Subfactors, Tensor Categories and Module Categories

Abstract: When Vaughan Jones started to think about the notion of an index for subfactors, the only known examples came from groups and their representations and from the embedding of groups H < G. In both cases the indices were integers. Vaughan's surprising examples with non-integer index were later connected to representations of the quantum group U_q(sl2) and to representations of the loop group LSU(2). This was subsequently generalized to the construction of a sequence of subfactors for every representation of a semisimple Lie algebra.The question remains to construct subfactors corresponding to analogs of subgroups of Lie groups; in modern language this amounts to classifying module categories of certain tensor categories. Again, Vaughan made important contributions for solving the problem for the sl_2 case constructing what is generally referred to as Goodman-de la Harpe-Jones subfactors. While complete classifications are known for several Lie groupsof small rank, the general problem is still far from being solved. We give an overview of the current state of knowledge, and present some explicit examples.

operator algebras

Audience: researchers in the topic

Conference on operator algebras and related topics in Istanbul, 2021