Integrable representations for toroidal extended affine Lie algebras

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