David Penneys
The Ohio State University
Title
Rigidity of non-negligible objects of moderate growth in braided categories
Abstract
We prove that in a Cauchy complete braided monoidal category C with simple unit, every non-negligible object of moderate growth is automatically rigid. In particular, if C is semisimple of moderate growth and weakly rigid, then C is rigid. As applications, we simplify Huang's proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category C, the data of a C-modular functor is equivalent to a modular fusion category structure on C, answering a question of Bakalov and Kirillov. Finally, we show that if C is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in C is zero. Hence C admits a semisimplification, which is a semisimple braided tensor category of moderate growth. This is joint work with Pavel Etingof [arXiv:2412.17681].
