Noncommutative Dunkl operators

Yuri Bazlov (University of Warwick)

Wednesday 13th May, 2009 16:00-17:00 Mathematics Building, Room 204


Suppose that V is a complex vector space, and let S(V) denote the free commutative algebra generated by V. If G is a finite group acting on V, one can form a semidirect product of S(V) and G. In 1985, Drinfeld introduced the notion of a degenerate affine Hecke algebra, which is a "flat deformation" of the semidirect product; the commutator of two elements of V in this new algebra is no longer zero but is a linear combination of elements of G. Drinfeld's idea was rediscovered at least twice, with rational Cherednik algebras of Etingof and Ginzburg being perhaps the most popular incarnation of it. The latter provide an algebraic realisation of Dunkl operators, which are differential-difference operators with an intricate property of commuting, associated to a reflection group. I will discuss Dunkl operators and relate them to braided doubles, introduced by Berenstein and myself, and will talk about a noncommutative generalisation of Dunkl operators and reflection groups.

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