## Finite dimensional Hopf actions on algebraic quantizations

##### Author(s)

Etingof, Pavel I; Walton, Chelsea
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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 ,… 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.

##### Date issued

2016-12##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Algebra & Number Theory

##### Publisher

Mathematical Sciences Publishers

##### Citation

Etingof, Pavel and Chelsea Walton. “Finite Dimensional Hopf Actions on Algebraic Quantizations.” Algebra & Number Theory 10, 10 (December 2016): 2287–2310 © 2016 Mathematical Sciences Publishers

Version: Author's final manuscript

##### ISSN

1944-7833

1937-0652