Clifford correspondence for algebras

We give a Clifford correspondence for an algebra A over an algebraically closed field, that is an algorithm for constructing some finite-dimensional simple A-modules from simple modules for a subalgebra and endomorphism algebras. This applies to all finite-dimensional simple A-modules in the case that A is finite-dimensional and semisimple with a given semisimple subalgebra. We discuss connections between our work and earlier results, with a view towards applications particularly to finite-dimensional semisimple Hopf algebras.Comment: 12 page

Hochschild cohomology and quantum Drinfeld Hecke algebras

Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups and we exploit computations from our paper with Shroff for diagonal actions. By combining our work with recent results of Levandovskyy and Shepler, we produce examples of quantum Drinfeld Hecke algebras. These algebras generalize the braided Cherednik algebras of Bazlov and Berenstein.Comment: 22 pages; v2: minor revisions as suggested by the refere

Support Varieties and Representation Type of Self-Injective Algebras

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.Comment: 21 page