}\rangle$ of $\mathbb{C}[Z]$. In Section 3, in case of $\dot{z}\neq 0$, using the modules $B_{\dot{z}}$ over the quantum Weyl algebra $H_{q}$, we show that the functor $-\otimes\tilde{B}_{\dot{z}}$ gives an equivalence between $\mathcal{O}^{(\mathfrak{sl}_{2})}$ and $\mathcal{O}[\dot{z}]$ , where $\mathcal{O}^{(\mathfrak{sl}_{2})}$ is the BGG category of $U_{q}(\mathfrak{sl}_{2})$, see Theorem 8. A Weight $U_{q}(\mathfrak{s})$-module $M$ is of type $1$ if the $\operatorname{Supp}(M)\subset q^{\mathbb{Z}}$. In Section 4, we study the category $\mathcal{A}$ of finite dimensional $U_{q}(\mathfrak{s})$-modules of type $1$ with zero action of $Z$. It is shown that there is an equivalence between $\mathcal{A}$ and the category of finite dimensional representations of an infinite quiver with some quadratic relations, see Theorem 17. In [DLMZ], the grading technique was used in the study of finite dimensional modules for the Schrödinger Lie algebra.

2. Basic properties of the category $\mathcal{O}$

2.1. The definition of Category $\mathcal{O}$

Let $U_{q}(\mathfrak{n}^{+})$ be the subalgebra of $U_{q}(\mathfrak{s})$ generated by the elements $E,X$ and let $U_{q}(\mathfrak{n}^{-})$ be the subalgebra generated by $F,Y$. Moreover let $U_{q}(\mathfrak{h})$ be the subalgebra generated by the elements $K,K^{-1},Z$. We write $\otimes$ for $\otimes_{\mathbb{C}}$.

Then we have the following triangular decomposition:

[TABLE]

A $U_{q}(\mathfrak{s})$-module $V$ is called a weight module if $K$ acts diagonally on $V$, i.e.,

[TABLE]

where $V_{\lambda}=\{v\in V\mid Kv=\lambda v\}$. For a weight module $V$, let

[TABLE]

Next, we introduce the category $\mathcal{O}$ for $U_{q}(\mathfrak{s})$.

Definition 1**.**

A left module $M$ over $U_{q}(\mathfrak{s})$ is said to belong to category $\mathcal{O}$ if

1. (1)

$M$* is finitely generated over $U_{q}(\mathfrak{s})$;* 2. (2)

$M$* is a weight module;* 3. (3)

The action of $U_{q}(\mathfrak{n}^{+})$ on $M$ is locally finite, i.e., $\dim U_{q}(\mathfrak{n}^{+})v<\infty$ for any $v\in M$.

For a weight module $V$, a weight vector $v_{\lambda}\in V_{\lambda}$ is called a highest weight vector if $Ev_{\lambda}=Xv_{\lambda}=0$. A module $M$ is called a highest weight module of highest weight $\lambda$ if there exists a highest weight vector $v_{\lambda}$ in $M$ which generates $M$. For $\lambda\in\mathbb{C}^{*}$ and $\dot{z}\in\mathbb{C}$, let $\Delta(\lambda,\dot{z})$ be the Verma module generated by $v_{\lambda}$, where $Kv_{\lambda}=\lambda v_{\lambda},Zv_{\lambda}=\dot{z}v_{\lambda}$. Then $\{Y^{k}F^{l}v_{\lambda}|k,l\in\mathbb{Z}_{+}\}$ is a basis of $\Delta(\lambda,\dot{z})$. Let $R(\lambda,\dot{z})$ be the largest proper submodule of $\Delta(\lambda,\dot{z})$. Hence $L(\lambda,\dot{z})=\Delta(\lambda,\dot{z})/R(\lambda,\dot{z})$ is the unique simple quotient module of $\Delta(\lambda,z)$.

2.2. Basic properties of $\mathcal{O}$

By the similar arguments as those in [Hu], we can see that every module $M$ in $\mathcal{O}$ has the following standard properties.

Lemma 2**.**

The category $\mathcal{O}$ is closed with respect to taking submodules, quotient modules and finite direct sums. That is, the category $\mathcal{O}$ is an abelian category.

Lemma 3**.**

Let $M$ be any module in $\mathcal{O}$.

