We define noncommutative deformations Wqsβ(G) of algebras of functions on
certain (finite coverings of) 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 Wqsβ(G) called q-W algebras are labeled
by (conjugacy classes of) elements s of the Weyl group of G. The algebra
Wqsβ(G) 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 Gβ dual to a
quasitriangular Poisson-Lie group. The algebras Wqsβ(G) can be regarded as
quantum group counterparts of W-algebras. However, in general they are not
deformations of the usual W-algebras.Comment: 48 pages; some arguments in the proof of Proposition 12.2 are
clarifie