Rings of skew polynomials and Gel'fand-Kirillov conjecture for quantum groups

TL;DR

This paper explores the action of quantum groups on skew polynomial rings, proposes a q-deformation of the Gel'fand-Kirillov conjecture, and investigates automorphisms and modules related to quantum groups, providing partial proofs and explicit calculations.

Contribution

It introduces a new framework for quantum group actions on skew polynomial rings and proposes a q-deformation of the Gel'fand-Kirillov conjecture with partial proof.

Findings

Partial proof of the q-deformation of the Gel'fand-Kirillov conjecture

Construction of automorphisms from complex powers of quantum group generators

Explicit calculation of singular vectors in Verma modules

Abstract

We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation'' of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutaive rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over $U_{q}(\gtsl_{n+1})$. We finally give a definition of a $q-$connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twisted by a $q-$connection.