1. $($1$)$

The module $M$ has a finite filtration

[TABLE]

such that each subquotient $M_{j}/M_{j-1}$ for $1\leqslant j\leqslant n$ is a highest weight module. 2. $($2$)$

Each weight space of $M$ is finite dimensional. 3. $($3$)$

Any simple module in $\mathcal{O}$ is isomorphic to some $L(\lambda,\dot{z})$, for $\lambda,\dot{z}\in\mathbb{C}$.

3. Blocks of nonzero central charge

In this section, we assume that $\dot{z}\neq 0$. We denote by $\mathcal{O}[\dot{z}]$ the full subcategory of $\mathcal{O}$ consisting of all modules which are annihilated by some power of the maximal ideal $\langle Z-\dot{z}\rangle$ of $\mathbb{C}[Z]$. Let $\mathcal{O}^{(\mathfrak{sl}_{2})}$ denote the BGG category for $U_{q}(\mathfrak{sl}_{2})$. We will show that there is an equivalence between $\mathcal{O}^{(\mathfrak{sl}_{2})}$ and $\mathcal{O}[\dot{z}]$ . Firstly, we found that the structure of Verma modules over $U_{q}(\mathfrak{s})$ is similar as that of Verma modules over $U_{q}(\mathfrak{sl}_{2})$.

3.1. The tensor product realizations of highest modules

In this subsection, we will give tensor product realizations of Verma modules using Verma modules over $U_{q}(\mathfrak{sl}_{2})$ and $H_{q}$. This construction is crucial to the study of the category $\mathcal{O}[\dot{z}]$ .

For a nonzero $\dot{z}\in\mathbb{C}$, let

[TABLE]

which is a simple $H_{q}$-module. Denote the image of $Y^{i}$ in $B_{\dot{z}}$ by $v_{i}$ for $i\in\mathbb{Z}_{+}$. We can see that

[TABLE]

Define the action of $U_{q}(\mathfrak{sl}_{2})$ on $B_{\dot{z}}$ by

[TABLE]

Then we can check that

[TABLE]

Thus the action (3.2) indeed makes $B_{\dot{z}}$ to be a module over $U_{q}(\mathfrak{s})$. We denote this $U_{q}(\mathfrak{s})$-module by $\widetilde{B}_{\dot{z}}$. In fact, $\widetilde{B}_{\dot{z}}\cong L(q^{-\frac{1}{2}},\dot{z})$.

We can make a $U_{q}(\mathfrak{sl}_{2})$-module $N$ to be a $U_{q}(\mathfrak{s})$-module be defining $H_{q}N=0$. We denote the resulting $U_{q}(\mathfrak{s})$-module by $\widetilde{N}$.

Lemma 4**.**

The following map

[TABLE]

defined by

[TABLE]

can define an algebra homomorphism.

Remark: There is no algebra homomorphism $\epsilon:U_{q}(\mathfrak{s})\rightarrow\mathbb{C}$ such that the following diagram commutes:

[TABLE]

So we can not define a bialgebra structure on $U_{q}(\mathfrak{s})$ from $\Delta$.

Via the map $\Delta$, the space $\widetilde{N}\otimes\widetilde{B}_{\dot{z}}$ can be defined as a $U_{q}(\mathfrak{s})$-module for any $U_{q}(\mathfrak{sl}_{2})$-module $N$. More precisely, for $u\in U_{q}(\mathfrak{s})$, if $\Delta(u)=\Sigma_{i}u_{i}\otimes w_{i}$, then the action of $U_{q}(\mathfrak{s})$ on $\widetilde{N}\otimes\widetilde{B}_{\dot{z}}$ is defined by

[TABLE]

Let $\Delta_{\mathfrak{sl}_{2}}(\lambda)$ be the Verma module over $U_{q}(\mathfrak{sl}_{2})$ with the highest weight $\lambda$ whose unique simple quotient module is $L_{\mathfrak{sl}_{2}}(\lambda)$. It is well known that the module $\Delta_{\mathfrak{sl}_{2}}(\lambda)$ is reducible if and only if $\lambda^{2}\in q^{2\mathbb{Z}_{+}}$, see [J]. For each $d\in\mathbb{Z}_{+}$, we have non-split short exact sequences

[TABLE]

and

[TABLE]

The structure of $\Delta(\lambda,\dot{z})$ was determined in [DDM2]. The following proposition give a constructive proof.

Proposition 5**.**

[CCL]** The following results hold.

