TY - JOUR TI - Dirac cohomology for Hopf-Hecke algebras DO - https://doi.org/doi:10.7282/t3-ss9m-we37 PY - 2018 AB - In this dissertation, a generalized version of Dirac cohomology is developed. It is shown that Dirac operators can be defined and their cohomology can be studied for a general class of algebras, which we call Hopf--Hecke algebras. A result relating the Dirac cohomology with central characters is established for a subclass of algebras, which we call Barbasch--Sahi algebras. This result simultaneously generalizes known results on such a relation for real reductive Lie groups and for various kinds of Hecke algebras, which all go back to a conjecture of David Vogan (1997). A variety of algebraic concepts and techniques is combined to create the general framework for Dirac cohomology, including central simple (super)algebras, Hopf algebras, smash products, PBW deformations, and Koszul algebras. Classification results on the studied classes of algebras are obtained, and infinitesimal Cherednik algebras of the general linear group are studied as novel examples for algebras with a Dirac cohomology theory. KW - Mathematics KW - Hecke algebras KW - Hopf algebras KW - Cohomology operations LA - eng ER -