On tensor categories (conferencia plenaria)

SONIA NATALE

Valparaíso

Encuentro; Primer Encuentro Conjunto SOMACHI-UMA; 2016

Tensor categories can be interpreted as a generalization of the notion of a group, where equalities become replaced by natural isomorphisms obeying appropriate conditions. They were introduced in the sixties by Mac Lane and Benabou. In the eighties, they started to emerge in different areas of mathematics and physics encoding symmetries of algebraic and geometrical structures, as part of the revolution caused by the discovery of quantum groups. In the nineties, tensor categories started to be considered as mathematical objects in their own and many adaptations of the usual questions and approaches in the classification of groups and Lie algebras were proposed in this more general context. We shall present an introduction to tensor and fusion categories and their representations and review some of their invariants. We shall then discuss some families of examples and classification results for these structures related to certain notions of extensions.