1. (1)

If $\dot{z}\neq 0$, then $\Delta(\lambda,\dot{z})\cong\widetilde{\Delta}_{\mathfrak{sl}_{2}}(\lambda q^{\frac{1}{2}})\otimes\widetilde{B}_{\dot{z}}$. Therefore the Verma module $\Delta(\lambda,\dot{z})$ is reducible if and only if $\lambda^{2}\in q^{2\mathbb{Z}_{+}-1}$. Moreover, $L(\lambda,z)\cong\widetilde{L}_{\mathfrak{sl}_{2}}(\lambda q^{\frac{1}{2}})\otimes\widetilde{B}_{\dot{z}}$. 2. (2)

For each $d\in\mathbb{Z}_{+}$, we have a non-split short exact sequences

[TABLE]

and

[TABLE]

3.2. Equivalence between

$\mathcal{O}^{(\mathfrak{sl}_{2})}$ and $\mathcal{O}[\dot{z}]$ .

Lemma 6**.**

If $\dot{z}\neq 0$, then any module in $\mathcal{O}[\dot{z}]$ has finite composition length.

Proof.

According to (1) in Lemma 3, any nonzero module $M$ in $\mathcal{O}[\dot{z}]$ has a finite filtration with sub-quotients given by highest weight modules. Hence it suffices to treat the case that $M$ is the Verma module $\Delta(\lambda,\dot{z})$. By Proposition 5, $\Delta(\lambda,\dot{z})$ has finite composition length if $\dot{z}\neq 0$. ∎

Proposition 7**.**

Suppose that $V$ is a module in $\mathcal{O}[\dot{z}]$ with nonzero central charge $\dot{z}$. Then $V\cong\widetilde{N}\otimes\widetilde{B}_{\dot{z}}$ for some $U_{q}(\mathfrak{sl}_{2})$-module $N$.

Proof.

By Lemma 6, $V$ has a finite composition length $l.$ We will proceed the proof by induction on $l$. Firstly, we consider the case $l=1$. The fact that $V\in\mathcal{O}$ forces that $V$ is a simple highest weight $U_{q}(\mathfrak{s})$-module. Then $V$ is a simple quotient module of some Verma module $\Delta(\lambda,z)$ . Note that $\Delta(\lambda,z)\cong\widetilde{\Delta}_{\mathfrak{sl}_{2}}(\lambda q^{\frac{1}{2}})\otimes\widetilde{B}_{\dot{z}}$. Thus $V\cong\widetilde{L}_{\mathfrak{sl}_{2}}(\lambda q^{\frac{1}{2}})\otimes\widetilde{B}_{\dot{z}}$.

Next, we consider the general case. Let $M$ be a maximal submodule of $V$. By the induction hypothesis, we see that

[TABLE]

where $N_{1},N_{2}$ are $U_{q}(\mathfrak{sl}_{2})$-modules.

As vector spaces, we can assume that $V=N\otimes{B}_{\dot{z}}$, where $N$ is a vector space such that $N_{1}\subseteq N$ and $N/N_{1}\cong N_{2}$. Moreover, $u(w\otimes v)=w\otimes uv,K(w\otimes v)=(Kw)\otimes(Kv)$, for $u\in H_{q},w\in N,v\in B_{\dot{z}}$.

For $w\in N$, we can find $w_{i}\in N,0\leqslant i\leqslant k$ such that $E(w\otimes v_{0})=\sum_{i=0}^{k}w_{i}\otimes v_{i}$, where $v_{i}$ is the image of $Y^{i}$ in $B_{\dot{z}}$ . From $X^{k}E=q^{-k}EX^{k},Xv_{0}=0$, we have

[TABLE]

By (3.1), when $k>0$, $X^{k}v_{k}\neq 0$ . So we must have that $k=0$. Denote $w_{E}=q^{\frac{1}{2}}w_{0}$. Then $E(w\otimes v_{0})=w_{E}\otimes Kv_{0}$.

From

[TABLE]

and

[TABLE]

we obtain that

[TABLE]

By the action of $F$ on $N/N_{1}\otimes\widetilde{B}_{\dot{z}}$, there exist $w_{F}\in N,w^{\prime}_{i}\in N$, $1\leqslant i\leqslant k$ satisfying

[TABLE]

where $\sum_{i=1}^{k}w^{\prime}_{i}\otimes v_{i}\in N_{1}\otimes\widetilde{B}_{\dot{z}}$.

