Extended affine Lie algebras, vertex algebras, and reductive groups

Abstract

In this paper, we explore natural connections among the representations of the extended affine Lie algebra $\widehat{s{l}_N}({\mathbb{C}}_q)$ with ${\mathbb{C}}_q={\mathbb{C}}_q[t_0^{\pm 1},t_1^{\pm 1}]$ an irrational quantum 2-torus, the simple affine vertex algebra $L_{\widehat{s{l}_{\infty }}}(\ell ,0)$ with $\ell$ a positive integer, and Levi subgroups $G$ of $G{L}_{\ell }(\mathbb{C})$. First, we give a canonical isomorphism between the category of integrable restricted $\widehat{s{l}_N}({\mathbb{C}}_q)$-modules of level $\ell$ and that of equivariant quasi $L_{\widehat{s{l}_{\infty }}}(\ell ,0)$-modules. Second, we classify irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{s{l}_{\infty }}}(\ell ,0)$-modules. Third, we establish a duality between irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{s{l}_{\infty }}}(\ell ,0)$-modules and irreducible regular $G$-modules on certain fermionic Fock spaces. Fourth, we obtain an explicit realization of every irreducible $\mathbb{N}$-graded equivariant quasi $L_{\widehat{s{l}_{\infty }}}(\ell ,0)$-module. Fifth, we completely determine the following branchings: 1 The branching from $L_{\widehat{s{l}_{\infty }}}(\ell ,0)\otimes L_{\widehat{s{l}_{\infty }}}({\ell }^{\prime },0)$ to $L_{\widehat{s{l}_{\infty }}}(\ell +{\ell }^{\prime },0)$ for quasi modules. 2 The branching from $\widehat{s{l}_N}({\mathbb{C}}_q)$ to its Levi subalgebras. 3 The branching from $\widehat{s{l}_N}({\mathbb{C}}_q)$ to its subalgebras $\widehat{s{l}_N}({\mathbb{C}}_q[t_0^{\pm M_0},t_1^{\pm M_1}])$.

54 Pages

Cited by in corpus (1)

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