Elliptic deformation of ${\mathcal{W}}_N$-algebras

We construct $q$-deformations of quantum ${\mathcal{W}}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}}_{q,p}(\widehat{gl}(N{)}_c)$). The closure of the algebras is insured by a critical surface condition relating the parameters $p,q$ and the central charge $c$. Further abelianity conditions are determined, either as $c=-N$ or as a second condition on $p,q,c$. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical $q$-deformed ${\mathcal{W}}_N$ algebras and quantum ${\mathcal{W}}_{q,p}({\mathfrak{s}\mathfrak{l}}_N)$.