Rigidity of quantum algebras

Abstract

Given an associative $C$-algebra $A$, we call $A$ strongly rigid if for any pair of finite subgroups of its automorphism groups $G,H$, such that $A^G\cong A^H$, then $G$ and $H$ must be isomorphic. In this paper, we show that a large class of filtered quantizations are strongly rigid. We also solve the inverse Galois problem for a wide class of rational Cherednik algebras that includes all (simple) classical generalized Weyl algebras, and also for quantum tori. Finally, we show that the Picard group of an $n$-dimensional quantum torus is isomorphic to the group of its outer automorphisms.