Classification of integrable representations for toroidal extended affine Lie algebras

TL;DR

This paper classifies irreducible integrable modules with finite-dimensional weight spaces for nullity 2 toroidal extended affine Lie algebras, advancing understanding of their representation theory.

Contribution

It provides a complete classification of such modules, focusing on the core action and nullity 2 case, which was previously not fully understood.

Findings

Classification of irreducible integrable modules achieved

Finite-dimensional weight spaces characterized

New insights into the structure of toroidal extended affine Lie algebras

Abstract

In this paper, we classify the irreducible integrable modules with finite dimensional weight spaces and non-trivial $\widetilde{\mathfrak g}_c$-action for the nullity $2$ toroidal extended affine Lie algebra $\widetilde{\mathfrak g}$, where $\widetilde{\mathfrak g}_c$ is the core of $\widetilde{\mathfrak g}$.

Equations

$$t^{\mathbf{m}}=t_0^{m_0}{t}_1^{m_1}.$$

$$\mathrm{d}({\Omega }_{\mathcal{R}}^1)={\text{Span}}_{\mathbb{C}}\{m_0{t}^{\mathbf{m}}{\mathrm{k}}_0+m_1{t}^{\mathbf{m}}{\mathrm{k}}_1\mid \mathbf{m}=(m_0,m_1)\in {\mathbb{Z}}^2\}$$

$${\tilde{\mathfrak{g}}}_c=\left(\mathcal{R}\otimes \dot{\mathfrak{g}}\right)\oplus \mathcal{K},$$

$$[t^{\mathbf{m}}\otimes x,t^{\mathbf{n}}\otimes y]=t^{\mathbf{m}+\mathbf{n}}\otimes [x,y]+⟨x,y⟩\sum _{i=0,1}{m}_i{t}^{\mathbf{m}+\mathbf{n}}{\mathrm{k}}_i,$$

$${\mathrm{d}}_{\mathbf{m}}=m_0{t}^{\mathbf{m}}{\mathrm{d}}_1-m_1{t}^{\mathbf{m}}{\mathrm{d}}_0,$$

$$\tilde{\mathfrak{g}}(\mu )={\tilde{\mathfrak{g}}}_c\oplus \mathcal{S}=\left(\mathcal{R}\otimes \dot{\mathfrak{g}}\right)\oplus \mathcal{K}\oplus \mathcal{S},$$

$$[{\mathrm{d}}_{\mathbf{m}},t^{\mathbf{n}}\otimes x]=\det \left(\genfrac{}{}{0ex}{}{\mathbf{m}}{\mathbf{n}}\right)t^{\mathbf{m}+\mathbf{n}}\otimes x,[{\mathrm{d}}_i,t^{\mathbf{n}}\otimes x]=n_i{t}^{\mathbf{n}}\otimes x,$$

$$[{\mathrm{d}}_{\mathbf{m}},t^{\mathbf{n}}{\mathrm{k}}_j]=\det \left(\genfrac{}{}{0ex}{}{\mathbf{m}}{\mathbf{n}}\right)t^{\mathbf{m}+\mathbf{n}}{\mathrm{k}}_j+(m_0{\delta }_{j,1}-m_1{\delta }_{j,0})\sum _{a=0,1}{m}_a{t}^{\mathbf{m}+\mathbf{n}}{\mathrm{k}}_a,$$

$$[{\mathrm{d}}_i,t^{\mathbf{n}}{\mathrm{k}}_j]=n_i{t}^{\mathbf{n}}{\mathrm{k}}_j,[{\mathrm{d}}_i,{\mathrm{d}}_{\mathbf{m}}]=m_i{\mathrm{d}}_{\mathbf{m}},[{\mathrm{d}}_0,{\mathrm{d}}_1]=0,$$

$$[{\mathrm{d}}_{\mathbf{m}},{\mathrm{d}}_{\mathbf{n}}]=\det \left(\genfrac{}{}{0ex}{}{\mathbf{m}}{\mathbf{n}}\right){\mathrm{d}}_{\mathbf{m}+\mathbf{n}}+\mu {\left(\det \left(\genfrac{}{}{0ex}{}{\mathbf{m}}{\mathbf{n}}\right)\right)}^2\sum _{a=0,1}{m}_a{t}^{\mathbf{m}+\mathbf{n}}{\mathrm{k}}_a,$$

$$\det \left(\genfrac{}{}{0ex}{}{\mathbf{m}}{\mathbf{n}}\right)=m_0{n}_1-n_0{m}_1.$$

$$\tilde{\mathfrak{h}}=\dot{\mathfrak{h}}\oplus \mathbb{C}{\mathrm{k}}_0\oplus \mathbb{C}{\mathrm{k}}_1\oplus \mathbb{C}{\mathrm{d}}_0\oplus \mathbb{C}{\mathrm{d}}_1,$$

$$⟨t^{\mathbf{m}}\otimes x,t^{\mathbf{n}}\otimes y⟩={\delta }_{m_0+n_0,0}{\delta }_{m_1+n_1,0}⟨x,y⟩,⟨{\mathrm{d}}_i,{\mathrm{k}}_j⟩={\delta }_{i,j},$$

$$⟨{\mathrm{d}}_{\mathbf{m}},t_0^{n_0}{t}_1^{n_1}{\mathrm{k}}_j⟩=(m_0{\delta }_{1,j}-m_1{\delta }_{0,j}){\delta }_{m_0+n_0,0}{\delta }_{m_1+n_1,0},$$

$$\alpha ({\mathrm{k}}_i)=0=\alpha ({\mathrm{d}}_i),$$

$${\delta }_i(\dot{\mathfrak{h}})=0,{\delta }_i({\mathrm{k}}_j)=0,{\delta }_i({\mathrm{d}}_j)={\delta }_{i,j},$$

$$\tilde{\Delta }=\{\alpha +m_0{\delta }_0+m_1{\delta }_1\mid \alpha \in \dot{\Delta },m_0,m_1\in \mathbb{Z}\}$$

$$\tilde{\mathfrak{g}}=\underset{\alpha \in \tilde{\Delta }}{⨁}{\tilde{\mathfrak{g}}}_{\alpha },$$

$${\tilde{\mathfrak{g}}}_{\alpha }=\{x\in \tilde{\mathfrak{g}}\mid [h,x]=\alpha (h)x,h\in \tilde{\mathfrak{h}}\}.$$

$$V={\oplus }_{\lambda \in {\tilde{\mathfrak{h}}}^{\ast }}{V}_{\lambda }$$

$$V_{\lambda }=\{v\in V\mid h.v=\lambda (h)v,\forall h\in \tilde{\mathfrak{h}}\}.$$

$${\tilde{\Delta }}^{\times }=\{\alpha \in \tilde{\Delta }\mid ⟨\alpha ,\alpha ⟩\ne 0\}\text{and}{\tilde{\Delta }}^0=\{\alpha \in \tilde{\Delta }\mid ⟨\alpha ,\alpha ⟩=0\}$$

$$\pi ({\mathrm{d}}_i)={\mathrm{d}}_i,\pi ({\mathrm{d}}_{\mathbf{m}})=\mathrm{d}(\mathbf{m}),$$

$$[{\mathrm{d}}_i,{\mathrm{d}}_j]=0,[{\mathrm{d}}_i,\mathrm{d}(\mathbf{m})]=m_i\mathrm{d}(\mathbf{m}),[\mathrm{d}(\mathbf{m}),\mathrm{d}(\mathbf{n})]=\det \left(\genfrac{}{}{0ex}{}{\mathbf{m}}{\mathbf{n}}\right)\mathrm{d}(\mathbf{m}+\mathbf{n}),$$

