Finite dimensional Hopf actions on algebraic quantizations

TL;DR

This paper extends previous results on semisimple Hopf actions, showing that such actions on various algebraic quantizations of commutative domains are essentially group actions, with applications to numerous algebraic structures.

Contribution

It generalizes earlier findings to a wide class of algebraic quantizations, demonstrating that finite dimensional Hopf actions are trivial or group actions in these contexts.

Findings

Hopf actions on universal enveloping algebras are group actions

Finite dimensional Hopf actions on quantum algebras factor through groups

Results apply to Sklyanin algebras and quantum polynomial rings

Abstract

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory, due to A. Perucca.