Finite dimensional modules over quantum toroidal algebras

TL;DR

This paper proves that for most types of quantum toroidal algebras, there are no nontrivial finite-dimensional simple modules when the parameter q is generic, filling a gap in the representation theory of these algebras.

Contribution

It establishes the nonexistence of finite-dimensional simple modules for quantum toroidal algebras of all types except A_1 at generic q, a previously unresolved question.

Findings

No nontrivial finite-dimensional simple modules for non-A_1 types at generic q

Completes the classification of finite-dimensional modules over quantum toroidal algebras

Provides a foundation for future research on representations of quantum toroidal algebras

Abstract

The representations of the quantum toroidal algebras have been widely studied by many authors. However, no one has constructed some finite dimensional modules for them while $q$ is generic. In this paper, for all $\mathfrak{g}$-generic $q$, if $\mathfrak{g}$ is not of type $A_1$, we prove that the quantum toroidal algebra $U_q(\mathfrak{g}_{\rm tor})$ has no nontrivial finite dimensional simple module.

Equations

$$D=diag(d_0,\cdots ,d_n)$$

$$q_i=q^{d_i},[m{]}_i=\frac{q_i^m-q_i^{-m}}{q_i-q_i^{-1}},m\in \mathbb{Z},$$

$$[m{]}_i!=\prod _{k=1}^m[k{]}_i,{\left[\genfrac{}{}{0ex}{}{m}{k}\right]}_i=\frac{[m{]}_i!}{[k{]}_i![m-k{]}_i!},m\ge 0.$$

$${\gamma }^{\pm \frac{1}{2}}\mathrm{are}\mathrm{central}\mathrm{and}{K}_i{K}_j=K_j{K}_i,K_i^{\pm 1}{K}_i^{\mp 1}=1,$$

$$[a_i(l),K_i^{\pm 1}]=0,$$

$$[a_i(l),a_j(l^{\prime })]={\delta }_{l+l^{\prime },0}\frac{[l{c}_{i,j}{]}_i}{l}\cdot \frac{{\gamma }^l-{\gamma }^{-l}}{q-q^{-1}},$$

$$K_i{x}_j^{\pm }(k)K_i^{-1}=q^{\pm c_{i,j}}{x}_j^{\pm }(k),$$

$$[a_i(l),x_j^{\pm }(k)]=\pm \frac{[l{c}_{i,j}{]}_i}{l}{\gamma }^{\mp \frac{|}{l}|2}{x}_j^{\pm }(l+k),$$

$$[x_i^{\pm }(k+1),x_j^{\pm }(k^{\prime }){]}_{q_i^{\pm c_{i,j}}}=-[x_j^{\pm }(k^{\prime }+1),x_i^{\pm }(k){]}_{q_i^{\pm c_{i,j}}},$$

$$[x_i^{+}(k),x_j^{-}(k^{\prime })]={\delta }_{i,j}\frac{{\gamma }^{\frac{k-k^{\prime }}{2}}{\varphi }_i^{+}(k+k^{\prime })-{\gamma }^{-\frac{k-k^{\prime }}{2}}{\varphi }_i^{-}(k+k^{\prime })}{q_i-q_i^{-1}},$$

$$\displaystyle Sym_{t_{1},\ldots,t_{m}}\sum_{k=0}^{m=1-c_{i,j}}(-1)^{k}\left[\begin{array}[]{c}1-c_{i,j}\\ k\end{array}\right]_{i}x_{i}^{\pm}(t_{1})\cdots x_{i}^{\pm}(t_{k})x_{j}(t)x_{i}^{\pm}(t_{k+1})\cdots x_{i}^{\pm}(t_{m})=0,$$

$${\varphi }_i^{\pm }(z)=\sum _{m=0}^{\infty }{\varphi }_i^{\pm }(\pm m)z^{\mp m}=K_i^{\pm 1}\exp \left(\pm (q_i-q_i^{-1})\sum _{l=1}^{\infty }{a}_i(\pm l)z^{\mp l}\right).$$

