Conjugacy classes in Weyl groups and q-W algebras

TL;DR

This paper introduces q-W algebras, noncommutative deformations of function algebras on slices related to conjugacy classes in algebraic groups, linking Poisson geometry and quantum groups.

Contribution

It defines and constructs q-W algebras as quantizations of Poisson structures on slices, extending the concept of W-algebras to a quantum group setting.

Findings

q-W algebras are labeled by Weyl group conjugacy classes

They serve as quantum counterparts to classical W-algebras

These algebras are not generally deformations of traditional W-algebras

Abstract

We define noncommutative deformations $W_q^s(G)$ of algebras of functions on certain (finite coverings of) transversal slices to the set of conjugacy classes in an algebraic group $G$ which play the role of Slodowy slices in algebraic group theory. The algebras $W_q^s(G)$ called q-W algebras are labeled by (conjugacy classes of) elements $s$ of the Weyl group of $G$. The algebra $W_q^s(G)$ is a quantization of a Poisson structure defined on the corresponding transversal slice in $G$ with the help of Poisson reduction of a Poisson bracket associated to a Poisson-Lie group $G^*$ dual to a quasitriangular Poisson-Lie group. The algebras $W_q^s(G)$ can be regarded as quantum group counterparts of W-algebras. However, in general they are not deformations of the usual W-algebras.