Simple ambiskew polynomial rings

TL;DR

This paper establishes simplicity criteria for a class of iterated skew polynomial rings, considering the existence of a canonical normal element and analyzing cases with inverses or quotients, with applications to quantum algebras.

Contribution

It provides new simplicity criteria for ambiskew polynomial rings in various characteristics, including conditions involving a canonical normal element.

Findings

Criteria for simplicity in characteristic 0 and p

Identification of obstructions due to normal elements

Applications to quantum and symplectic reflection algebras

Abstract

We determine simplicity criteria in characteristics 0 and $p$ for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal element $z$. In the case where this element exists we give simplicity criteria for the rings obtained by inverting $z$ and the rings obtained by factoring out the ideal generated by $z$. The results are illustrated by numerous examples including higher quantized Weyl algebras and generalizations of some low-dimensional symplectic reflection algebras.