A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

Abstract

This paper extends an algorithm and canonical embedding in [6] to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchons original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space A from the original quantum algebra A via a series of change of variables within the division ring of fractions Frac(A). The canonical embedding takes a completely prime ideal P A to a completely prime ideal Q A such that when A is a PI algebra, PI-deg(A/P) = PI-deg(A/Q). When the quantum parameter is a root of unity, combining our construction with results from [2] allows us to state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of A/P given a suitable irreducible representation of A/Q when A is PI. (c) The authors, 2024.