## Algebra & Number Theory

### Finite dimensional Hopf actions on algebraic quantizations

#### 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 $k[z1,…,zs]$ 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.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 10 (2016), 2287-2310.

Dates
Revised: 1 August 2016
Accepted: 22 October 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842638

Digital Object Identifier
doi:10.2140/ant.2016.10.2287

Mathematical Reviews number (MathSciNet)
MR3582020

Zentralblatt MATH identifier
1355.16030

#### Citation

Etingof, Pavel; Walton, Chelsea. Finite dimensional Hopf actions on algebraic quantizations. Algebra Number Theory 10 (2016), no. 10, 2287--2310. doi:10.2140/ant.2016.10.2287. https://projecteuclid.org/euclid.ant/1510842638

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