$$\tilde{\Delta }={\tilde{\Delta }}^{+}\cup {\tilde{\Delta }}^0\cup {\tilde{\Delta }}^{-}$$

$${\tilde{\Delta }}^{\pm }=\{\alpha +m_0{\delta }_0+m_1{\delta }_1\mid \alpha \in {\dot{\Delta }}_{\pm },m_0,m_1\in \mathbb{Z}\}.$$

$$\tilde{\mathfrak{g}}={\tilde{\mathfrak{g}}}^{+}\oplus {\tilde{\mathfrak{g}}}^0\oplus {\tilde{\mathfrak{g}}}^{-}$$

$${\tilde{\mathfrak{g}}}^0={\oplus }_{\alpha \in {\tilde{\Delta }}^0}{\tilde{\mathfrak{g}}}_{\alpha }=\left(\mathcal{R}\otimes \dot{\mathfrak{h}}\right)\oplus \mathcal{K}\oplus \mathcal{S}.$$

$$M(T)=\mathcal{U}(\tilde{\mathfrak{g}}){\otimes }_{\mathcal{U}({\tilde{\mathfrak{g}}}^{+}\oplus {\tilde{\mathfrak{g}}}^0)}T.$$

$$T=\{v\in V\mid {\tilde{\mathfrak{g}}}^{+}.v=0\}$$

$${\dot{P}}_{+}=\{\lambda \in {\dot{\mathfrak{h}}}^{\ast }\mid \lambda ({\alpha }_i^{\vee })\in \mathbb{N},i=1,\cdots ,\ell \}$$

$$t^{\mathbf{m}}\otimes h.t^{\mathbf{n}}\otimes u$$

Full text

1. Partially supported by China NSF grants (No.11501478)

2. Partially supported by China NSF grants (Nos.11471268, 11531004)

1. Introduction

In this paper we study the integrable representation for the nullity $2$ toroidal extended affine Lie algebras (EALAs for short). The notion of EALAs was first introduced by Hoegh-Krohn and Torresani in [H-KT] under the name of quasi-simple Lie algebras. Since then the structure theory of EALAs has been intensively studied for the past twenty years (see [AABGP, BGK, N] and the references therein). An EALA $E$ is by definition a complex Lie algebra, together with a non-zero finite dimensional ad-diagonalizable subalgebra $H$ and a non-degenerate invariant symmetric bilinear form $(\cdot|\cdot)$, that satisfies a list of natural axioms. A crucial consequence of these axioms is that the form $(\cdot|\cdot)$ on $E$ induces a semi-positive bilinear form on the $\mathbb{R}$-span of the root system of $E$ relative to $H$. Then the root system of $E$ divides into a disjoint union of the sets of isotropic and non-isotropic roots. The subalgebra $E_{c}$ of $E$ generated by non-isotropic root vectors is called the core of $E$, which is in fact an ideal of $E$. An $E$-module is said to be integrable if it admits a weight space decomposition relative to $H$ and all non-isotropic root vectors in $E$ act locally nilpotent on it. We remark that the integrability of a weight $E$-module $V$ depends only on the action of $E_{c}$. In particular, if $E_{c}.V=0$, then $V$ is automatically integrable.

One of the axioms for EALAs requires that the rank of the group generated by the isotropic roots is finite, and this rank is called the nullity of EALAs. The nullity [math] EALAs are nothing but the finite dimensional simple Lie algebras, and the affine Kac-Moody algebras are precisely the nullity $1$ EALAs [ABGP]. It is well-known that in the study of the representation theory for finite or affine Kac-Moody algebras, one of the main ingredients is the classification of irreducible integrable representations. The irreducible integrable modules (with finite dimensional weight spaces) for the finite dimensional simple Lie algebras are precisely the finite dimensional irreducible modules. And, such modules for the affine Kac-Moody algebras are the highest weight integrable modules, their dual or the loop modules [C, CP1, CP2]. Similar to the nullity [math] and nullity $1$ cases, one of the fundamental problems in the representation theory of extended affine Lie algebras with higher nullities is to classify their irreducible integrable modules.

Toroidal extended affine Lie algebras are a class of important EALAs that provide examples with arbitrary nullities. The construction of toroidal EALAs parallels with the untwisted affine Lie algebras. Let $\dot{\mathfrak{g}}$ be a finite dimensional simple Lie algebras, let $\mathcal{R}=\mathbb{C}[t_{0}^{\pm 1},\cdots,t_{N-1}^{\pm 1}]$ be the ring of Laurent polynomials in $N$ variables and let $\mathcal{S}$ be the subspace of divergence zero derivations on $\mathcal{R}$ (which is also called the set of skew-derivations [BGK]). The nullity $N$ toroidal EALA $\widetilde{\mathfrak{g}}=\widetilde{\mathfrak{g}}_{c}\oplus\mathcal{S}$ is defined by adding the space $\mathcal{S}$ (possibly twisted with a $2$-cocycle) to the universal central extension $\widetilde{\mathfrak{g}}_{c}=(\mathcal{R}\otimes\dot{\mathfrak{g}})\oplus\mathcal{K}$ of the multi-loop Lie algebra $\mathcal{R}\otimes\dot{\mathfrak{g}}$. We recall that $\widetilde{\mathfrak{g}}_{c}$ is the core of $\widetilde{\mathfrak{g}}$ and is often called the toroidal Lie algebra. One may see Section 2 or [B, Section 1] for details.

Let $\mathcal{D}$ be the space of derivations over $\mathcal{R}$. As a left $\mathcal{R}$-module, $\mathcal{D}$ has a basis $\mathrm{d}_{0},\cdots,\mathrm{d}_{N-1}$, where $\mathrm{d}_{i}=t_{i}\frac{\partial}{\partial t_{i}}$. Besides toroidal EALAs, there are some other interesting extensions of the toroidal Lie algebra $\widetilde{\mathfrak{g}}_{c}$ that are obtained by adding various subspace of $\mathcal{D}$. For examples, one can add the subspaces $\mathcal{D}_{0}=\mathbb{C}\mathrm{d}_{0}\oplus\cdots\oplus\mathbb{C}\mathrm{d}_{N-1}$, $\mathcal{D}^{*}=\mathbb{C}\mathrm{d}_{0}\oplus\mathcal{R}\mathrm{d}_{1}\oplus\cdots\oplus\mathcal{R}\mathrm{d}_{N-1}$ of $\mathcal{D}$, or the space $\mathcal{D}$ itself. The irreducible integrable representations of the Lie algebras $\widetilde{\mathfrak{g}}_{c}\oplus\mathcal{D}_{0}$, $\widetilde{\mathfrak{g}}_{c}\oplus\mathcal{D}^{*}$ and $\widetilde{\mathfrak{g}}_{c}\oplus\mathcal{D}$ with finite dimensional weight spaces have been studied respectively in [E1, E2],[JM] and [EJ]. This paper is aimed to classify the irreducible integrable modules for the nullity $2$ toroidal EALAs.

We denote by $\widetilde{\mathcal{S}}$ the quotient algebra of $\widetilde{\mathfrak{g}}$ obtained by modulo its core $\widetilde{\mathfrak{g}}_{c}$, which is isomorphic to the Lie algebra of divergence zero derivations. When $N=2$, the derived subalgebra of $\widetilde{\mathcal{S}}$ is often called the (centerless) Virasoro-like algebra. As we mention before, any weight $\widetilde{\mathfrak{g}}$-module with trivial $\widetilde{\mathfrak{g}}_{c}$-action is integrable. Thus the category of integrable $\widetilde{\mathfrak{g}}$-modules with trivial $\widetilde{\mathfrak{g}}_{c}$-action is equivalent to the category of weight $\widetilde{\mathcal{S}}$-modules. In recent years, the classification problem on the irreducible weight $\widetilde{\mathcal{S}}$-modules with finite dimensional weight spaces has been studied by several authors, see [BT, GL, JL, LS, LT] for details.

