Algebra & Number Theory

Finite dimensional Hopf actions on algebraic quantizations

Pavel Etingof and Chelsea Walton

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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[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

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

Received: 19 May 2016
Revised: 1 August 2016
Accepted: 22 October 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16T05: Hopf algebras and their applications [See also 16S40, 57T05]
Secondary: 16S80: Deformations of rings [See also 13D10, 14D15] 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22] 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22]

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


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.

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