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

TL;DR

This paper constructs elliptic-deformed quantum $ ext{W}_N$ algebras with explicit generators and explores their algebraic closure, abelianity conditions, and connections to classical and quantum deformations.

Contribution

It introduces a new elliptic deformation of quantum $ ext{W}_N$ algebras with explicit generator construction and analyzes their algebraic properties and relations to existing structures.

Findings

Constructed elliptic $q$-deformed $ ext{W}_N$ algebras with explicit generators.

Identified conditions for algebra closure and abelianity.

Established connections with classical and quantum $ ext{W}_N$ algebras.

Abstract

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 $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$.