In this paper, we give a complete classification of irreducible integrable representations for the nullity $2$ toroidal EALA $\widetilde{\mathfrak{g}}$ with finite dimensional weight spaces and non-trivial $\widetilde{\mathfrak{g}}_{c}$-action. We remark that the classification of the irreducible integrable modules for the nullity $N$ toroidal EALAs with $N\geq 3$ is different to that of the $N=2$ case, and will be given in a subsequent paper. So, from now on, in the rest of the paper we always assume that $N=2$.

To be more precisely about our result, we say that an irreducible integrable $\widetilde{\mathfrak{g}}$-module $V$ has zero (resp. non-zero) central charge if the central elements $\mathrm{k}_{0},\mathrm{k}_{1}$ of $\widetilde{\mathfrak{g}}$ act trivially (resp. non-trivially) on $V$. In [EJ], by using the method developed in [JM], the classification of irreducible integrable modules for $\widetilde{\mathfrak{g}}_{c}\oplus\mathcal{D}$ with zero central charge is reduced to the classification of irreducible modules for the semi-product algebra $\mathcal{R}\rtimes\mathcal{D}$ with certain conditions. As in [EJ], in the present work we reduce the classification problem for integrable $\widetilde{\mathfrak{g}}$-modules with zero central charge to the classification problem for certain $\mathcal{R}\rtimes\widetilde{\mathcal{S}}$-modules. Such $\mathcal{R}\rtimes\widetilde{\mathcal{S}}$-modules have been classified in [JL] (see also [GL, BT]). We emphasise that Jiang-Meng’s reduction method can not be applied directly to the toroidal EALAs, as almost all lemmas used in this reduction need certain distinguished derivations that do not belong to $\mathcal{S}$. See [JM, Lemma 3.1, 3.2, 3.4, 3.9] for details. To make Jiang-Meng’s reduction work for the toroidal EALAs, we use rather different tricks to fulfill each step. We first prove in Subsection 4.2 that the whole space $\mathcal{K}$ acts trivially on any irreducible integrable $\widetilde{\mathfrak{g}}$-module with zero central charge. The proof of this assertion is based on a result of Chari-Pressely [CP2] on integrable modules for extended loop algebras (see Lemma 4.1). Then, by using the technique developed in [GL], we give a classification of the irreducible uniformly bounded $(\mathcal{R}\otimes\dot{\mathfrak{h}})\rtimes\widetilde{\mathcal{S}}$-modules in Subsection 4.3, where $\dot{\mathfrak{h}}$ stands for the Cartan subalgebra of $\dot{\mathfrak{g}}$. Finally, by putting the above results together, in Subsection 4.4 we classify the irreducible integrable modules for $\widetilde{\mathfrak{g}}$ in Theorem 3.2 for the zero central charge case. A realization of such irreducible integrable $\widetilde{\mathfrak{g}}$-modules is also presented in Subsection 4.4.

In [E2], after a change of coordinates, the classification of irreducible integrable $\widetilde{\mathfrak{g}}_{c}\oplus\mathcal{D}_{0}$-modules with non-zero central charge is reduced to determine certain linear functionals on a distinguished Heisenberg algebra. As done in [E2], after a change of coordinates, we show that each irreducible integrable $\widetilde{\mathfrak{g}}$-module with non-zero central charge is a highest weight module associated to certain linear functionals on a Heisenberg algebra $\widehat{\mathcal{H}}$ (see (3.5)). The sufficient and necessary condition for such a highest weight module to be integrable and has finite dimensional weight spaces is given in Theorem 3.5 and proved in Section 5. It turns out that the characterization of these linear functionals on $\widehat{\mathcal{H}}$ is more subtle than that given in [E2]. In particular, the notion of exp-polynomial function is needed in the toroidal EALAs case. One may see Subsection 3.2 for details. As a loop generalization of Billig’s work [B], a class of irreducible integrable highest weight $\widetilde{\mathfrak{g}}$-modules with non-zero central charge were constructed in [CLT]. This construction plays a key role in our proof of Theorem 3.5.

We now outline the structure of the paper. In Section 2, we review some basics on nullity $2$ toroidal EALAs and their integrable modules. The main result, Theorem 3.7, of our paper is stated and proved in Section 3. The proof of Theorem 3.7 is based on two classification results (Theorem 3.2 and Theorem 3.5) on certain irreducible integrable highest weight $\widetilde{\mathfrak{g}}$-modules. Section 4 and Section 5 are respectively devoted to the proofs of Theorem 3.2 and Theorem 3.5.

We denote the sets of integers, non-negative integers, complex numbers, nonzero complex numbers and rational numbers by $\mathbb{Z}$, $\mathbb{N}$, $\mathbb{C}$, $\mathbb{C}^{\times}$ and $\mathbb{Q}$, respectively.

2. Nullity $2$ toroidal extended affine Lie algebras

In this section we collect some basic results about nullity $2$ toroidal extended affine Lie algebras and their integrable representations, which will be used later on.

2.1. The toroidal EALA $\widetilde{\mathfrak{g}}$

In this subsection we review the definition of the torodial EALA $\widetilde{\mathfrak{g}}$.

Let $\dot{\mathfrak{g}}$ be a finite dimensional complex simple Lie algebra and let $\dot{\mathfrak{h}}$ be a Cartan subalgebra of it. We denote by $\ell=\dim\dot{\mathfrak{h}}$ the rank of $\dot{\mathfrak{g}}$. Let $\dot{\Delta}$ (containing [math]) the root system of $\dot{\mathfrak{g}}$ relative to the Cartan subalgebra $\dot{\mathfrak{h}}$. Let ${\langle}\cdot,\cdot{\rangle}$ be a non-degenerate invariant symmetric bilinear form on $\dot{\mathfrak{g}}$, and we will usually identify $\dot{\mathfrak{h}}^{\ast}$ with $\dot{\mathfrak{h}}$ by means of it. We further assume that the invariant form ${\langle}\cdot,\cdot{\rangle}$ is normalized so that the square length of long roots equal to $2$. For each $\alpha\in\dot{\Delta}\setminus\{0\}$, let $\alpha^{\vee}=\frac{2\alpha}{{\langle}\alpha,\alpha{\rangle}}$ indicate the corresponding coroot of $\alpha$.

Let $\mathcal{R}=\mathbb{C}[t_{0}^{\pm 1},t_{1}^{\pm 1}]$ be the Laurent polynomial ring in two variables. For $\bm{m}=(m_{0},m_{1})\in\mathbb{Z}^{2}$, we write

[TABLE]

Denote by $\Omega_{\mathcal{R}}^{1}$ the space of Kähler differentials on $\mathcal{R}$. As a left $\mathcal{R}$-module, $\Omega_{\mathcal{R}}^{1}$ has a basis $\mathrm{k}_{0},\mathrm{k}_{1}$, where $\mathrm{k}_{i}$, $i=0,1$, stands for the $1$-forms $t_{i}^{-1}\mathrm{d}t_{i}$. Form the quotient space $\mathcal{K}=\Omega^{1}_{\mathcal{R}}/\mathrm{d}(\Omega_{\mathcal{R}}^{1})$, where

[TABLE]

is the space of exact $1$-forms in $\Omega_{\mathcal{R}}^{1}$. The universal central extension of the loop algebra $\mathcal{R}\otimes\dot{\mathfrak{g}}$, called $2$-toroidal Lie algebra, can be realized as

[TABLE]

and its Lie bracket is given by (see [MRY],[EM])

[TABLE]

where $x,y\in\dot{\mathfrak{g}}$, $\bm{m}=(m_{0},m_{1}),\bm{n}=(n_{0},n_{1})\in\mathbb{Z}^{2}$, and $\mathcal{K}$ is central.