$$\{x_i^{\pm }(0),K_i^{\pm 1},(i=0,1,\cdots ,n)\}.$$

$$x_i^{\pm }(k),a_i(l),K_i^{\pm 1},{\gamma }^{\pm \frac{1}{2}}(i=1,\cdots ,n,k\in \mathbb{Z},l\in \mathbb{Z}\setminus \{0\}.$$

$$\Theta (x_i^{\pm }(0))=x_i^{\pm }(0),i=1,\cdots ,n,$$

$$\Theta ({\Gamma }^{\frac{1}{2}})={\gamma }^{\frac{1}{2}}.$$

$$\pi :x_i^{\pm }(0)\mapsto x_i^{\pm }(\pm m_i),K_i^{\pm 1}\mapsto ({\gamma }^{m_i}{K}_i{)}^{\pm 1}$$

$$U_q(\hat{\mathfrak{g}})$$

$$\det (C(q,l))$$

$$[\xi (l),x_j^{\pm }(k)]$$

$$({\xi }_0,\cdots ,{\xi }_n)C(q,l)=(l,\cdots ,l),$$

$$[\xi (l),x_j^{\pm }(k)]=\pm \sum _{i=0}^n{\xi }_i\frac{[l{C}_{i,j}{]}_i}{l}{x}_j^{\pm }(l+k)=\pm x_j^{\pm }(l+k).$$

$$x_i^{\pm }\otimes f(z)$$

$$x_i^{\pm }\otimes g_i^{\pm }(z)V=0.$$

$$\Xi \cdot (x_{i_1}^{\pm }(m_1)\cdots x_{i_k}^{\pm }(m_k)v)=(\lambda \pm k)x_{i_1}^{\pm }(m_1)\cdots x_{i_k}^{\pm }(m_k)v.$$

$$[\Xi {|}_V,x_i^{\pm }(m){|}_V]=\pm (x_i^{\pm }\otimes z^{m}f(z)+x_i^{\pm }(m)){|}_V=\pm x_i^{\pm }(m){|}_V.$$

$$x_i^{+}(m)v^{\prime }$$

$$a_i(l)v^{\prime }=0,l\in \mathbb{Z}\setminus \{0\},i=0,\cdots ,n.$$

Full text

E-mail address: xialimeng@ujs.edu.cn

Institute of Applied System Analysis, Jiangsu University

Zhenjiang 212013, Jiangsu Prov. China

1. Introduction

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra and $\widehat{\mathfrak{g}}=\mathfrak{g}\otimes{\mathbb{C}}[t,t^{-1}]\oplus{\mathbb{C}}c$ be the associated affine Lie algebra. Both quantum groups $U_{q}(\mathfrak{g})$ and $U_{q}(\widehat{\mathfrak{g}})$ have finite dimensional simple modules which can be viewed as the quantization of simple modules over classic Lie algebras ([2], [3], [4]).

In 1987, Drinfeld gave an extremely important realization of quantum affine algebras, then it was applied to construct the affinization of quantum affine algebras. Such new algebras are called quantum toroidal algebras. Let $\mathfrak{g}_{\rm tor}$ be the toroidal Lie algebra of $\mathfrak{g}$ with nullity $2$. The quantum toroidal algebra $U_{q}(\mathfrak{g}_{\rm tor})$ can be regarded as a quantum deformation of the enveloping algebra $U(\mathfrak{g}_{\rm tor})$ of $\mathfrak{g}_{\rm tor}$. In the past decades, the quantum toroidal algebras and their representations have been researched by many authors (see [6], [7], [8], [10], [11], [12], [13], [14]).

Certainly, $\mathfrak{g}_{\rm tor}$ has nontrivial finite dimensional modules. However, no one has constructed some finite dimensional modules for $U_{q}(\mathfrak{g}_{\rm tor})$ while $q$ is generic. In this paper, we prove the following result.

Theorem 1.1**.**

If $\mathfrak{g}$ is not of type $A_{1}$ and $q\in{\mathbb{C}}^{*}$ is generic, then quantum toroidal algebra $U_{q}(\mathfrak{g}_{\rm tor})$ has no nontrivial finite dimensional simple module.

