Discriminant formulas and applications

Abstract

The discriminant is a classical invariant associated to algebras which are finite over their centers. It was shown recently by several authors that if the discriminant of $A$ is “sufficiently nontrivial” then it can be used to answer some difficult questions about $A$. Two such questions are: What is the automorphism group of $A$? Is $A$ Zariski cancellative?

We use the discriminant to study these questions for a class of (generalized) quantum Weyl algebras. Along the way, we give criteria for when such an algebra is finite over its center and prove two conjectures of Ceken, Wang, Palmieri and Zhang.