Algebraic group actions on noncommutative spectra

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 set Spec R of all prime ideals of R, viewed as a topological space with the Jacobson–Zariski topology, and on the subspace Rat R ⊆ Spec R consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid $$ \mathcal{C}\left( {R/P} \right) $$ is equal to the base field. Our results generalize the work of Mœglin and Rentschler and of Vonessen to arbitrary associative algebras while also simplifying some of the earlier proofs. The map P ↦ ⋂g ∈ G g.P gives a surjection from Spec R onto the set G-Spec R of all G-prime ideals of R. The fibers of this map yield the so-called G-stratification of Spec R which has played 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 of Spec R in terms of certain commutative spectra. Furthermore, we show that if a rational ideal P is locally closed in Spec R then the orbit G.P is locally closed in Rat R. This generalizes a standard result on G-varieties. Finally, we discuss the situation where G-Spec R is a finite set.