Tensor Categories and Subfactors

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Abstract: Wenzl is continuing his study of tensor categories related to quantum groups and conformal field theory, with a particular emphasis on their connection with subfactors. Braided tensor categories related to classical Lie groups can be conveniently studied via Hecke algebras and q-versions of Brauer algebras, also called BMW algebras. Wenzl plans to study the question whether categories related to exceptional Lie groups can similarly be described via suitable braid representations. Moreover, a detailed knowledge of such braid representations, combined with additional categorical constructions, such as the double construction, might also be useful for understanding various exotic subfactors. Physical systems usually have symmetries which can be expressed via groups and their representations. In recent years, one has encountered new situations in confomal field theory with a highly nontrivial tensor product operation called fusion. Unlike the group case, it is no longer symmetric, but only braided. Moreover, the numbers which naturally occur as 'dimensions' of representations of the symmetries are no longer integers. This is a situation very familiar in the theory of von Neumann algebras. Wenzl plans to study the above mentioned symmetries in the context of von Neumann algebras. He will mainly consider algebras generated by unitary representations of braid groups. Such representations are comparatively rare, and hence provide a good tool for classifying models of the type mentioned above.