Rigidity of quantum algebras

Abstract

Given an associative $\mathbb{C}$-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 prove several other rigidity type results for various quantum algebras. For example, we show that given two non-isomorphic complex semi-simple Lie algebras $\mathfrak{g}_1, \mathfrak{g}_2$ of equal dimension, there are no injective $\mathbb{C}$-algebra homomorphisms between their enveloping algebras. We also show that any finite subgroup of automorphisms of a central reduction of a finite $W$-algebra $W_{\chi}(\mathfrak{g}, e)$ must be isomorphic to a subgroup of $Aut(\mathfrak{g}(e)).$ We 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 $A_q$ (with $q$ not a root of unity) is isomorphic to the group of outer automorphisms of $A_q.$