Homological properties of quantized coordinate rings of semisimple groups

TL;DR

This paper establishes important homological properties of quantized coordinate rings of semisimple groups, showing they are Auslander-regular, Cohen-Macaulay, and catenary, thus advancing understanding of their algebraic structure.

Contribution

It proves that the generic quantized coordinate ring of any connected semisimple Lie group has key homological properties, answering longstanding questions in the field.

Findings

Quantized coordinate rings are Auslander-regular.

They are Cohen-Macaulay.

They are catenary.

Abstract

We prove that the generic quantized coordinate ring $\mathcal{O}_q(G)$ is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group $G$. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a new set of positive cases for a question of Brown and the first author.