Kenichi Shimizu

Given an algebra A in a finite tensor category C, the category C_A$ of right A-modules in C$ is defined. 

Since C_A is a left C-module category, one can define C-injective objects in C_A in terms of the internal Hom functor of C_A.

An algebra A in C is said to be quasi-Frobenius (QF) if A in C_A is C-injective. 

It turns out that Frobenius algebras in C, exact algebras in C, and Hopf algebras in C (when C is braided) are QF. 

In this talk, I will give several criteria for an algebra to be QF. 

One of important characterizations is that, as one might expect, an algebra in C is QF if and only if it is Morita equivalent to a Frobenius algebra in C (here, two algebras A and B in C are Morita equivalent if C_A \approx C_B as module categories over C). 

The techniques for proving this also allow us to show that the class of symmetric Frobenius algebras in C is closed under Morita equivalence, provided that C is pivotal so that symmetricity of an algebra in C makes sense.