Conjugacy classes in Weyl groups and q-W algebras

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We define noncommutative deformations W_q^s(G) 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 W_q^s(G) called q-W algebras are labeled by (conjugacy classes) of elements s of the Weyl group of G. The algebra W_q^s(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 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 W_q^s(G) can be regarded as quantum group counterparts of W-algebras. However, in general they are not deformations of the usual W-algebras.
Original languageEnglish
Pages (from-to)1315-1376
Number of pages61
JournalAdvances in Mathematics
Issue number3
Early online date24 Jun 2011
Publication statusPublished - 20 Oct 2011


  • Quantum group
  • W-algebra


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