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

2023/04/04 by Launois, Stéphane, Lopes, Samuel A., Rogers, Alexandra · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2304.02151

Abstract

This paper extends an algorithm and canonical embedding by Cauchon to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon's 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\lhd A to a completely prime ideal Q\lhd A' such that when A is a PI algebra, \rm PI-\rm deg(A/P) = \rm PI-\rm deg(A'/Q). When the quantum parameter is a root of unity we can 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.

Cited by

• Simple Modules and PI Structure of the Two-Parameter Quantized Algebra U⁺_r,s(B₂)

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