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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}]). [ABSTRACT FROM AUTHOR]

We show that one-parameter deformation A q , t of the skein algebra S k q (Σ 2) of a genus two surface suggested in Arthamonov and Shakirov (Sel Math New Ser 25(2):17, 2019) is flat. We solve the word problem in the algebra and describe monomial basis. In addition, we calculate the classical limit A q = 1 , t of the algebra and prove that it is a one-parameter flat Poisson deformation of the coordinate ring A q = t = 1 of an S L (2 , C) -character variety of a genus two surface. As a byproduct, we obtain a remarkably simple presentation in terms of generators and relations for the coordinate ring A q = t = 1 of a genus two character variety. [ABSTRACT FROM AUTHOR]