Extended affine Lie algebras, affine vertex algebras, and general linear groups

Abstract:

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