Drinfeld-hecke Algebras over Cocommutative Algebras

If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra SymV with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have been deformed by Drinfel’d and more recently, Crawley-Boevey and Holland, Etingof and Ginzburg (and Gan), and others. These algebras include symplectic reflection algebras and infinitesimal Hecke algebras. We introduce a family of deformations Hβ of the smash product algebras mentioned at the beginning of the abstract, by deforming the relations V ∧ V . Thus β : V ∧ V → A ⊕ V ; we characterize the β’s for which the PBW property holds. We then analyse in detail the case where A = NCW is the nilCoxeter algebra, and β : V ∧ V → A. In the case where A is a cocommutative Hopf algebra, that β is A-compatible, is equivalent to some other conditions that β is an Amodule map, or the Yetter-Drinfeld condition. We examine what further conditions are needed on β to achieve a Hopf algebra structure on the deformed algebra (with V primitive). Finally, we provide a Hopf-theoretic analogue of symplectic reflections.