TY - JOUR

T1 - Finite dimensional Hopf actions on algebraic quantizations

AU - Etingof, Pavel

AU - Walton, Chelsea

N1 - Funding Information:
We thank Bjorn Poonen for many useful discussions and for the number-theoretic reference [Perucca 2009], which is crucial for our arguments. We are also grateful to R. Bezrukavnikov, I. Losev, and H. Nakajima for useful discussions and explanations. We thank the referee for many useful comments that improved greatly the quality of this manuscript. The authors were supported by the National Science Foundation: NSF-grants DMS-1502244 and DMS-1550306.
Publisher Copyright:
© 2016 Mathematical Sciences Publishers.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Algebraic quantization

KW - Filtered deformation

KW - Hopf algebra action

KW - Quantum polynomial algebra

KW - Sklyanin algebra

KW - Twisted coordinate ring

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U2 - 10.2140/ant.2016.10.2287

DO - 10.2140/ant.2016.10.2287

M3 - Article

AN - SCOPUS:85006410588

VL - 10

SP - 2287

EP - 2310

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 10

ER -