Integrable representations for toroidal extended affine Lie algebras

TL;DR

This paper constructs and analyzes a new class of integrable, completely reducible modules for toroidal extended affine Lie algebras, expanding understanding of their representation theory with explicit module constructions.

Contribution

It introduces a novel class of modules for toroidal extended affine Lie algebras and proves their complete reducibility and integrability under certain conditions.

Findings

Modules are completely reducible.

Modules are integrable with dominant integral weights.

New irreducible integrable modules are constructed.

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 $\wt\fg(\mu)$.