The Poisson geometry of the conjugation quotient map for simple algebraic groups and deformed Poisson W-algebras

TL;DR

This paper introduces new Poisson structures on slices related to conjugacy classes in complex simple algebraic groups, connecting them to W-algebras and revealing novel structures on Kleinian singularities.

Contribution

It defines Poisson structures on transversal slices associated with Weyl group elements, linking algebraic group theory, Poisson geometry, and W-algebras in a novel way.

Findings

New Poisson structures on slices related to conjugacy classes.

Connections established between Poisson structures and W-algebras.

Identification of novel Poisson structures on Kleinian singularities.

Abstract

We define Poisson structures on certain transversal slices to conjugacy classes in complex simple algebraic groups introduced in arXiv:0809.0205. These slices are associated to the elements of the Weyl group, and the Poisson structures on them are analogous to the Poisson structures introduced by J. de Boer, T. Tjin and A. Premet in papers arXiv:hep-th/9211109 and http://www.maths.man.ac.uk/DeptWeb/Homepages/aap/Reprints/Transverse.ps on the Slodowy slices in complex simple Lie algebras. The quantum deformations of these Poisson structures are known as W-algebras of finite type. As an application of our definition we obtain some new Poisson structures on the coordinate rings of simple Kleinian singularities.