From

[TABLE]

we have $w^{\prime}_{i}=0$, for any $1\leqslant i\leqslant k.$ Then

[TABLE]

Consequently, from $YF=FY$ and $Y^{i}v_{0}=v_{i}$, we have

[TABLE]

We can define the action of $U_{q}(\mathfrak{sl}_{2})$ on $N$ as follows:

[TABLE]

From (3.5) and (3.6), we can see that $V\cong\widetilde{N}\otimes\widetilde{B}_{\dot{z}}$. The proof is complete. ∎

By Lemma 7, we have the following category equivalence.

Theorem 8**.**

If $\dot{z}\neq 0$, using the algebra homomorphism $\Delta$ in (3.3), we can define a functor

[TABLE]

Moreover, this functor is an equivalence of categories.

Proof.

By the definition of category $\mathcal{O}$, the functor $F:=-\otimes\tilde{B}_{\dot{z}}$ maps modules in $\mathcal{O}^{(\mathfrak{sl}_{2})}$ to modules in $\mathcal{O}[\dot{z}]$. By Lemma 7, the functor $F$ is essentially surjective.

Next, we will show that for any $M,N\in\text{Obj}(\mathcal{O}^{(\mathfrak{sl}_{2})}$), the map

[TABLE]

is a bijection.

For $f\in\text{Hom}_{\mathcal{O}^{(\mathfrak{sl}_{2})}}(M,N)$ such that $F_{M,N}(f)=f\otimes 1=0$, we must have $f=0$. So $F_{M,N}$ is injective.

For any $g\in\text{Hom}_{\mathcal{O}[\dot{z}]}(F(M),F(N))$, suppose that

[TABLE]

where $v_{i}$ is the image of $Y^{i}$ in $B_{\dot{z}}$ . Since $\tilde{B}_{\dot{z}}$ is a simple $H_{q}$-module, by the density theorem, there exists  $u\in H_{q}$ such that

[TABLE]

Form

[TABLE]

we have:

[TABLE]

Define the map $f:M\rightarrow N$ such that $f(w)=w_{0}$, i.e., $g(w\otimes v_{0})=f(w)\otimes v_{0}.$

From $Y^{i}g(w\otimes v_{0})=g(w\otimes Y^{i}v_{0})$, we have

[TABLE]

By $Eg(w\otimes v_{0})=g(E(w\otimes v_{0}))$ and $Ev_{0}=0$, we have that

[TABLE]

i.e., $Ef(w)=f(Ew)$. Similarly, we can check that $Ff(w)=f(Fw)$, $Kf(w)=f(Kw)$. Then $f\in\text{Hom}_{\mathcal{O}^{(\mathfrak{sl}_{2})}}(M,N)$. So $F_{M,N}(f)=g$, $F_{M,N}$ is surjective.

Therefore $F_{M,N}$ is bijective, and hence $F:=-\otimes\tilde{B}_{\dot{z}}$ is an equivalence. ∎

3.3. The description of $\mathcal{O}[\xi,\dot{c},\dot{z}]$

Let $C=FE+\frac{qK+q^{-1}K^{-1}}{(q-q^{-1})^{2}}$ be the Casimir element of $U_{q}(\mathfrak{sl}_{2})$. Define the following element in $U_{q}(\mathfrak{s})$:

[TABLE]

In [CCL], the following lemma was proved.

Lemma 9**.**

The element $\tilde{C}$ belongs to the center of $U_{q}(\mathfrak{s})$.

For a module $M$ in $\mathcal{O}$, from that the weight spaces of $M$ are finite dimensional, we can see that the action of $\mathbb{C}[\tilde{C},Z]$ on $M$ is locally finite.

For $\lambda\in\mathbb{C},\dot{z}\in\mathbb{C}$, let $v_{\lambda}$ be a highest vector of the $U_{q}(\mathfrak{s})$-module $L(\lambda,\dot{z})$ . We denote by $\tilde{c}_{\lambda}$ the scalar corresponding to the action of the central element $\tilde{C}$ on $v_{\lambda}$. Similarly, for the $U_{q}(\mathfrak{sl}_{2})$-module $L_{\mathfrak{sl}_{2}}(\lambda)$, we denote scalar corresponding to the action of $C$ by $c_{\lambda}$.