We denote by $\mathcal{D}=\textnormal{{Der}}(\mathcal{R})$ the space of derivations over $\mathcal{R}$. As a left $\mathcal{R}$-module, $\textnormal{{Der}}(\mathcal{R})$ has a basis $\mathrm{d}_{0},\mathrm{d}_{1}$, where $\mathrm{d}_{i}=t_{i}\frac{\partial}{\partial t_{i}}$, $i=0,1$. Let $\mathcal{S}$ be the subspace of $\mathcal{D}$ consisting of skew derivations (also called divergence zero derivations). So, by definition, it is the subspace of $\mathcal{D}$ spanned by the degree zero derivations $\mathrm{d}_{0},\mathrm{d}_{1}$ and the following skew derivations

[TABLE]

for $\bm{m}\in\mathbb{Z}^{2}$.

Let $\mu$ be a fixed complex number. We define an extension $\widetilde{\mathfrak{g}}(\mu)$ of $\widetilde{\mathfrak{g}}_{c}$ by the skew derivations as follows. The underlying vector space of $\widetilde{\mathfrak{g}}(\mu)$ is

[TABLE]

and the remaining bracket relations on $\widetilde{\mathfrak{g}}$ are given by

[TABLE]

for $x\in\dot{\mathfrak{g}}$, $\bm{m},\bm{n}\in\mathbb{Z}^{2}$ and $i,j=0,1$, where

[TABLE]

The resulting Lie algebra $\widetilde{\mathfrak{g}}(\mu)$ is an extended affine Lie algebra with nullity $2$, and is often called the nullity $2$ toroidal EALA. For the sake of simplicity, in what follows we will write $\widetilde{\mathfrak{g}}=\widetilde{\mathfrak{g}}(\mu)$ for convenience. Note that the Cartan subalgebra of $\widetilde{\mathfrak{g}}$ is given by

[TABLE]

and the non-degenerate invariant symmetric bilinear form ${\langle}\cdot,\cdot{\rangle}$ on $\widetilde{\mathfrak{g}}$ can be determined as follows

[TABLE]

and the others are trivial.

For each $\alpha\in\dot{\Delta}$, we extend it to a linear functional in $\widetilde{\mathfrak{h}}^{*}$ so that

[TABLE]

for $i=0,1$, and define linear functionals $\delta_{i}$, $i=0,1$ on $\widetilde{\mathfrak{h}}$ by setting

[TABLE]

for $j=0,1$. Then the set

[TABLE]

is the root system of $\widetilde{\mathfrak{g}}$ with respect to the Cartan subalgebra $\widetilde{\mathfrak{h}}$. And we have the following root space decomposition of $\mathfrak{g}$

[TABLE]

where

[TABLE]

2.2. Integrable representations of $\widetilde{\mathfrak{g}}$

In this subsection we introduce the definition of integrable modules for the extended affine Lie algebra $\widetilde{\mathfrak{g}}$.

We say that a $\widetilde{\mathfrak{g}}$-module $V$ is a weight module if it admits a weight space decomposition

[TABLE]

relative to its Cartan subalgebra $\widetilde{\mathfrak{h}}$, where

[TABLE]

Note that the restriction of ${\langle}\cdot,\cdot{\rangle}$ on $\widetilde{\mathfrak{h}}$ is non-degenerate. Thus, by duality, it induces a non-degenerate bilinear form on $\widetilde{\mathfrak{h}}^{*}$. We denote by

[TABLE]

the sets of non-isotropic and isotropic roots in $\widetilde{\Delta}$, respectively.

Definition 2.1**.**

A $\widetilde{\mathfrak{g}}$-module $V$ is said to be integrable if it is a weight module and for every $\alpha\in\widetilde{\Delta}^{\times}$, $\widetilde{\mathfrak{g}}_{\alpha}$ acts locally nilpotent on $V$.

We say that a $\widetilde{\mathfrak{g}}$-module $V$ has central charge $(c_{0},c_{1})\in\mathbb{C}^{2}$ if $\mathrm{k}_{i}.v=c_{i}v$ for $v\in V$, $i=0,1$. Furthermore, we say that $V$ has non-zero (reps. zero) central charge if $(c_{0},c_{1})\neq(0,0)$ (resp. $=(0,0)$). The following is obvious.

Lemma 2.2**.**

Let $V$ be an irreducible integrable $\widetilde{\mathfrak{g}}$-module. Then $V$ has central charge $(c_{0},c_{1})$ for some $(c_{0},c_{1})\in\mathbb{Z}^{2}$.

We recall that $\widetilde{\mathfrak{g}}_{c}$ is the core of $\widetilde{\mathfrak{g}}$, namely, the subalgebra of $\widetilde{\mathfrak{g}}$ generated by non-isotropic root vectors. As in the Introduction, let $\widetilde{\mathcal{S}}$ be the quotient algebra of $\widetilde{\mathfrak{g}}$ obtained by modulo its core $\widetilde{\mathfrak{g}}_{c}$, and let $\pi:\widetilde{\mathfrak{g}}\rightarrow\widetilde{\mathcal{S}}$ be the quotient map. Set

[TABLE]

for $i=1,2$ and $\bm{m}\in\mathbb{Z}^{2}$. Then the elements $\mathrm{d}_{i},i=0,1$ and $\mathrm{d}(\bm{m})$, $\bm{m}\in\mathbb{Z}^{2}\setminus\{\bm{0}\}$ $(\bm{0}=(0,0)$) form a basis of $\widetilde{\mathcal{S}}$, and the Lie relations are given by

[TABLE]

where $i,j=0,1$, $\bm{m},\bm{n}\in\mathbb{Z}^{2}$. The derived subalgebra of $\widetilde{\mathcal{S}}$ is often called the centerless Virasoro-like algebra (see [LT] and the references therein). An $\widetilde{\mathcal{S}}$-module is called a weight module if it admits a weight space decomposition relative to $\mathbb{C}\mathrm{d}_{0}\oplus\mathbb{C}\mathrm{d}_{1}$, and a weight $\widetilde{\mathcal{S}}$-module is called quasi-finite if its weight spaces are all finite dimensional.

We denote by $\vartheta_{\mathrm{fin}}$ the category of integrable $\widetilde{\mathfrak{g}}$-modules with finite dimensional weight spaces, and denote by $\vartheta_{\mathrm{fin}}^{\times}$ and $\vartheta_{\mathrm{fin}}^{0}$ respectively the subcategories of $\vartheta_{\mathrm{fin}}$ whose objects have non-trivial and trivial $\widetilde{\mathfrak{g}}_{c}$-action. Let $V$ be a weight $\widetilde{\mathfrak{g}}$-module such that $\widetilde{\mathfrak{g}}_{c}.V=0$. Then it is integrable and admits a natural weight $\widetilde{\mathcal{S}}$-module structure. Conversely, via the quotient map $\pi$, any weight $\widetilde{\mathcal{S}}$-module $V$ is an integrable $\widetilde{\mathfrak{g}}$-module such that $\widetilde{\mathfrak{g}}_{c}.V=0$. In particular, this gives that the category $\vartheta_{\mathrm{fin}}^{0}$ is isomorphic to the category of quasi-finite weight $\widetilde{\mathcal{S}}$-modules.

