Pivotal categories, matrix units, and towers of biadjunctions

Abstract

In this thesis we study a tower of biadjunctions coming from a pivotal tensor category with a self dual object. In order to do this, we present some relevant parts of the standard theory of monoidal categories, tensor categories, and pivotal tensor categories. We recall a method for constructing matrix units for the algebras End(X n) for any object X in a semisimple linear monoidal category. Using these matrix units, we then prove our main result, Theorem 4.2.14. In a linear monoidal category, endomorphism algebras for tensor powers of a distinguished object X can be used to build a tower of algebras. We prove that when the category is a pivotal tensor category and the object X is self dual, the induction and restriction functors associated to this tower form biadjoint pairs. Inspired by [Kho14], we use the data of these biadjunctions to construct a graphical category GX. The morphisms in this category are various planar diagrams, modulo some local relations. The construction in [Kho14] has been a rich source of interesting mathematics. The hope is that our category might prove to be similarly interesting.