Lemma 10**.**

We have the following:

1. (1)

\tilde{c}_{\lambda}=\frac{\dot{z}}{(q-q^{-1})^{2}}\Big{(}(q+q^{2})\lambda+(q^{-1}+q^{-2})\lambda^{-1}\Big{)}. 2. (2)

$\tilde{c}_{\lambda}=\tilde{c}_{\lambda q^{-k}}$* iff $\lambda^{2}=q^{k-3}$.*

Let $\xi\in\mathbb{C}^{*}/q^{\mathbb{Z}}$. We denote by $\mathcal{O}[\xi]$ the full subcategory of $\mathcal{O}$ consisting of all $M$ such that $\mathrm{supp}(M)\subset\xi$. For $\dot{z},\dot{c}\in\mathbb{C}$, we denote by $\mathcal{O}[\xi,\dot{c},\dot{z}]$ the full subcategory of $\mathcal{O}[\xi]$ consisting of all $M$ such that $M$ is annihilated by some power of the maximal ideal $\langle\tilde{C}-\dot{c},Z-\dot{z}\rangle$ of $\mathbb{C}[Z,\tilde{C}]$. Since the action of $\mathbb{C}[\tilde{C},Z]$ on $M$ is locally finite, we have

[TABLE]

Similarly, we can define the subcategory $\mathcal{O}^{(\mathfrak{sl}_{2})}[\xi,\dot{c}]$ of $\mathcal{O}^{(\mathfrak{sl}_{2})}$, where $\dot{c}$ is defined by the action of Casimir element $C$ of $U_{q}(\mathfrak{sl}_{2})$.

Using the equivalence between $\mathcal{O}^{(\mathfrak{sl}_{2})}$ and $\mathcal{O}[\dot{z}]$ given in Theorem 8, we have the following equivalence.

Lemma 11**.**

The restriction functor

[TABLE]

is an equivalence of categories, where $\lambda\in\xi,c_{\lambda}=\frac{q\lambda+q^{-1}\lambda^{-1}}{(q-q^{-1})^{2}}$.

Using the structure of $\mathcal{O}^{(\mathfrak{sl}_{2})}$ (see Section 5.3 in [Maz]), we give descriptions of each block $\mathcal{O}[\xi,\dot{c},\dot{z}]$ as follows.

Proposition 12**.**

Let $\xi=q^{\frac{1}{2}+\mathbb{Z}},\dot{z}\in\mathbb{C}^{*}$. Then the following claims hold.

1. $($1$)$

The module $\Delta(q^{-\frac{3}{2}},\dot{z})$ is the unique simple object in $\mathcal{O}[q^{\frac{1}{2}+\mathbb{Z}},\tilde{c}_{q^{-\frac{3}{2}}},\dot{z}]$. Moreover, the block $\mathcal{O}[q^{\frac{1}{2}+\mathbb{Z}},\tilde{c}_{q^{-\frac{3}{2}}},\dot{z}]$ is semisimple. 2. $($2$)$

For $n\in\mathbb{Z}_{+}$, $\Delta(q^{-n-\frac{5}{2}},\dot{z})$ , $L(q^{n-\frac{1}{2}},\dot{z})$ are all the simple objects in $\mathcal{O}[q^{\frac{1}{2}+\mathbb{Z}},\tilde{c}_{q^{n-\frac{1}{2}}},\dot{z}]$. 3. $($3$)$

For $n\in\mathbb{Z}_{+}$, the subcategory $\mathcal{O}[q^{\frac{1}{2}+\mathbb{Z}},\tilde{c}_{q^{n-\frac{1}{2}}},\dot{z}]$ is equivalent to the category of finite dimensional representations over $\mathbb{C}$ of the following quiver with relations:

[TABLE]

Proposition 13**.**

Let $\xi=q^{\mathbb{Z}},\dot{z}\in\mathbb{C}^{*}$. Then we have the following:

1. $($1$)$

For any $n\in\mathbb{Z}$, the Verma module $\Delta(q^{n},\dot{z})$ is simple. 2. $($2$)$

The modules $\Delta(q^{n},\dot{z})$, $\Delta(q^{2n+3},\dot{z})$ are simple objects in $\mathcal{O}[q^{\mathbb{Z}},\tilde{c}_{q^{n}},\dot{z}]$. 3. $($3$)$

The block $\mathcal{O}[q^{\mathbb{Z}},\tilde{c}_{q^{n}},\dot{z}]$ is equivalent to $\mathbb{C}\oplus\mathbb{C}$-mod for any $n\in\mathbb{Z}$.

Proposition 14**.**

For $\xi\in\mathbb{C}^{*}/q^{\mathbb{Z}},\dot{z}\in\mathbb{C}^{*},\lambda\in\xi$. If $\xi\neq\pm q^{\mathbb{Z}},\pm q^{\frac{1}{2}+\mathbb{Z}}$, then the block $\mathcal{O}[\xi,\tilde{c}_{\lambda},\dot{z}]$ is semisimple with the unique simple object $\Delta(\lambda,\dot{z})$ .

