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

Abstract

Let $k$ be a field, and let $\mathcal{C}$ be a Cauchy complete $k$-linear braided category with finite dimensional morphism spaces and $\mathrm{End}(1)=k$. We call an indecomposable object $X$ of $\mathcal{C}$ non-negligible if there exists $Y\in \mathcal{C}$ such that $1$ is a direct summand of $Y\otimes X$. We prove that every non-negligible object $X\in \mathcal{C}$ such that $\dim \mathrm{End}(X^{\otimes n})<n!$ 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. Finally, 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.

17 pages, latex; new section 4 of rigidity of braided non-semisimple r-categories added in v2

References in corpus (1)

• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels