Conjugacy classes in Weyl groups and q-W algebras

2010/11/10 by A. Sevostyanov, Sevostyanov, A. · 1 citation
Mathematics · Physics and Astronomy · #17B37 (primary) #17B63 #20F55 #20G20 (secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #hep-th #math.QA #math.RT #msc:17B37 #msc:17B63 #msc:20F55 #msc:20G20

paper · pdf · doi:10.48550/arxiv.1011.2431

48 pages; some arguments in the proof of Proposition 12.2 are clarified

openalex publication_date 2010/11/10 · arxiv created 2015/06/26 · arxiv updated 2015/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define noncommutative deformations W_q^s(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 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 G^* dual to a quasitriangular Poisson-Lie group. 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.

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