Rational Group Actions on Affine Pi-Algebras

Abstract

ABSTRACT. Let R be an affine PI-algebra over an algebraically closed field and let G be an affine algebraic -group that acts rationally by algebra automorphisms on R. For R prime and G a torus, we show that R has only finitely many G-prime ideals if and only if the action of G on the center of R is multiplicity free. This extends a standard result on affine algebraic G-varieties. Under suitable hypotheses on R and G, we also prove a PI-version of a well-known result on spherical varieties and a version of Schelter's catenarity theorem for G-primes.

Tensor Triangular Geometry for Quantum Groups

“…by [Lor2,Proposition 1]. Furthermore, ρ defines an inclusion preserving bijection between the G-stable closed sets of N and the closed sets of G-Spec(C[N ]) (see [BKN,Section 2.3]).…”

“…Let R = C[N ]. Following the work in[Lor1] and[Lor2], the rational ideals of R are precisely the maximal ideals of R. Furthermore, the points in G-MaxSpec(C[N ]) are in bijective correspondence with G-orbits in MaxSpec(C[N ])) (i.e., the nilpotent orbits in N ). It follows that G-MaxSpec(C[N ]) is finite and…”