Algebraic Group Actions on Noncommutative Spectra

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 C(R/P ) is equal to the base field. Our results generalize work of Mœglin & Rentschler and Vonessen to arbitrary associative algebras while also simplifying some of the earlier proofs. The map P 7→ T g∈G g.P gives a surjection from Spec R onto the set G-Spec R of all G-prime ideals of R. The fibres 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.