  In this paper, we explore natural connections among the representations of
the extended affine Lie algebra $\widehat{sl_N}(\mathbb{C}_q)$ with
$\mathbb{C}_q=\mathbb{C}_q[t_0^{\pm1},t_1^{\pm1}]$ an irrational quantum
2-torus, the simple affine vertex algebra $L_{\widehat{sl_{\infty}}}(\ell,0)$
with $\ell$ a positive integer, and Levi subgroups $G$ of
$GL_\ell(\mathbb{C})$. First, we give a canonical isomorphism between the
category of integrable restricted $\widehat{sl_N}(\mathbb{C}_q)$-modules of
level $\ell$ and that of equivariant quasi
$L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Second, we classify irreducible
$\mathbb{N}$-graded equivariant quasi
$L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Third, we establish a duality
between irreducible $\mathbb{N}$-graded equivariant quasi
$L_{\widehat{sl_{\infty}}}(\ell,0)$-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
$L_{\widehat{sl_{\infty}}}(\ell,0)$-module. Fifth, we completely determine the
following branchings: 1 The branching from
$L_{\widehat{sl_{\infty}}}(\ell,0)\otimes L_{\widehat{sl_{\infty}}}(\ell',0)$
to $L_{\widehat{sl_{\infty}}}(\ell+\ell',0)$ for quasi modules. 2 The branching
from $\widehat{sl_N}(\mathbb{C}_q)$ to its Levi subalgebras. 3 The branching
from $\widehat{sl_N}(\mathbb{C}_q)$ to its subalgebras
$\widehat{sl_N}(\mathbb{C}_q[t_0^{\pm M_0},t_1^{\pm M_1}])$.

Quantum Algebra;Representation Theory