2. Quantum toroidal algebras

Let ${C}=(c_{i,j})_{0\leq i,j\leq n}$ be a symmetrizable Cartan matrix. So there exists a diagonal matrix

[TABLE]

with $d_{i}\in{\mathbb{Z}}_{+}$ such that $\gcd(d_{0},\cdots,d_{n})=1$ and $DC$ is symmetric.

In this paper, we always assume that $(c_{i,j})_{1\leq i,j\leq n}$ is of finite type $X_{n}$ ($\not=A_{1}$) and $C$ is of affine type $X_{n}^{(1)}$. They are the Cartan matrices of Lie algebras $\mathfrak{g}$ and $\widehat{\mathfrak{g}}$, respectively.

For convenience, we use the following standard notations:

[TABLE]

2.1. quantum toroidal algebras

Definition 2.1**.**

The quantum toroidal algebra $U_{q}(\mathfrak{g}_{\rm tor})$ is an associative ${\mathbb{C}}(q)$-algebra generated by elements $x_{i}^{\pm}(k),a_{i}(l),K_{i}^{\pm 1},\gamma^{\pm\frac{1}{2}}\,(i=0,1,\cdots,n,k\in{\mathbb{Z}},l\in{\mathbb{Z}}\setminus\{0\}$ satisfying

[TABLE]

and the $q$-Serre relations

[TABLE]

where $Sym_{t_{1},\ldots,t_{m}}$ denotes the symmetrization with respect to the indices $(t_{1},\cdots,t_{m})$ and

[TABLE]

2.2. quantum affine algebras

Definition 2.2**.**

The horizontal quantum affine algebra $U_{q}(\widehat{\mathfrak{g}})$ is the subalgebra of $U_{q}(\mathfrak{g}_{\rm tor})$ generated by elements

[TABLE]

Let $\alpha_{i}$ be the simple root associated to $x_{i}^{+}(0)$ and $\delta=\sum_{i=0}^{n}s_{i}\alpha_{i}$ the primitive imaginary root. Then $\Gamma=\prod_{i=0}^{n}K_{i}^{s_{i}}$ is a central element.

Definition 2.3**.**

The vertical quantum affine algebra $\mathcal{U}_{q}(\widehat{\mathfrak{g}})$ is the subalgebra of $U_{q}(\mathfrak{g}_{\rm tor})$ generated by elements

[TABLE]

Adding $\Gamma^{\frac{1}{2}}$ to $U_{q}(\widehat{\mathfrak{g}})$, it is well known that the extended algebra is isomorphic to $\mathcal{U}_{q}(\widehat{\mathfrak{g}})$. In particular, there exists an isomorphism $\Theta$ such that

[TABLE]

3. Highest weight modules over quantum affine algebras

In this section, we introduce the notion highest weight module of Kac-Moody type.

For arbitrary given $m_{0},\cdots,m_{n}\in{\mathbb{Z}}$, let $A(m_{0},\cdots,m_{n})$ denote the subalgebra generated by $\{x_{i}^{\pm}(\pm m_{i}),(\gamma^{m_{i}}K_{i})^{\pm 1},(i=0,1,\cdots,n)\}$.

Lemma 3.1**.**

$A(m_{0},\cdots,m_{n})$* is isomorphic to $U_{q}(\widehat{\mathfrak{g}})$.*

Proof.

In fact, the map

[TABLE]

defines an isomorphism of algebras. In the following we shall identify $A(m_{0},\cdots,m_{n})$ as $U_{q}(\widehat{\mathfrak{g}})$ by this isomorphism. ∎

Let $N^{\pm}$ be the subalgebra generated by $\{x_{i}^{\pm}(0),(i=0,1,\cdots,n)\}$ and let $N^{0}$ be the Laurent polynomial algebra ${\mathbb{C}}(q)[K_{0}^{\pm 1},\cdots,K_{n}^{\pm 1}]$, then

[TABLE]

Definition 3.2**.**

(a) Suppose that $V$ is a $U_{q}(\widehat{\mathfrak{g}})$-module and $v\in V$. If $x_{i}^{+}(0)v=0,K_{i}^{\pm}v=a_{i}^{\pm 1}v$ for all $i$, then $v$ is called a highest weight vector of Kac-Moody type.

