0 Introduction

We start with the following important formula from partition theory:

$ \prod\limits_{n=1}^\infty\dfrac{1}{(1-q^n)}=1+\sum\limits_{n=1}^\infty P(n)q^n, $

where $P(n)$ is the number of partitions of $n$. For convenience, we write $P(0)=1$ and $P(n) = 0$ for negative integers $n$.

The three-dimensional simple Lie algebra ${\frak{sl}}_2$ over ${\bf C}$ is a vector space spanned by elements $e_\alpha, e_{-\alpha}, h$ with brackets

$ [e_\alpha, e_{-\alpha}]=h, \quad [h, e_{\pm\alpha}]=\pm2e_{\pm\alpha}. $

It has an invariant non-degenerate bilinear form determined by

$ (e_\alpha, e_{-\alpha}) = 1, \quad (h, h) = 2. $

Then, the affine algebra $\widehat{{\frak{sl}}_2}$ can be realized as

$ \widehat{{\frak{sl}}_2}={{\frak{sl}}_2}\otimes{\bf C}[t, t^{-1}]\oplus {\bf C} c\oplus{\bf C} d, $

where $c$ is the center, $d$ is the degree derivative associated with $t$, and

$ {}[x\otimes t^m, y\otimes t^n]=[x, y]\otimes t^{m+n} + m\delta_{m+n, 0}(x, y)c. $

Regarding ${\frak{sl}}_2$ as the subalgebra of $\widehat{{\frak{sl}}_2}$ under embedding $x\mapsto x\otimes1$, we define a four-dimensional Lie subalgebra $L ={\frak{sl}}_2\oplus {\bf C} d$ of $\widehat{{\frak{sl}}_2}$. It is well-known that $V$ is an irreducible $L$-module if and only if it is an irreducible ${{\frak{sl}}_2}$-module. Our main result can be stated as the following theorem.

Theorem 1    Let $V$ be the irreducible highest weight $\widehat{{\frak{sl}}_2}$-module with highest weight $\Lambda_0$. Then as an $L$-module, $V$ has a decomposition

$ V=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty V_{n, k}^{\oplus A_{n, k}}, $

where $V_{n, k}$ is ($2n+1$)-dimensional and irreducible as ${\frak{sl}}_2$-module, and $d|_{V_{n, k}}=(-k-n^2){\rm id}_{V_{n, k}}$, $A_{n, k}=P(k)-P(k-2n-1)$ is the multiplicity of $V_{n, k}$.

1 Preliminaries

In this section, we first recall the construction of a vertex operator representation for $\widehat{{\frak{sl}}_2}$ (see [1] for more details, [2] and [3] can also be referenced). Then, two useful lemmas are introduced.

1.1 Vertex representation of $L(\Lambda_0)$

Define the Lie algebra

$ \hat{H}=\bigoplus\limits_{n\in{\bf Z}}{\bf C}\alpha(n)\oplus{\bf C} c, $

with bracket

$ [\hat{H}, c]=0, $

$ [\alpha(m), \alpha(n)]=2m\delta_{m+n, 0}c. $

Let $S(\widehat{H}^-)$ be the polynomial algebra generated by

$ \widehat{H}^-=\bigoplus\limits_{n=1}^\infty{\bf C}\alpha(-n). $

Let ${\bf C}[Q]$ be the group algebra of $Q$, which is spanned by $e^\gamma$ ($\gamma\in Q={\bf Z} \alpha$). Also, let $V(Q)=S(\widehat{H}^-)\otimes{\bf C}[Q]$.

For any $v\otimes e^\gamma=\alpha(-n_1)\alpha(-n_2)\cdots\alpha(-n_k)\otimes e^\gamma$, the action of $\widehat{H}$ and $d$ on $V(Q)$ is defined by

$ \begin{align*} \alpha(-n)\cdot(v\otimes e^\gamma)=&\alpha(-n)\alpha(-n_1)\alpha(-n_2)\cdots\alpha(-n_k)\otimes e^\gamma, \\ \alpha(n)\cdot(1\otimes e^\gamma)=&0, \\ \alpha(n)\cdot(v\otimes e^\gamma)=&\alpha(-n_1)\alpha(n)\alpha(-n_2)\cdots\alpha(-n_k)\otimes e^\gamma\\ &+2n\alpha(-n_2)\cdots\alpha(-n_k)\otimes e^\gamma, \\ \alpha(0)\cdot(v\otimes e^\gamma)=& (\alpha, \gamma)v\otimes e^\gamma, \\ c=&{\rm id}_{V(Q)}, \\ d\cdot(v\otimes e^\gamma)=&-\Big(n_1+n_2+\cdots+n_k+\dfrac{ (\gamma, \gamma)}{2}\Big)v\otimes e^\gamma, \end{align*} $

where $n, n_1, n_2, \cdots, n_k$ are positive integers.

Also define operators

$ \begin{align*} E^\pm(\gamma, z)&=\exp\Big(\mp\sum\limits_{n=1}^\infty\dfrac{\gamma(\pm n)} {n}z^{\mp n}\Big), \\ z^\beta\cdot(v\otimes e^\gamma)&=z^{(\beta, \gamma)}v\otimes e^\gamma, \\ E(\gamma, z)&=E^-(\gamma, z)E^+(\gamma, z), \\ X(\pm\alpha, z)&=E(\pm\alpha, z)z^{\pm\alpha+1}e^{\pm\alpha}, \end{align*} $

where the expansions

$ E(\gamma, z)=\sum\limits_{n\in{\bf Z}}E_n(\gamma)z^{-n}, $

$ X(\pm\alpha, z)=\sum\limits_{n\in{\bf Z}}X_n(\pm\alpha)z^{\mp n} $

define operators $E_n(\gamma), X_n(\pm\alpha)$ on $V(Q)$.

