### Abstract

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 language | English |
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Pages (from-to) | 1315-1376 |

Number of pages | 61 |

Journal | Advances in Mathematics |

Volume | 228 |

Issue number | 3 |

Early online date | 24 Jun 2011 |

DOIs | |

Publication status | Published - 20 Oct 2011 |

### Keywords

- Quantum group
- W-algebra

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## Cite this

Sevastyanov, A. (2011). Conjugacy classes in Weyl groups and q-W algebras.

*Advances in Mathematics*,*228*(3), 1315-1376. https://doi.org/10.1016/j.aim.2011.06.018