Rigidity of quantum algebras

TL;DR

The paper introduces and develops the notion of strong rigidity for quantum algebras, showing that a broad class of filtered quantizations are rigid or strongly rigid via reduction modulo large primes and Poisson-center analysis. It proves a suite of rigidity results, including nonexistence of injections between enveloping algebras of nonisomorphic semisimple Lie algebras of equal dimension, structural bounds on automorphism groups of W-algebras, and inverse Galois results for spherical Cherednik algebras and quantum tori. A central methodological theme is the Z-infinity (dequantization) functor, which connects good quantizations to Poisson geometry and constrains automorphisms through centers in characteristic $p$. The paper also provides complete classifications for fixed rings in two important families: spherical Cherednik algebras (and generalized Weyl algebras) and quantum tori, yielding cyclic Galois groups in many cases and identifying the Picard group with outer automorphisms for quantum tori. Collectively, these results advance understanding of when quantum algebras retain or reveal rigid structural features under symmetry and fixed-ring constructions, with implications for automorphisms, Galois-type problems, and noncommutative geometry.

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 $W_χ(\mathfrak{g}, e)$ 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 $A_q.$

Theorems & Definitions (44)

• Definition 1.1
• Theorem 1.1
• Definition 1.2
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.1
• Lemma 2.1
• Lemma 2.2
• Proposition 2.1