4. Finite dimensional $U_{q}(\mathfrak{s})$-modules

In this section, we study finite dimensional $U_{q}(\mathfrak{s})$-modules. A Weight $U_{q}(\mathfrak{s})$-module $M$ is of type $1$ if the $\operatorname{Supp}(M)\subset q^{\mathbb{Z}}$. Note that $L(\lambda,\dot{z})$ is infinite dimensional when $\dot{z}\neq 0$. So $Z$ acts trivially on any finite dimensional simple $U_{q}(\mathfrak{s})$-module. Consequently $Z$ acts nilpotently on any finite dimensional $U_{q}(\mathfrak{s})$-module. Let $\mathcal{A}$ denote the category of finite dimensional $U_{q}(\mathfrak{s})$-modules of type $1$ with zero action of $Z$. Thus any module in $\mathcal{A}$ is a module over the smash product algebra $A:=\mathbb{C}_{q}[X,Y]\rtimes U_{q}(\mathfrak{sl}_{2})$, where $\mathbb{C}_{q}[X,Y]=\mathbb{C}\langle X,Y|XY=qYX\rangle$ is the quantum plane. The quantum plane $\mathbb{C}_{q}[X,Y]$ is a $U_{q}(\mathfrak{sl}_{2})$-module on the following action.

[TABLE]

We will use the completely reducibility of finite dimensional $U_{q}(\mathfrak{sl}_{2})$-modules and the Clebsch-Gordon rule to discuss the category $\mathcal{A}$ . For convenience, let $L(i)$ denote the finite dimensional $U_{q}(\mathfrak{sl}_{2})$-module with the highest weight $q^{i},i\in\mathbb{Z}_{+}$. In fact, we can assume that $L(1)=\mathbb{C}X+\mathbb{C}Y$, whose $U_{q}(\mathfrak{sl}_{2})$-module structure was defined by (4.1). It is well known that the tensor product $L(m)\otimes L(n)$ is a $U_{q}(\mathfrak{sl}_{2})$-module under the action defined by the following co-multiplication:

[TABLE]

**Remark: ** The above co-multiplication $\Delta^{\prime}$ is different from $\Delta$ in (3.3). Now $\Delta^{\prime}$ can guarantee that $\tau(\theta_{i})$ defined in Lemma 16 is a $U_{q}(\mathfrak{sl}_{2})$-module homomorphism, however $\Delta$ in (3.3) can not. Because we consider the left action of $L(1)$ on $L(i)$ in Lemma 16 .

Next, we will introduce two lemmas which are used in the proof of Theorem 17.

Lemma 15**.**

1. $($1$)$

$L(1)\otimes L(i)\cong L(i+1)\oplus L(i-1)$, for $i\geqslant 1$; 2. $($2$)$

Suppose that $v_{i}$ is a highest weight vector of $L(i)$. Then $X\otimes v_{i}$ is a highest weight vector of $L(1)\otimes L(i)$ whose highest weight is $q^{i+1}$, and $[i]Y\otimes v_{i}-q^{-1}X\otimes Fv_{i}$ is a highest weight vector of $L(1)\otimes L(i)$ whose highest weight is $q^{i-1}$; 3. $($3$)$

In $L(1)\otimes L(1)\otimes L(i)$, the elements

[TABLE]

and

[TABLE]

are highest weight vectors with the highest weight $q^{i}$.

Proof.

(1) follows from the Clebsch-Gordon rule [J]:

[TABLE]

(2) We can check that $\begin{aligned} E(X\otimes v_{i})=(E\cdot X)\otimes v_{i}+(K\cdot X)\otimes Ev_{i}=0,\end{aligned}$

[TABLE]

Then (2) holds.

(3) follows from (1) and (2). ∎

Lemma 16**.**

For any module $V\in\mathcal{A}$, $\theta_{i}\in\text{Hom}_{U_{q}(\mathfrak{sl}_{2})}(L(i),V)$, the following map

[TABLE]

a $U_{q}(\mathfrak{sl}_{2})$-module homomorphism, where $\ a,b\in\mathbb{C},v\in L(i)$.

Proof.

For $a,b\in\mathbb{C},v\in L(i)$, we can check that

[TABLE]

[TABLE]

[TABLE]

So $\tau(\theta_{i})$ is a $U_{q}(\mathfrak{sl}_{2})$-module homomorphism.

