The structure of q-W algebras

TL;DR

This paper provides two explicit descriptions of Poisson q-W algebras related to algebraic group slices in complex semisimple Lie algebras, introducing projection operators and new formulas for coordinates on Bruhat cells.

Contribution

It introduces novel explicit descriptions of Poisson q-W algebras using projection operators and derives new formulas for coordinates on Bruhat cells.

Findings

Two explicit descriptions of Poisson q-W algebras.

Introduction of projection operators analogous to Zhelobenko and extremal projections.

New formulas for natural coordinates on Bruhat cells.

Abstract

We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. To obtain the first description we introduce certain projection operators which are analogous to the quasi-classical versions of the so-called Zhelobenko and extremal projection operators. As a byproduct we obtain some new formulas for natural coordinates on Bruhat cells in algebraic groups.