Cocycle and Galois cocycle twists of algebras, representations and orders

Abstract: In a construction known as Drinfeld twist, a 2-cocycle on a Hopf algebra H is used to modify the coproduct on H as well as the associative product in any H-module algebra A. I am interested to know to what extent the representation theory of the twist of A can be recovered from that of A; the A#H-module category, unchanged under the twist, plays a role here. I will talk about an application of this idea to rational Cherednik-type algebras, which led, in a joint work with E. Jones-Healey, to establishing nontrivial isomorphisms between braided and classical versions of these algebras. Twists also help to approach representation theory of the so-called mystic reflection groups, defined by the Chevalley-Serre-Shephard-Todd property of their action on a quantum polynomial ring. An important source of twists, motivated by torsors in geometry, should be cocycles arising from (Hopf-)Galois extensions of algebras, and I will discuss this in the context of constructing orders and normal integral bases in central simple algebras over a number field.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home