Semi-simple commutative algebras in non-semisimple braided tensor categories

Speaker's Name: Azat Ganuitdinov
Speaker's Affiliation: CNRS
Speaker's Research Theme(s): Pure Mathematics,
Abstract:
Vertex-Operator Algebras are certain generalizations of both Lie algebras and commutative algebras, and became an important subject in pure mathematics, while they have originally appeared in mathematical physics in 80’s in an attempt to formulate the locality axiom in 2d conformal field theories. Their representation categories have naturally a braided tensor structure and provide representations of Artin's braid groups that don’t factor through symmetric groups. Motivated by representation theory of the VOAs, we study simple commutative algebras living in braided tensor categories of modules over the famous Virasoro algebra at certain specific values of central charges. The internal representation theory of such simple algebras is surprisingly rich and has deep connections with Lie theory. In particular, we obtain a new class of T-crossed braided tensor categories, graded by T - the maximal torus of a simple Lie group G - and with associators given by a 3-cocycle from non-zero cohomology class in the group of locally continuous cohomologies of T. In the case of G=SL(2), this construction provides an interesting non-semisimple rigid monoidal 2-category structure on the group $\mathbb{C}^*$.

Venue: Maths A17