In recent years, the classification problem on irreducible quasi-finite weight $\widetilde{\mathcal{S}}$-modules has been investigated in several papers, see [BT, GL, JL, LT, LS] for details. As mentioned in the Introduction, the main goal of this paper is to classify the irreducible objects in the category $\vartheta_{\mathrm{fin}}^{\times}$. In particular, we present a classification of the irreducible objects in $\vartheta_{\mathrm{fin}}$ with non-zero central charges.

3. Classification of irreducible integrable $\widetilde{\mathfrak{g}}$-modules

In this section we state the main results of the paper.

3.1. Integrable highest weight $\widetilde{\mathfrak{g}}$-modules of type I

In this subsection we introduce a notion of highest weight module for the algebra $\widetilde{\mathfrak{g}}$, and state the sufficient and necessary condition for such an irreducible highest weight $\widetilde{\mathfrak{g}}$-module belongs to the category $\vartheta_{\mathrm{fin}}^{\times}$.

Let $\dot{\Pi}=\{\alpha_{1},\cdots,\alpha_{\ell}\}$ be a fixed simple root system of $\dot{\mathfrak{g}}$. Denote by $\dot{\Delta}_{+}$ and $\dot{\Delta}_{-}$ respectively the corresponding positive and negative root systems of $\dot{\mathfrak{g}}$. Then we have a decomposition

[TABLE]

of $\widetilde{\Delta}$, where

[TABLE]

This gives a triangular decomposition

[TABLE]

for the nullity $2$ toroidal EALA $\widetilde{\mathfrak{g}}$, where $\widetilde{\mathfrak{g}}^{\pm}=\oplus_{\alpha\in\widetilde{\Delta}^{\pm}}\widetilde{\mathfrak{g}}_{\alpha}$ and

[TABLE]

We say that a $\widetilde{\mathfrak{g}}^{0}$-module is a level [math] weight module if $\mathrm{d}_{0},\mathrm{d}_{1}$ act semi-simple and $\mathrm{k}_{0},\mathrm{k}_{1}$ act trivially. By using the decomposition (3.1), we define the following notion of highest weight module for $\widetilde{\mathfrak{g}}$.

Definition 3.1**.**

A $\widetilde{\mathfrak{g}}$-module $V$ is called a highest weight module of type I if there exists a non-zero vector $v$ in $V$ such that

1. (1)

$V$* is generated by $v$;* 2. (2)

$\widetilde{\mathfrak{g}}^{+}.v=0$; 3. (3)

$\mathcal{U}(\widetilde{\mathfrak{g}}^{0}).v$* is an irreducible level [math] weight $\widetilde{\mathfrak{g}}^{0}$-module.*

Given an irreducible level [math] weight $\widetilde{\mathfrak{g}}^{0}$-module $T$. We extend $T$ to be a $(\widetilde{\mathfrak{g}}^{+}\oplus\widetilde{\mathfrak{g}}^{0})$-module by letting $\widetilde{\mathfrak{g}}^{+}$ acts trivially on it. Form the induced $\widetilde{\mathfrak{g}}$-module

[TABLE]

We denote by $V(T)$ the unique irreducible quotient of the $\widetilde{\mathfrak{g}}$-module $M(T)$. If $V$ is an irreducible highest weight $\widetilde{\mathfrak{g}}$-module of type I, then

[TABLE]

is an irreducible level [math] weight $\widetilde{\mathfrak{g}}^{0}$-module and $V$ is isomorphic to $V(T)$.

Now we are going to construct a class of irreducible level [math] weight $\widetilde{\mathfrak{g}}^{0}$-modules. We denote by

[TABLE]

the set of dominant integral weights of $\dot{\mathfrak{g}}$. Let $U$ be an irreducible finite dimensional $\mathfrak{sl}_{2}$-module, $\lambda\in\dot{P}_{+}\setminus\{0\}$ and $\bm{\gamma}=(\gamma_{1},\gamma_{2}),\bm{\gamma}^{\prime}=(\gamma_{1}^{\prime},\gamma_{2}^{\prime})\in\mathbb{C}^{2}$. We define a $\widetilde{\mathfrak{g}}^{0}$-module $T_{U,\lambda,\bm{\gamma},\bm{\gamma}^{\prime}}$ associated to the quadruple $(U,\lambda,\bm{\gamma},\bm{\gamma}^{\prime})$ as follows. The underlying space of $T_{U,\lambda,\bm{\gamma},\bm{\gamma}^{\prime}}$ is the tensor product space $\mathcal{R}\otimes U=\mathbb{C}[t_{0}^{\pm 1},t_{1}^{\pm 1}]\otimes U$ and the actions are given by

[TABLE]

where $\bm{m},\bm{n}\in\mathbb{Z}^{2}$, $u\in U$, $h\in\dot{\mathfrak{h}}$ and $i=0,1$. It is obvious that $T_{U,\lambda,\bm{\gamma},\bm{\gamma}^{\prime}}$ is an irreducible level [math] weight $\widetilde{\mathfrak{g}}^{0}$-module. The following classification result will be proved in Section 4.

Theorem 3.2**.**

Let $U$ be an irreducible finite dimensional $\mathfrak{sl}_{2}$-module, $\dot{\lambda}\in\dot{P}_{+}\setminus\{0\}$ and $\bm{\gamma},\bm{\gamma}^{\prime}\in\mathbb{C}^{2}$. Then the irreducible highest weight $\widetilde{\mathfrak{g}}$-module $V(T_{U,\lambda,\bm{\gamma},\bm{\gamma}^{\prime}})\in\vartheta_{\mathrm{fin}}^{\times}$. Conversely, if $V\in\vartheta_{\mathrm{fin}}^{\times}$ is an irreducible highest weight $\widetilde{\mathfrak{g}}$-module of type I, then $V$ has the form $V(T_{U,\lambda,\bm{\gamma},\bm{\gamma}^{\prime}})$.

3.2. Integrable highest weight $\widetilde{\mathfrak{g}}$-modules of type II

In this subsection we introduce another kind of highest weight module for the Lie algebra $\widetilde{\mathfrak{g}}$, and also state the sufficient and necessary condition for such an irreducible highest weight $\widetilde{\mathfrak{g}}$-module belongs to the category $\vartheta_{\mathrm{fin}}^{\times}$.

We denote by

[TABLE]

which is an affine subalgebra of $\widetilde{\mathfrak{g}}$. Let $\mathfrak{h}=\dot{\mathfrak{h}}\oplus\mathbb{C}\mathrm{k}_{0}\oplus\mathbb{C}\mathrm{d}_{0}$ be the usual Cartan subalgebra of $\mathfrak{g}$. Then the set

[TABLE]

is the root system of $\mathfrak{g}$ relative to the Cartan subalgebra $\mathfrak{h}$. And the set $\Pi=\{\alpha_{0}=\delta_{0}-\theta,\,\alpha_{i}\mid i=1,\cdots,\ell\}$ is a simple root system of $\mathfrak{g}$, where $\theta$ indicates the highest root in $\dot{\Delta}$. Denote by $\Delta_{+}$ and $\Delta_{-}$ respectively the corresponding positive and negative root systems of $\mathfrak{g}$. Then we have another decomposition

[TABLE]

of $\widetilde{\Delta}$, where

[TABLE]

As in the previous subsection, this decomposition of $\widetilde{\Delta}$ induces a triangular decomposition

[TABLE]

for the nullity $2$ toroidal EALA $\widetilde{\mathfrak{g}}$, where $\widetilde{\mathfrak{g}}_{\pm}=\oplus_{\alpha\in\widetilde{\Delta}_{\pm}}\widetilde{\mathfrak{g}}_{\alpha}$, and

[TABLE]

We say that an $\widetilde{\mathcal{H}}$-module is a level [math] weight module if $\mathrm{d}_{1}$ acts semi-simple and $\mathrm{k}_{1}$ acts trivially. Similar to Definition 3.1, by using the decomposition (3.2) of $\widetilde{\mathfrak{g}}$, we have the following definition.

