Algebraic group actions on noncommutative spectra

TL;DR

This paper investigates how algebraic group actions influence the structure of prime spectra in associative algebras, focusing on orbit properties and stratification, with applications to quantum groups and generalizations beyond previous work.

Contribution

It extends the theory of algebraic group actions on spectra to arbitrary associative algebras and simplifies existing proofs, enhancing understanding of G-orbits and stratification.

Findings

Characterization of G-strata in terms of commutative spectra

Analysis of local closedness of G-orbits in Spec R

Generalization of previous results to broader algebra classes

Abstract

Let G be an affine algebraic group and let R be an associative algebra with a rational action of G by algebra automorphisms. We study the induced G-action on the spectrum Spec R of all prime ideals of R, viewed as a topological space with the Jacobson-Zariski topology, and on the subspace consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid of R/P is equal to the base field. The main themes of the article are local closedness of G-orbits in Spec R and the so-called G-stratification of Spec R. This stratification plays a central role in the recent investigation of algebraic quantum groups, in particular in the work of Goodearl and Letzter. We describe the G-strata in terms of certain commutative spectra. Our principal results are based on prior work of Moeglin & Rentschler and Vonessen. We generalize the theory arbitrary associative algebras while also simplifying some of the earlier proofs.