Noncommutative discriminants via Poisson primes

TL;DR

This paper introduces a novel method for computing discriminants of noncommutative algebras by linking them to Poisson geometry, enabling explicit calculations for quantum algebras at roots of unity.

Contribution

It establishes a general approach connecting noncommutative discriminants with Poisson primes, applicable to quantum algebras and their specializations.

Findings

Discriminants expressed as products of Poisson primes

Method applied to quantum matrices at roots of unity

Extended to quantum Schubert cell algebras

Abstract

We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It is illustrated with the specializations of the algebras of quantum matrices at roots of unity and more generally all quantum Schubert cell algebras.