Definition 3.3**.**

A $\widetilde{\mathfrak{g}}$-module $V$ is called a highest weight module of type II if there exists a non-zero vector $v$ in $V$ such that

1. (1)

$V$* is generated by $v$;* 2. (2)

$\widetilde{\mathfrak{g}}_{+}.v=0$; 3. (3)

$\mathcal{U}(\widetilde{\mathcal{H}}).v$* is an irreducible level [math] weight $\widetilde{\mathcal{H}}$-module.*

Note that the algebra $\widetilde{\mathfrak{g}}$ is $\mathbb{Z}$-graded with respect to the action of $\mathrm{d}_{1}$. Namely,

[TABLE]

where $\widetilde{\mathfrak{g}}_{n}=\{x\in\widetilde{\mathfrak{g}}\mid[\mathrm{d}_{1},x]=nx\}$. For any graded subalgebra $\mathcal{L}$ of $\widetilde{\mathfrak{g}}$, the corresponding homogenous subspaces will be denoted by $\mathcal{L}_{n}$ for $n\in\mathbb{Z}$.

We denote by

[TABLE]

which is a Heisenberg subalgebra of $\widetilde{\mathfrak{g}}$ and $\widetilde{\mathcal{H}}=\widehat{\mathcal{H}}\oplus\mathbb{C}\mathrm{d}_{1}$. Let $\psi$ be a linear functional on $\widehat{\mathcal{H}}$ with $\psi(\mathrm{k}_{1})=0$. Using the functional $\psi$, we can define an $\widetilde{\mathcal{H}}$-module structure on the Laurent polynomial ring $\mathbb{C}[t_{1},t_{1}^{-1}]$ with action given by

[TABLE]

for $m,n\in\mathbb{Z}$ and $h\in\widehat{\mathcal{H}}_{n}$. The resulting $\widetilde{\mathcal{H}}$-module is a level [math] weight $\widetilde{\mathcal{H}}$-module and will be denoted by $L(\psi)$. We write $L_{\psi}=\mathcal{U}(\widetilde{\mathcal{H}}).1$ for the $\widetilde{\mathcal{H}}$-submodule of $L(\psi)$ generated by $1$. For any $b\in\mathbb{C}$, one can modify the action of $\mathrm{d}_{1}$ on $L_{\psi}$ by adding the scalar action $b\,\mathrm{Id}$. The resulting $\widetilde{\mathcal{H}}$-module will be denoted by $L_{\psi,b}$.

For any non-negative integer $r$, set

[TABLE]

The following results are well-known ([C]).

Lemma 3.4**.**

(i). Let $\psi$ be a linear functional on $\widehat{\mathcal{H}}$ with $\psi(\mathrm{k}_{1})=0$, and $b\in\mathbb{C}$. Then the $\widetilde{\mathcal{H}}$-module $L_{\psi,b}$ is irreducible if and only if as vector spaces, $L_{\psi}=L_{r}$ for some $r\in\mathbb{N}$. Furthermore, if $L_{\psi}=L_{r}$ for some $r>0$, then the $\widetilde{\mathcal{H}}$-module $L(\psi)$ is completely reducible

[TABLE]

and each irreducible component $\mathcal{U}(\widetilde{\mathcal{H}}).t_{1}^{i}=t_{1}^{i}L_{r}$ is isomorphic to $L_{\psi,i}$.

(ii). Any irreducible level [math] weight $\widetilde{\mathcal{H}}$-module has the form $L_{\psi,b}$.

We denote by $\mathcal{E}$ the set of linear functionals $\psi$ on $\widehat{\mathcal{H}}$ such that $\psi(\mathrm{k}_{1})=0$ and that the associated $\widetilde{\mathcal{H}}$-module $L_{\psi}$ is irreducible. For any $\psi\in\mathcal{E}$ and $b\in\mathbb{C}$, we extend the $\widetilde{\mathcal{H}}$-module $L_{\psi,b}$ to be a $(\widetilde{\mathfrak{g}}_{+}\oplus\widetilde{\mathcal{H}})$-module by letting $\widetilde{\mathfrak{g}}_{+}$ acts trivially on it. Then we have the induced $\widetilde{\mathfrak{g}}$-module

[TABLE]

Let $\widetilde{V}(\psi,b)$ stand for the unique irreducible quotient of the $\widetilde{\mathfrak{g}}$-module $\widetilde{M}(\psi,b)$. Suppose now that $V$ is a highest weight $\widetilde{\mathfrak{g}}$-module of type II. Then, by Lemma 3.4 (ii), $V$ is a quotient of $\widetilde{M}(\psi,b)$ for some $\psi\in\mathcal{E}$ and $b\in\mathbb{C}$. Furthermore, if $V$ is also irreducible, then $V$ must be isomorphic to $\widetilde{V}(\psi,b)$.

Denote by

[TABLE]

the set of dominant integral weights of $\mathfrak{g}$, where $\alpha_{0}^{\vee}=\mathrm{k}_{0}-\theta^{\vee}$. Recall from [BZ] that a function $f:\mathbb{Z}\rightarrow\mathbb{C}$ is called exp-polynomial if there exist finitely many $c_{1},\cdots,c_{p},b_{1},\cdots,b_{p}\in\mathbb{C}^{\times}$ and $m_{1},\cdots,m_{p}\in\mathbb{N}$ such that

[TABLE]

for $n\in\mathbb{Z}$.

Let $k$ be a fixed positive integer, and let $(\bm{\lambda},\bm{a},\varphi)$ be a triple which satisfies the following conditions

1. (c1)

$\bm{\lambda}=(\lambda_{1},\cdots,\lambda_{k})\in(P_{+})^{k}$ with $\lambda_{i}\neq 0$ for all $i$; 2. (c2)

$\bm{a}=(a_{1},\cdots,a_{k})\in(\mathbb{C}^{\times})^{k}$ with $a_{1},\cdots,a_{k}$ distinct; 3. (c3)

$\varphi:\mathbb{Z}\rightarrow\mathbb{C}$ is an exp-polynomial function such that $\varphi(0)=0$.

We now define a linear functional $\psi_{\bm{\lambda},\bm{a},\varphi}:\widehat{\mathcal{H}}\rightarrow\mathbb{C}$ associated to the triple $(\bm{\lambda},\bm{a},\varphi)$ as follows

[TABLE]

where $n\in\mathbb{Z}$, $h\in\dot{\mathfrak{h}}$ and $m\in\mathbb{Z}\setminus\{0\}$. It is easy to see that the linear functional $\psi_{\bm{\lambda},\bm{a},\varphi}\in\mathcal{E}$. The following result will be proved in Section 5.

Theorem 3.5**.**

Let $(\bm{\lambda},\bm{a},\varphi)$ be a triple which satisfies the conditions (c1)-(c3) and let $b\in\mathbb{C}$. Then the irreducible highest weight $\widetilde{\mathfrak{g}}$-module $\widetilde{V}(\psi_{\bm{\lambda},\bm{a},\varphi},b)\in\vartheta_{\mathrm{fin}}^{\times}$. Conversely, if $V\in\vartheta_{\mathrm{fin}}^{\times}$ is an irreducible highest weight $\widetilde{\mathfrak{g}}$-module of type II, then it has the form $\widetilde{V}(\psi_{\bm{\lambda},\bm{a},\varphi},b)$.