∎

Consider the following quiver.

[TABLE]

The following theorem is inspired by the ideas in [DLMZ, Mak].

Theorem 17**.**

The category $\mathcal{A}$ is equivalent to the category $\mathcal{B}$ of finite dimensional representations for the quiver $\mathbf{Q}_{\infty}$ satisfying the following condition:

[TABLE]

Proof.

By Lemma 15, we can define $U_{q}(\mathfrak{sl}_{2})$-module homomorphisms

[TABLE]

such that

[TABLE]

where each $v_{i}$ is a fixed highest weight vector of $L(i)$.

We will prove the theorem in three steps.

Step 1. We define a functor $F$ from $\mathcal{A}$ to $\mathcal{B}$. Let $V$ be a $U_{q}(\mathfrak{s})$-module which belongs to $\mathcal{A}$.

(1) For every $i$, we can associate it with a vector space $V_{i}:=\text{Hom}_{U_{q}(\mathfrak{sl}_{2})}(L(i),V)$.

(2) For arrows $a_{i},b_{i}$, we can define linear maps $V(b_{i}):V_{i+1}\rightarrow V_{i},V(a_{i}):V_{i}\rightarrow V_{i+1}$ as follows:

[TABLE]

Next we check that:

[TABLE]

For $\theta_{0}\in V_{0},v\in L(0)$, from $Fv=0$, we have

[TABLE]

For $\theta_{i}\in V_{i}$, from $XY=qYX,[i+1]q^{-1}-q^{-i-1}=[i]$, we have

[TABLE]

By the fact that the $U_{q}(\mathfrak{sl}_{2})$-module $L(i)$ is generated by $v_{i}$ and $V(b_{i})V(a_{i})(\theta_{i})$, $V(a_{i-1})V(b_{i-1})(\theta_{i})$ are $U_{q}(\mathfrak{sl}_{2})$-module homomorphisms, we see that $V(b_{i})V(a_{i})(\theta_{i})=V(a_{i-1})V(b_{i-1})(\theta_{i})$.

Thus $(V_{i},V(a_{i}),V(b_{i}),i\in\mathbb{Z}_{+})$ is a representation of $\mathbf{Q}_{\infty}$ satisfying the relation (4.2).

(3) We define a functor $F$ from $\mathcal{A}$ to $\mathcal{B}$. For $V,W\in\text{Obj}(\mathcal{A}),f\in\text{Hom}_{U_{q}(\mathfrak{s})}(V,W)$, define

[TABLE]

[TABLE]

where $f^{*}_{i}$ satisfies $f^{*}_{i}(\theta_{i})=f\theta_{i},\theta_{i}\in V_{i}$.

We check the following diagram

[TABLE]

is commutative.

Since $\tau(\theta_{i})$ is a $U_{q}(\mathfrak{sl}_{2})$-module homomorphism,

[TABLE]

and

[TABLE]

So

[TABLE]

Therefore, $F(f)=(f^{*}_{i}:V_{i}\rightarrow W_{i},\ i\in\mathbb{Z}_{+})$ is a morphism from the representation $F(V)$ to $F(W)$.

Step 2. For a representation $(V_{i},V(a_{i}),V(b_{i}),i\in\mathbb{Z}_{+})$ of the quiver $\mathbf{Q}_{\infty}$ satisfying the relation (4.2), there is a $U_{q}(\mathfrak{s})$-module $V$ such that $F(V)\cong(V_{i},V(a_{i}),V(b_{i}),i\in\mathbb{Z}_{+})$.

Let $V=\bigoplus_{i\in\mathbb{Z}_{+}}V_{i}\otimes L(i)$. Next we define the action of $U_{q}(\mathfrak{s})$ on $V$. Since $v_{i}$ is a highest weight vector of $L(i)$, $F^{s}v_{i},s\in\mathbb{Z}_{+}$ is a basis of $L(i)$. For $u\in U_{q}(\mathfrak{sl}_{2}),\theta_{i}\in V_{i},s\in\mathbb{Z}_{+}$, define

[TABLE]

The reason for defining the action of $U_{q}(\mathfrak{s})$ on $V$ by (4.3) coming from the definitions of $V(a_{i}),V(b_{i})$ in Step 1 . We can check that the action (4.3) indeed defines a $U_{q}(\mathfrak{s})$-module through a little cumbersome calculation. For the verification process, one can see the appendix of the present paper.

