Reflection functors and symplectic reflection algebras for wreath products

Reflection Functors and Symplectic Reflection Algebras for Wreath Products

We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.

Quantum algebras and symplectic reflection algebras for wreath products

To a finite subgroup Γ of SL2(C), we associate a new family of quantum algebras which are related to symplectic reflection algebras for wreath products Sl o Γ via a functor of Schur-Weyl type. We explain that they are deformations of matrix algebras over rank-one symplectic reflection algebras for Γ and construct for them a PBW basis. When Γ is a cyclic group, we are able to give more informati...

Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflect...

Symplectic Reflection Algebras

Continuous symplectic reflection algebras

The theory of PBW properties of quadratic algebras, to which this paper aims to be a modest contribution, originates from the pioneering work of Drinfeld (see [Dr1]). In particular, as we learned after publication of [EG] (to the embarrassment of two of us!), symplectic reflection algebras and even more general reflection algebras considered in Section 3.6 below, as well as PBW theorems for the...