Conjugacy classes in Weyl groups and q-W algebras

We define noncommutative deformations of algebras of regular functions on certain transversal slices to the set of conjugacy classes in an algebraic group G which play the role of Slodowy slices in algebraic group theory. The algebras called q-W algebras are labeled by (conjugacy classes of) elements s of the Weyl group of G. The algebra is a quantization of a Poisson structure defined on the corresponding transversal slice in G with the help of Poisson reduction of a Poisson bracket associated to a Poisson–Lie group G⁎ dual to a quasitriangular Poisson–Lie group. We also define a quantum group counterpart of the category of generalized Gelfand–Graev representations and establish an equivalence between this category and the category of representations of the corresponding q-W algebra. The algebras can be regarded as quantum group counterparts of W-algebras. However, in general they are not deformations of the usual W-algebras.