(b) A module generated by a highest weight vector of Kac-Moody type is called a highest weight module of Kac-Moody type.

(c) If $v$ is a highest weight vector of Kac-Moody type, then ${\mathbb{C}}v$ is a one dimensional module over $N^{0}\cdot N^{+}$. The induced module $M(v)=U_{q}(\widehat{\mathfrak{g}})\otimes_{N^{0}\cdot N^{+}}{\mathbb{C}}v$ is called a Verma module of Kac-Moody type.

Lemma 3.3**.**

Any highest weight module of Kac-Moody type is a quotient of some Verma module of Kac-Moody type.

Proof.

It follows by the definition. ∎

Lemma 3.4**.**

Assume that $v$ is a highest weight vector of Kac-Moody type and $V$ is a simple $U_{q}(\widehat{\mathfrak{g}})$-module generated by $v$. If ${\hbox{\rm dim}}V<\infty$, then $V={\mathbb{C}}v$ is trivial.

Proof.

First we claim that any simple finite dimensional $U_{q}(\widehat{\mathfrak{g}})$-module $V$ is a simple $\mathcal{U}_{q}(\widehat{\mathfrak{g}})$-module.

Note that $\gamma$ is central and it acts as a scalar over $V$. By relation $[a_{1}(1),a_{1}(-1)]=[2]_{1}\frac{\gamma-\gamma^{-1}}{q-q^{-1}}$, if $\gamma\not=\pm 1$, the subalgebra generated by $a_{1}(1)|_{V},a_{1}(-1)|_{V}$ is isomorphic to the Weyl algebra $\mathfrak{A}$, which has no finite dimensional module ([1]). So $\Gamma|_{V}=\gamma|_{V}=\pm 1$.

For each $i$, $v$ is a highest weight vector of the quantum group of type $A_{1}$ generated by $x_{i}^{\pm}(0),K_{i}^{\pm 1}$. So ${\hbox{\rm dim}}V<\infty$ implies $K_{i}v=\varepsilon_{i}q^{\lambda_{i}}v$ such that $\lambda_{i}\in\mathbb{N},\varepsilon_{i}\in\{1,-1\}$ (see Theorem 2.6 of [9]).

Moreover, we infer that $q^{s_{0}\lambda_{0}+\cdots+s_{n}\lambda_{n}}\in\{1,-1\}$. Since $q$ is generic, we have $s_{0}\lambda_{0}+\cdots+s_{n}\lambda_{n}=0$ and $\lambda_{0}=\cdots=\lambda_{n}=0$. Then the Verma module $M(v)$ has a unique maximal submodule $J$ generated by $\{x_{i}^{-}(0)v|i=0,1,\cdots\}$. So $V=M(v)/J\cong{\mathbb{C}}(q)v$. ∎

4. Proof for main theorem

Throughout this section, we always assume that $q$ is generic and $V$ is a finite-dimensional simple $U_{q}(\mathfrak{g}_{\rm tor})$-module. So $\gamma|V=\pm 1$. For convenience, we assume $\gamma|V=1$. The proof for $\gamma|V=-1$ is very similar.

Define matrix $C(q,l)=([lc_{i,j}]_{i})_{0\leq i,j\leq n}$ for all $l\not=0$. Then there exits a polynomial $\det_{\mathfrak{g}}$ such that

[TABLE]

The polynomial $\det_{\mathfrak{g}}$ is explicitly give by the following tabular:

[TABLE]

4.1. some useful lemmas

Lemma 4.1**.**

$C(q,l)$* is invertible for all $l\not=0$.*

Proof.

Straightforward. ∎

Lemma 4.2**.**

There exist elements $\xi(l)$ for all $l\in{\mathbb{Z}}\setminus\{0\}$ such that

[TABLE]

Proof.

