Finite dimensional Hopf actions on algebraic quantizations

Etingof, Pavel; Walton, Chelsea

doi:10.2140/ant.2016.10.2287

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Finite dimensional Hopf actions on algebraic quantizations

Pavel Etingof and Chelsea Walton

Vol. 10 (2016), No. 10, 2287–2310

DOI: 10.2140/ant.2016.10.2287

Abstract

Let $k$ be an algebraically closed field of characteristic zero. In joint work with J. Cuadra, we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra $k [z_{1}, \dots, z_{s}]$ 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.

Keywords

algebraic quantization, filtered deformation, Hopf algebra action, quantum polynomial algebra, Sklyanin algebra, twisted coordinate ring

Mathematical Subject Classification 2010

Primary: 16T05

Secondary: 16S80, 13A35, 16S38

Milestones

Received: 19 May 2016

Revised: 1 August 2016

Accepted: 22 October 2016

Published: 9 December 2016

Authors

Pavel Etingof
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307 United States

Chelsea Walton
Department of Mathematics Temple University 1805 N. Broad Street Philadelphia, PA 19122-6094 United States

Vol. 10, No. 10, 2016