On De Concini-Kac forms of quantum groups

TL;DR

This paper introduces a modified even part algebra of De Concini-Kac quantum groups at roots of unity, providing new insights into their centers, Azumaya loci, and connections to reflection equation algebras.

Contribution

It proposes the even part algebra as a uniform and manageable form of De Concini-Kac quantum groups, extending classical results and establishing new structural properties.

Findings

Complete description of the center in terms of Frobenius and Harish-Chandra centers

Identification of the locally finite part with the reflection equation algebra

Analysis of the Azumaya locus over the center

Abstract

Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, hardest to study. In this paper, we propose a suitable modification to the De Concini-Kac form, namely the even part algebra, which has some appealing features. Notably, it behaves uniformly with respect to the order of the roots of unity and admits an adjoint action of the Lusztig form. We revisit several results due to De Concini-Kac-Procesi and Tanisaki for the even part algebra. Namely, we give conceptual definitions of the Frobenius and Harish-Chandra centers and describe the entire center in terms of these two subalgebras getting a complete quantum analog of the Veldkamp theorem on the center of the universal enveloping algebras in positive characteristic. We investigate the Azumaya locus of the even part algebra over its center. We also show that the locally finite part of the even part algebra under the adjoint action of the Lusztig form is isomorphic to the reflection equation algebra, which is the quantized coordinate algebra with the product twisted by $R$-matrix. Some results on Lusztig forms at roots of unity are revisited and proved in greater generality including Kempf vanishing theorem and good filtrations on the quantized coordinate algebra.