From the definition of $F$, we can see that $F(V)\cong(V_{i},V(a_{i}),V(b_{i}),i\in\mathbb{Z}_{+})$.

Step 3. The functor $F$ is completely faithful, i.e., the map

[TABLE]

is a bijection.

If $f\in\text{Hom}_{U_{q}(\mathfrak{s})}(V,W)$ satisfying that $F_{V,W}(f)=0$, then for any $\theta_{i}\in\text{Hom}_{U_{q}(\mathfrak{sl}_{2})}(L(i),V)$, $i\in\mathbb{Z}_{+}$, we have: $f\theta_{i}=0$. Since $V$ is a sum of simple submodules $L(i)$. So $f=0$, and $F_{V,W}$ is injective.

From the completely reducibility of finite dimensional $U_{q}(\mathfrak{sl}_{2})$-modules,

[TABLE]

For $g=(g_{i}:V_{i}\rightarrow W_{i},\ i\in\mathbb{Z}_{+})\in\text{Hom}_{\mathcal{B}}(F(V),F(W))$, we define $f:V\rightarrow W$ as follows:

[TABLE]

Since $\theta_{i},g_{i}(\theta_{i})$ is a $U_{q}(\mathfrak{sl}_{2})$-module homomorphism, for $u\in U_{q}(\mathfrak{sl}_{2})$, we have

[TABLE]

At the same time, using the following commutative diagram:

[TABLE]

we obtain that

[TABLE]

and

[TABLE]

So $f$ is a $U_{q}(\mathfrak{s})$-module homomorphism such that $F(f)=g$.

Therefore, $F$ is an equivalence. ∎

Let $\mathbb{C}\langle x_{1},x_{2}\rangle$ be the free associative algebra over $\mathbb{C}$ generated by two variables $x_{1},x_{2}$. Recall that an abelian category $\mathcal{C}$ is wild if there exists an exact functor from the category of representations of the algebra $\mathbb{C}\langle x_{1},x_{2}\rangle$ to $\mathcal{C}$ which preserves indecomposability and takes nonisomorphic modules to nonisomorphic ones, see Definition 2 in [Mak].

By [DLMZ], the category $\mathcal{B}$ for the quiver $\mathbf{Q}_{\infty}$ is wild. Hence we have the following corollary.

Corollary 18**.**

The representation type of the category $\mathcal{A}$ is wild.

Let $\mathcal{O}[0]$ be the full subcategory $\mathcal{O}$ consisting of modules with locally nilpotent action of $Z$. Since $\mathcal{A}$ is a subcategory of $\mathcal{O}[0]$ , $\mathcal{O}[0]$ is also wild.

5. Appendix

In this appendix, we check that the action (4.3) indeed defines a $U_{q}(\mathfrak{s})$-module. Let $v_{i}$ be a fixed highest weight vector of $L(i)$, $\theta_{i}\in V_{i},s\in\mathbb{Z}_{+}$.

From the action of $U_{q}(\mathfrak{sl}_{2})$ on $V$, it suffices to check the relations $XY=qYX$ and the relations between $H_{q}$ and $U_{q}(\mathfrak{sl}_{2})$.

Firstly it is easy to see that $FY(\theta_{i}\otimes F^{s}v_{i})=YF(\theta_{i}\otimes F^{s}v_{i})$ from $FY=YF$.

Next we can compute that:

[TABLE]

[TABLE]

Then using $V(a_{i-1})V(b_{i-1})=V(b_{i})V(a_{i})$ and

[TABLE]

we have  $XY(\theta_{i}\otimes F^{s}v_{i})=qYX(\theta_{i}\otimes F^{s}v_{i})$.

Furthermore, from $EF^{s}v_{i}=[s][i+1-s]F^{s-1}v_{i}$ , we have

[TABLE]

[TABLE]

Then $EX(\theta_{i}\otimes F^{s}v_{i})=qXE(\theta_{i}\otimes F^{s}v_{i})$.

Using $EF^{s}=F^{s}E+[s]F^{s-1}\frac{q^{-(s-1)}K-q^{s-1}K^{-1}}{q-q^{-1}}$, we have

[TABLE]

[TABLE]

Then from $[i+1-s]-q^{s-i}=q[i-s]$ and $q^{s}+q^{-1}[s]=[s+1]$, we have

[TABLE]

Next we have

[TABLE]

[TABLE]

Using

[TABLE]

we have

[TABLE]