Theorem 2^[1]    The Lie algebra generated by operators $X_n(\pm\alpha), \alpha(n), {\rm id}_{V(Q)}, d$ on $V(Q)$ is isomorphic to $\widehat{{\frak{sl}}_2}$ via homomorphism $\pi: \widehat{{\frak{sl}}_2}\rightarrow {\rm End}(V(Q))$,

$ \begin{align*} \pi(e_{\pm\alpha}\otimes t^n)&=X_n(\pm\alpha), \\ \pi(h\otimes t^n)&=\alpha(n), \\ \pi(c)&={\rm id}_{V(Q)}, \\ \pi(d)&=d. \end{align*} $

In particular, $V(Q)$ is isomorphic to $L(\Lambda_0)$ as $\widehat{{\frak{sl}}_2}$-modules.

1.2 Two useful lemmas

Lemma 1    The following identity holds:

$ \sum\limits_{n, k=0}^\infty(2n+1)(P(k)-P(k-2n-1))q^{n^2+k}= \dfrac{\sum_{n\in{\bf Z}}q^{n^2}}{\prod_{n=1}^\infty(1-q^n)}. $

Proof    By direct computation,

$ \begin{align*} &\sum\limits_{n\geqslant 0, k\geqslant 0}(2n+1)(P(k)-P(k-2n-1))q^{n^2+k} \\ =&\sum\limits_{n\in{\bf Z}, k\geqslant0}\sum\limits_{i\geqslant|n|, j\geqslant0\atop i^2+j=n^2+k}(P(j)-P(j-2i-1))q^{n^2+k} \\ =&\sum\limits_{n\in{\bf Z}, k\geqslant0}P(k)q^{n^2+k}\\ =&\dfrac{\sum_{n\in{\bf Z}}q^{n^2}}{\prod_{n=1}^\infty(1-q^n)}. \end{align*} $

Lemma 2    Regard $V(Q)$ as a ${\frak{sl}}_2$-module, then $v\otimes e^{n\alpha}$ is a highest weight vector if and only if $n$ is nonnegative and $E_{2n+1}(\alpha)(v\otimes1)=0$.

Proof    It is well known that $V(Q)$ is integrable as a $\widehat{{\frak{sl}}_2}$-module. Then as a ${\frak{sl}}_2$-module, $V(Q)$ is a direct sum of the finite dimensional irreducible submodules (highest weight modules). If $v\otimes e^{n\alpha}$ is the highest weight vector, then $n\geqslant0$. Further,

$ X_0(\alpha)(v\otimes e^{n\alpha})=E_{2n+1}(\alpha)(v\otimes e^{n\alpha})= 0 $

implies $E_{2n+1}(\alpha)(v\otimes 1)=0$.

2 Poof for Theorem 1

Suppose that $S(\widehat{H}^-) =\sum_{k=0}^\infty S_k$, where $S_k$ is the homogeneous space, i.e., $d|_{S_k}=-k{\rm id}$. For any $v\in S_k$, define ${\rm deg}(v\otimes e^{n\alpha}) =-k-n^2$. Then, $V(Q)$ has decomposition

$ V(Q)=\sum\limits_{j=0}^\infty V_j(Q), $

where $V_j(Q)={\rm span}\{x=v\otimes e^{n\alpha}\mid{\rm deg}(x) =-j\}$. Moreover, both $X_0(\alpha)$ and $X_0(-\alpha)$ preserve $V_j(Q)$ for any $j$.

Now, we are ready to prove Theorem 1.

Proof    Define $q$-series as

$ {\rm ch}_q(V(Q))=\sum\limits_{j=0}^\infty\dim V_j(Q)q^j, $

then by definition of $V(Q)$, we have

$ {\rm ch}_q(V(Q))=\sum\limits_{n\in{\bf Z}, k\geqslant0}P(k)q^{n^2+k}. $

We also define the linear map $f_{n, k}: S_k\otimes 1\rightarrow S_{k-2n-1}\otimes 1$ by

$ v\otimes 1\mapsto E_{2n+1}(\alpha)(v\otimes 1), $

then

$ \dim{\rm Ker}(f_{n, k})\geqslant P(k)-P(k-2n-1). $

By Lemma 2, for any non-negative integer $n$, $\bigoplus_{k\geqslant0}{\rm Ker}(f_{n, k})\otimes e^{n\alpha}$ is the highest weight vector space as ${\frak{sl}}_2$-modules such that any irreducible part is isomorphic to $L(n\alpha)$, which is $(2n+1)$-dimensional. Hence,

$ {\rm ch}_q(V(Q))=\sum\limits_{n\geqslant0, k\geqslant0}(2n+1)\dim{\rm Ker}(f_{n, k}). $

Explicitly, $\dim{\rm Ker}(f_{n, k})=A_{n, k}$ is the multiple of the irreducible submodules isomorphic to $V_{n, k}$.

Accordingly, by Lemma 1 we have

$ \sum\limits_{n\geqslant0, k\geqslant0}(2n+1)(P(k)-P(k-2n-1))q^{n^2+k}= \sum\limits_{n\in{\bf Z}, k\geqslant0}P(k)q^{n^2+k}, $

then $\dim{\rm Ker}(f_{n, k})=P(k)-P(k-2n-1)$ for all $n, k$.

This marks completion of the proof.