Tuesday, June 16, 14:00, 7.527.

Mateusz Stroinski (Hamburg)
Developing the fd story inside a tensor category.

Abstract:
Generalizing representation categories of finite-dimensional Hopf algebras, a finite tensor category is an abelian category C equivalent to one of the form A-mod for a finite-dimensional algebra A, together with a rigid monoidal structure, i.e. a well-behaved tensor product on C.

The tensor product allows for formalization of basic algebraic concepts such as an associative algebra or its modules, completely internally to C. The categories of such internal modules play an important role in the study of tensor categories and their applications - a fundamental insight of Etingof--Ostrik is that "internally semisimple" categories are not semisimple in the ordinary sense. I will present results building on this observation, introducing internal analogues of the Jacobson radical, quivers and relations and the associated Morita representatives, largely resembling the classical theory, but also deviating from it and exhibiting new interesting behaviour. Time permitting, I will indicate potential applications to (non-internal) representation theory and ongoing work on more advanced topics.

This talk is based on joint work and work in progress with Edmund Heng (U Sydney), and joint work with Kevin Coulembier (U Sydney) and Tony Zorman (U Hamburg).