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Lie Group Quantum Mathematics Seminar

Dirac cohomology for Hopf--Hecke algebras

Johannes Flake, Rutgers University

Location:  Hill 705
Date & time: Friday, 07 September 2018 at 12:00PM - 1:00PM

  • Abstract Dirac operators have played an important role in representation theory, especially for real reductive Lie groups and their unitary representations. As a generalization, David Vogan suggested considering the cohomology of Dirac operators for possibly non-unitary representations. He conjectured a connection between this Dirac cohomoloy and central characters which was not only proved in the original context but more recently, versions of it for various kinds of Hecke algebras were established. We explain how these instances can be treated uniformly using superalgebras, Hopf algebras, smash products, and PBW deformations, and we prove a version of Vogan's conjecture in this generalized framework, generalizing the known results for special cases. We will discuss the concepts and techniques used in our generalized approach, results on the classification of the algebraic objects covered by our theory, new examples, and open questions.

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