Hochschild cohomology for finite groups acting linearly and graded Hecke algebras
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Thu, 26/11/2009 14:30 |
Anne Shepler (Denton, Texas and RWTH, Aachen) |
Representation Theory Seminar  |
L3 |
| Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a critical role. We explore this structure for a finite group G acting on an algebra S by automorphisms. We capture the group together with its action with the natural semi-direct product, S#G, known as the "skew group algebra" or "smash product algebra". For example, when G acts linearly on a complex vector space V, it induces an action on the symmetric algebra S(V), a polynomial ring. The semi-direct product S(V)#G is a surrogate for the ring of invariant polynomials on V; it serves as the coordinate ring for the orbifold arising from the action of G on V. Deformations of this skew group algebra S(V)#G play a prominent role in representation theory. Such deformations include graded Hecke algebras (originally defined independently by Drinfeld and by Lusztig), symplectic reflection algebras (investigated by Etingof and Ginzburg in the study of orbifolds), and rational Cherednik algebras (introduced to solve Macdonald's inner product conjectures). We explore the graded Lie structure (or Gerstenhaber bracket) of the Hochschild cohomology of skew group algebras with an eye toward deformation theory. For abelian groups acting linearly, this structure can be described in terms of inner products of group characters. (Joint work with Sarah Witherspoon.) | |||