A proof of the Brown--Goodearl Conjecture for weak Hopf algebras

TL;DR

This paper proves the Brown--Goodearl Conjecture for a class of weak Hopf algebras that are finitely generated over their affine center, establishing finite self-injective dimension in this context.

Contribution

It demonstrates that weak Hopf algebras finitely generated over their affine center satisfy the Brown--Goodearl Conjecture, a significant step in understanding their homological properties.

Findings

Weak Hopf algebras have finite self-injective dimension under given conditions.

The Brown--Goodearl Conjecture holds for this class of weak Hopf algebras.

Abstract

Let $H$ be a weak Hopf algebra that is a finitely generated module over its affine center. We show that $H$ has finite self-injective dimension and so the Brown--Goodearl Conjecture holds in this special weak Hopf setting.