The lowest discriminant ideal of central extensions of abelian groups

Abstract

In a previous joint paper with Wu and Yakimov, we gave an explicit description of the lowest discriminant ideal in Cayley-Hamilton Hopf algebras $(H,C,\mathrm{tr})$ with basic identity fiber, i.e. all irreducible representations over the kernel of the counit of the central Hopf subalgebra $C$ are one-dimensional. We first show in this work that the zero set of lowest discriminant ideal of the group algebra of any central extension of an arbitrary finite group is the whole maximal spectrum of $C$. Then we specialize to central extensions of abelian groups and study the fiber algebras and orbits under automorphism groups in this zero set of the lowest discriminant ideal. An example is given, in which isomorphisms of fiber algebras do not always lift to automorphisms of $H$.