3.3. The main result

In this subsection we state and prove the main result of our paper by using Theorem 3.2 and Theorem 3.5. We begin with the following result whose proof is standard, see [E2, Proposition 2.4] or [JM, Theorem 2.1] for example.

Proposition 3.6**.**

Let $V\in\vartheta_{\mathrm{fin}}$ be irreducible with central charge $(c_{0},c_{1})$.

(i) If $c_{0}=c_{1}=0$, then there exists a non-zero weight vector $v\in V$ such that $\widetilde{\mathfrak{g}}^{+}.v=0.$

(ii). If $c_{0}>0$ and $c_{1}=0$, then there exists a non-zero weight vector $v\in V$ such that $\widetilde{\mathfrak{g}}_{+}.v=0.$

Let $A=(a_{ij})_{i,j=0,1}$ be a matrix in $\mathrm{GL}_{2}(\mathbb{Z})$. As in [E2, Section 4], there is an automorphism $\chi_{A}$ of $\widetilde{\mathfrak{g}}_{c}$ such that

[TABLE]

where $x\in\dot{\mathfrak{g}}$, $\bm{m}\in\mathbb{Z}^{2}$, $j=0,1$ and $A^{t}$ stands for the transpose of $A$. Write $B=(b_{ij})_{i,j=0,1}$ for the inverse matrix of $A$. Then the map $\chi_{A}$ can be extended to an automorphism of $\widetilde{\mathfrak{g}}$, still call $\chi_{A}$, such that

[TABLE]

where $j=0,1$ and $\bm{m}\in\mathbb{Z}^{2}$.

For any $\widetilde{\mathfrak{g}}$-module $W$, by the automorphism $\chi_{A}$ twisting, one obtains another module structure on $W$. The resulting $\widetilde{\mathfrak{g}}$-module will be denoted by $W_{A}$. We say that two $\widetilde{\mathfrak{g}}$-modules $W$ and $W^{\prime}$ are isomorphism after a change of coordinates if $W^{\prime}$ is isomorphic to $W_{A}$ for some $A\in\mathrm{GL}_{2}(\mathbb{Z})$.

Now we are in a position to prove the main result of this paper.

Theorem 3.7**.**

Let $V$ be an irreducible integrable $\widetilde{\mathfrak{g}}$-module with finite dimensional weight spaces.

(i). If $V$ has zero central charge and $\widetilde{\mathfrak{g}}_{c}$ acts non-trivially on it, then $V$ is isomorphic to the type I highest weight $\widetilde{\mathfrak{g}}$-module $V(T_{U,\lambda,\bm{\gamma},\bm{\gamma}^{\prime}})$, where $U$ is an irreducible finite dimensional $\mathfrak{sl}_{2}$-module, $\lambda\in\dot{P}_{+}\setminus\{0\}$ and $\bm{\gamma},\bm{\gamma}^{\prime}\in\mathbb{C}^{2}$.

(ii). If $V$ has non-zero central charge, then after a change of coordinates, $V$ is isomorphic to the type II highest weight $\widetilde{\mathfrak{g}}$-module $\widetilde{V}(\psi_{\bm{\lambda},\bm{a},\varphi},b)$, where $(\bm{\lambda},\bm{a},\varphi)$ is a triple which satisfies the conditions (c1)-(c3), and $b\in\mathbb{C}$.

Proof.

Assume first that $V\in\vartheta_{\mathrm{fin}}^{\times}$ is irreducible and has zero central charge. By applying Proposition 3.6 (i), we know that

[TABLE]

is a non-zero $\widetilde{\mathfrak{g}}^{0}$-submodule of $V$. The irreducibility of $V$ implies that the $\widetilde{\mathfrak{g}}^{0}$-module $T$ is also irreducible. Therefore, $V$ is isomorphic to the type I highest weight $\widetilde{\mathfrak{g}}$-module $V(T)$. Then the assertion (i) follows from Theorem 3.2.

Secondly, let $V\in\vartheta_{\mathrm{fin}}$ be irreducible and has non-zero central charge $(c_{0},c_{1})$. Note that one can choose a suitable matrix $A\in\mathrm{GL}_{2}(\mathbb{Z})$ such that the $\widetilde{\mathfrak{g}}$-module $V_{A}$ has central charge $(c_{0},c_{1})A=(r,0)$ for some $r>0$. Thus, after a change of coordinates, we may assume that $c_{0}>0$ and $c_{1}=0$. Then, by Proposition 3.6 (ii), one finds a non-zero $\widetilde{\mathcal{H}}$-submodule

[TABLE]

of $V$, which is obviously irreducible. This gives that $V$ is a type II highest weight module for $\widetilde{\mathfrak{g}}$. Therefore the assertion (ii) follows from Theorem 3.5. ∎

4. Proof of Theorem 3.2

This section is devoted to the proof of Theorem 3.2.

4.1. Integrable representations for extended loop algebras

In this subsection we collect some basic results on integrable representations of the extended loop algebra of $\mathfrak{sl}_{2}$ for later use, see [CP2] for details.

The extended loop algebra of $\mathfrak{sl}_{2}$ is the Lie algebra

[TABLE]

with commutator given by

[TABLE]

for $m,n\in\mathbb{Z}$, $x,y\in\mathfrak{sl}_{2}$. Let $L(\mathfrak{sl}_{2})=\mathbb{C}[t,t^{-1}]\otimes\mathfrak{sl}_{2}$ be the loop subalgebra of $L^{e}(\mathfrak{sl}_{2})$, and let $\{\mathrm{e},\mathrm{h},\mathrm{f}\}$ be a standard basis of $\mathfrak{sl}_{2}$ such that

[TABLE]

For $m\in\mathbb{Z}$, we define an elements $\Lambda_{m}\in\mathcal{U}(L^{e}(\mathfrak{sl}_{2}))$ as follows

[TABLE]

A module $V$ of $L^{e}(\mathfrak{sl}_{2})$ (resp. $L(\mathfrak{sl}_{2})$) is called integrable if $V$ admits a weight space decomposition relative to $\mathbb{C}\mathrm{h}\oplus\mathbb{C}\mathrm{d}$ (resp. $\mathbb{C}\mathrm{h}$), and $t^{m}\otimes\mathrm{e}$, $t^{m}\otimes\mathrm{f}$ act locally nilpotent on $V$ for $m\in\mathbb{Z}$. Let $V$ be an integrable module for $L^{e}(\mathfrak{sl}_{2})$ or $L(\mathfrak{sl}_{2})$. For $n\in\mathbb{Z}$, we set

[TABLE]

Note that $V_{n}^{+}\neq 0$ only if $n\in\mathbb{N}$.

Lemma 4.1**.**

Let $V$ be an integrable $L^{e}(\mathfrak{sl}_{2})$-module, and $0\neq v\in V_{n}^{+}$ for some $n\in\mathbb{N}$. Then

1. (i)

$\Lambda_{m}.v=0$* for $|m|>n$;* 2. (ii)

$\Lambda_{n}\Lambda_{-n}.v=v$; 3. (iii)

if $t^{m}\otimes\mathrm{h}.v=0$ for all $m>0$, then $n=0$.

Proof.

The assertion (i) follows from [CP2, Proposition 1.1 (i), (iv)], the assertion (ii) follows from [CP2, Proposition 1.1 (v)], and the assertion (iii) can be deduced from (ii). ∎

Lemma 4.2**.**

Let $V$ be an integrable module for $L(\mathfrak{sl}_{2})$ with finite dimensional weight spaces. Assume that $V$ is generated by a vector $v\in V_{n}^{+}$ for some $n\in\mathbb{N}$ and $\dim V_{n}=1$. Then there exist some non-negative integers $n_{1},\cdots,n_{k}$ and distinct non-zero complex numbers $p_{1},\cdots,p_{k}$ such that

[TABLE]

Proof.

