Integrable representations for toroidal extended affine Lie algebras

Abstract

Let $\fg$ be any untwisted affine Kac-Moody algebra, $\mu$ any fixed complex number, and $\wt\fg(\mu)$ the corresponding toroidal extended affine Lie algebra of nullity two. For any $k$-tuple $\bm{\lambda}=({\lambda}_1, \cdots, {\lambda}_k)$ of weights of $\fg$, and $k$-tuple $\bm{a}=(a_1,\cdots, a_k)$ of distinct non-zero complex numbers, we construct a class of modules $\wt V(\bm{\lambda},\bm{a})$ for the extended affine Lie algebra $\wt\fg(\mu)$. We prove that the $\wt\fg(\mu)$-module $\wt V(\bm{\lambda},\bm{a})$ is completely reducible. We also prove that the $\wt\fg(\mu)$-module $\wt V(\bm{\lambda},\bm{a})$ is integrable when all weights $\lambda_i$ in $\bm{\lambda}$ are dominant integral. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra $\wt\fg(\mu)$.