Because $C(q,l)$ is invertible, let $(\xi_{0},\cdots,\xi_{n})$ be the unique solution of

[TABLE]

and let $\xi(l)=\sum_{i=0}^{n}\xi_{i}a_{i}(l)$. Then

[TABLE]

∎

For convenience, we write

[TABLE]

for all $0\leq i\leq n$ and $f(z)=\sum f_{j}z^{j}\in{\mathbb{C}}[z,z^{-1}]$.

Lemma 4.3**.**

There exists $f(z)=\sum_{j=0}^{n}f_{j}z^{j}\in{\mathbb{C}}[z]$ such that $f_{0}=-1$ and $x_{i}^{\pm}\otimes f(z)V=0$ for each $i$.

Proof.

Since ${\hbox{\rm dim}}V<\infty$, there are polynomials $g^{\pm}_{i}(z)\in{\mathbb{C}}[z,z^{-1}]$ such that

[TABLE]

If $p(z)=g_{i}^{\pm}(z)h(z)$, then $x_{i}^{\pm}\otimes p(z)=\pm[\sum h_{l}\xi(l),x_{i}^{\pm}\otimes g_{i}^{\pm}(z)]$, then $x_{i}^{\pm}\otimes p(z)V=0$. Let $f(z)=cz^{m}g_{0}^{-}(z)\cdots g_{n}^{-}(z)g_{0}^{+}(z)\cdots g_{n}^{+}(z)$, where $c\in{\mathbb{C}}^{*}$ and $m\in{\mathbb{Z}}$ such that $f(z)\in{\mathbb{C}}[z]$ and $f(0)=-1$. ∎

Let $\Xi=\sum_{l=1}^{\hbox{\rm deg}(f)}f_{l}\xi(l)$. Assume $v\in V$ is an eigenvector of $\Xi$ with eigenvalue $\lambda$.

Lemma 4.4**.**

For all $0\leq i_{1},\cdots,i_{k}\leq n$ and $m_{1},\cdots,m_{k}\in{\mathbb{Z}}$, we have

[TABLE]

Proof.

It follows from $x_{i}^{\pm}\otimes z^{m}f(z)V=0$ and

[TABLE]

∎

4.2. proof of main theorem

Because $V$ is finite-dimensional, by (4.3), there exists $v^{\prime}$ of $U_{q}(\widehat{\mathfrak{g}})$ such that $\Xi\cdot v^{\prime}=\lambda^{\prime}v^{\prime}$ and

[TABLE]

By (4.4), $U_{q}(\widehat{\mathfrak{g}})v^{\prime}$ is a finite dimensional highest weight module of $A(m_{0},m_{1},\cdots,m_{n})$. By Lemma 3.4, ${\hbox{\rm dim}}A(m_{0},m_{1},\cdots,m_{n})v^{\prime}<\infty$ and Theorem 2.6 of [9], we have $K_{i}v^{\prime}=\epsilon_{i}v^{\prime},\epsilon_{i}\in\{1,-1\}$ for all $i=0,\cdots,n$.

If there exists $v^{\prime\prime}:=x_{i}^{-}(m_{i})v^{\prime}\not=0$ for some index $i$ and integer $m_{i}$, then $v^{\prime\prime}$ is also a highest weight vector of $U_{q}(\widehat{\mathfrak{g}})$ and $K_{i}v^{\prime\prime}=\pm q_{i}^{-2}v^{\prime\prime}\not=\pm v^{\prime\prime}$, this forces ${\hbox{\rm dim}}V=\infty$, a contradiction. So we also have $x_{i}^{-}(m)v^{\prime}=0$ for all $m\in{\mathbb{Z}}$ and $i=0,1,\cdots,n$. This by (2.7) also implies

[TABLE]

So $V={\mathbb{C}}v^{\prime}$ is trivial.

ACKNOWLEDGMENTS

The author gratefully acknowledges the partial financial support from the NNSF (Nos. 11871249, 11771142) and the Jiangsu Natural Science Foundation (No. BK20171294). Part of this work was done during the author’s visiting Paris Diderot. The author would like to thank Prof. Marc Rosso for warm hospitality and helpful discussions.