Deformed Poisson W-algebras of Type A

TL;DR

This paper describes the structure of deformed Poisson W-algebras of type A related to conjugacy classes in SL_{l+1}(C), providing explicit descriptions of their algebraic properties and invariants.

Contribution

It introduces a new explicit description of the algebra of regular functions on slices to conjugacy classes in SL_{l+1}(C), linking to graded parabolic invariants and W-algebras.

Findings

Explicit polynomial algebra of invariants for slices in SL_{l+1}(C)

Connection between algebraic group invariants and graded W-algebras

Description of positive roots associated to conjugacy classes in the Weyl group

Abstract

For the algebraic group $SL_{l+1}(\mathbb{C})$ we describe a system of positive roots associated to conjugacy classes in its Weyl group. Using this we explicitly describe the algebra of regular functions on certain transverse slices to conjugacy classes in $SL_{l+1}(\mathbb{C})$ as a polynomial algebra of invariants. These may be viewed as an algebraic group analogue of certain graded parabolic invariants that generate the (graded) W-algebra in type A.