Rigidity of non-negligible objects of moderate growth in braided categories

Abstract Let $\mathbb {k}$ be a field, and let $\mathcal {C}$ be a Cauchy complete $\mathbb {k}$ -linear braided category with finite-dimensional morphism spaces and . We call an indecomposable object X of $\mathcal C$ non-negligible if there exists $Y\in \mathcal {C}$ such that is a direct summand of $Y\otimes X$ . We prove that every non-negligible object $X\in \mathcal {C}$ such that $\dim \operatorname {End}(X^{\otimes n}) for some n is automatically rigid. In particular, if $\mathcal {C}$ is semisimple of moderate growth and weakly rigid, then $\mathcal {C}$ 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 $\mathcal {C}$ , the data of a $\mathcal {C}$ -modular functor is equivalent to a modular fusion category structure on $\mathcal {C}$ , answering a question of Bakalov and Kirillov. Furthermore, we show that if $\mathcal {C}$ is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in $\mathcal {C}$ is zero. Hence $\mathcal {C}$ admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig.