Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms

TL;DR

This paper investigates reversible skew Laurent polynomial rings, focusing on automorphisms that invert the variable, and explores their invariants as deformations of Poisson automorphisms, with applications to quantum algebra structures.

Contribution

It introduces invariants for reversing automorphisms in skew Laurent polynomial rings and applies these to specific algebraic structures related to Poisson deformations.

Findings

Identified invariants for reversing automorphisms in specific skew Laurent rings

Connected reversing automorphisms to deformations of Poisson automorphisms

Determined rings of invariants for key algebraic examples

Abstract

A skew Laurent polynomial ring R[x^{\pm 1};\alpha] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of two simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus. These two skew Laurent polynomial rings are deformations of Poisson algebras and we interpret their reversing automorphisms and their invariants as deformations of Poisson automorphisms and their invariants.