Note that the integrability of $V$ forces that $V=\oplus_{|m|\leq n}V_{m}$. Thus, $V$ is a finite dimensional $L(\mathfrak{sl}_{2})$-module. Then the lemma is implied by [CP2, Proposition 2.1 (iii)]. ∎

4.2. Vanishing of central operators

The main goal of this subsection is to prove that the subalgebra $\mathcal{K}$ acts trivially on any irreducible integrable highest weight $\widetilde{\mathfrak{g}}$-module of type I. Throughout this subsection, let $T$ be an irreducible weight $\widetilde{\mathfrak{g}}^{0}$-module such that the associated irreducible highest weight $\widetilde{\mathfrak{g}}$-module $V(T)$ is integrable.

Lemma 4.3**.**

There exists a dominant integral weight $\lambda\in\dot{P}_{+}$ such that

[TABLE]

for $h\in\dot{\mathfrak{h}}$, $v\in T$.

Proof.

We remark that $\dot{\mathfrak{h}}$ lies in the center of $\widetilde{\mathfrak{g}}^{0}$. The irreducibility of $T$ yields a linear functional $\lambda$ on $\dot{\mathfrak{h}}$ such that $h.v=\lambda(h)v$ for $h\in\dot{\mathfrak{h}}$, $v\in T$. Furthermore, the integrability of $V$ forces that $\lambda\in\dot{P}_{+}$, as desired. ∎

For any $\bm{m}\in\mathbb{Z}^{2}$ and $k\in\mathbb{Z}$, set

[TABLE]

where

[TABLE]

We remark that the set

[TABLE]

form a basis of $\mathcal{K}$. Moreover, the following two lemmas can be easily checked.

Lemma 4.4**.**

For any $\bm{m},\bm{n}\in\mathbb{Z}^{2}$, the following holds in $V(T)$

[TABLE]

For every $\alpha\in\dot{\Delta}_{+}$, $\bm{m}\in\mathbb{Z}^{2}\setminus\{\bm{0}\}$ and $n\in\mathbb{Z}$, we set

[TABLE]

where $x_{\alpha}^{\pm}$ are root vectors in $\widetilde{\mathfrak{g}}_{\pm\alpha}$ such that $\{x_{\alpha}^{+},\alpha^{\vee},x_{\alpha}^{-}\}$ is a $\mathfrak{sl}_{2}$-triple and

[TABLE]

Lemma 4.5**.**

For every $\alpha\in\dot{\Delta}_{+}$, $\bm{m}\in\mathbb{Z}^{2}\setminus\{\bm{0}\}$ and $n\in\mathbb{Z}$, $\widetilde{\mathfrak{sl}_{2}}(\alpha,\bm{m},n)$ is a subalgebra of $\widetilde{\mathfrak{g}}$ and is isomorphic to the affine Kac-Moody Lie algebra of type $A_{1}^{(1)}$.

Let $\lambda\in\dot{P}_{+}$ be as in Lemma 4.3 and set

[TABLE]

Lemma 4.6**.**

For all $\bm{m}\in\mathbb{Z}^{2}$ and $v\in T$, one has that

[TABLE]

Proof.

Let $\alpha\in\dot{\Delta}_{+}$, $\bm{m}\in\mathbb{Z}^{2}\setminus\{\bm{0}\}$ and $n\in\mathbb{Z}$. Since $\mathrm{k}_{0}$, $\mathrm{k}_{1}$ act trivially on $V(T)$, it follows from Lemma 4.5 that there is an integrable $L^{e}(\mathfrak{sl}_{2})$-module structure on $V(T)$ by using the subalgebra $\widetilde{\mathfrak{sl}}_{2}(\alpha,\bm{m},n)$. The resulting $L^{e}(\mathfrak{sl}_{2})$-module is denoted by $V(T,\alpha,\bm{m},n)$. We remark that $T\subset V(T,\alpha,\bm{m},n)_{\lambda(\alpha^{\vee})}^{+}$. Thus, by applying Lemma 4.1 (i), one obtains

[TABLE]

for $v\in T$ and $b>\lambda(\alpha^{\vee})$, where the operators $\Lambda_{b}^{\alpha,\bm{m},n}\in\mathrm{End}(V(T))$ are defined as follows

[TABLE]

We now introduce more operators $\Lambda_{1,b}^{\alpha,\bm{m}},\Lambda_{2,b}^{\alpha,\bm{m},n},\Lambda_{2,b}^{\bm{m}}(s)$ on $V(T)$ for $b,s\in\mathbb{N}$ as follows

[TABLE]

and

[TABLE]

Then it is easy to see that

[TABLE]

for $b\in\mathbb{N}$. Using the above two equations, one gets

[TABLE]

for $m\in\mathbb{N}$, $n\in\mathbb{Z}$, where

[TABLE]

Now, for any given $\alpha\in\dot{\Delta}_{+}$, $\bm{m}\in\mathbb{Z}^{2}\setminus\{\bm{0}\}$, $b>\lambda(\alpha^{\vee})$ and $v\in T$, by applying (4.5) and (4.6), we have the following system of $b+1$ equations

[TABLE]

By solving this system of equations, we get that

[TABLE]

for $0\leq s\leq b$. This together with the fact that

[TABLE]

gives

[TABLE]

as required.

∎

Proposition 4.7**.**

One has that

[TABLE]

Proof.

Notice first that $\mathcal{K}$ commutes with $\widetilde{\mathfrak{g}}^{-}$ and $V(T)=\mathcal{U}(\widetilde{\mathfrak{g}}^{-})T$. Therefore it suffices to prove that $\mathcal{K}.T=0$. Given an $\bm{n}\in\mathbb{Z}^{2}\setminus\{\bm{0}\}$. Using (4.2), for any $\bm{m}_{1}\in\mathbb{Z}^{2}$, the following holds in $\mathrm{End}(V(T))$

[TABLE]

This together with Lemma 4.6 gives that, if $\epsilon_{1}\bm{m}_{1}\notin\mathbb{Q}\bm{n}$, $\epsilon_{1}\in\{0,1\}$, then

[TABLE]

Again by (4.2), for any $\bm{m}_{2}\in\mathbb{Z}^{2}$, one has that

[TABLE]

as operators on $V(T)$. Thus, by (4.9), if $\epsilon_{1}\bm{m}_{1}+\epsilon_{2}\bm{m}_{2}\not\in\mathbb{Q}\bm{n}$, $\epsilon_{1},\epsilon_{2}\in\{0,1\}$, then

[TABLE]

By repeating the above argument, one easily gets that

[TABLE]

for all $\bm{m}_{1},\cdots,\bm{m}_{N}\in\mathbb{Z}^{2}$ with $\sum_{i=1}^{N}\epsilon_{i}\bm{m}_{i}\notin\mathbb{Q}\bm{n}$, $\epsilon_{i}\in\{0,1\}$.

Let $\bm{n}_{1},\cdots,\bm{n}_{N}\in\mathbb{Z}^{2}\setminus\{0\}$. Note that one can choose an $\bm{n}\in\mathbb{Z}^{2}$ such that $\sum_{i=1}^{N}\epsilon_{i}\bm{n}_{i}\notin\mathbb{Q}\bm{n}$, $\epsilon_{i}\in\{0,1\}$. By taking $\bm{m}_{i}=\bm{n}_{i}-\bm{n}$