Integrable representations for toroidal extended affine Lie algebras

Abstract

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