Finite dimensional Hopf actions on algebraic quantizations

Abstract

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], 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.Comment: v3: Removed package bbold as it is not compatible with the arxiv's compiler. Still 28 pages; to appear in Algebra and Number Theor

Actions of quantum linear spaces on quantum algebras

“…By [12,Theorem 5.2], under certain conditions on p, the action of a finite-dimensional Hopf algebra on k p [u 1 , . .…”

Partial (co)actions of Taft and Nichols Hopf algebras on their base fields

“…However, calculating and classifying such actions is a difficult task. There exist several works that calculate (global) actions of certain Hopf algebras on different types of algebras, such as fields, domains, Weyl algebras, path algebras of quivers, among others [8,9,11,12,13,14,15].…”

A class of semisimple Hopf algebras acting on quantum polynomial algebras

“…In order to do that, we will construct a class of semisimple Hopf algebras H 2n 2 , which are not group algebras, and show that there exist inner faithful actions of those algebras on the quantum polynomial algebras, in particular on the quantum plane. In a recent paper, [9], P. Etingof and C. Walton say that there is no finite quantum symmetry when the action of any finite-dimensional Hopf algebra factors through a group action. In this way, we give examples of algebras where there is quantum symmetry.…”