Q-W-algebras, Zhelobenko operators and a proof of De Concini-Kac-Procesi conjecture

TL;DR

This paper presents a comprehensive theory of q-W-algebras, introduces Zhelobenko operators, and uses these tools to prove the De Concini-Kac-Procesi conjecture regarding the dimensions of irreducible modules over quantum groups at roots of unity.

Contribution

It provides a new description of q-W-algebras via Zhelobenko operators and proves a significant conjecture in quantum group representation theory.

Findings

Description of q-W-algebras in terms of Zhelobenko operators

Construction of algebraic group analogues of Slodowy slices

Proof of the De Concini-Kac-Procesi conjecture on module dimensions

Abstract

This monograph, along with a self-consistent presentation of the theory of q-W-algebras including the construction of algebraic group analogues of Slodowy slices, contains a description of q-W-algebras in terms of Zhelobenko type operators introduced in the book. This description is applied to prove the De Concini-Kac-Procesi conjecture on the dimensions of irreducible modules over quantum groups at roots of unity.