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

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

💡 Research Summary

The paper studies braided monoidal categories C over a field k that are Cauchy complete, have finite‑dimensional Hom spaces, and satisfy End(1)=k. An indecomposable object X is called “non‑negligible’’ if there exists some Y with the unit object 1 appearing as a direct summand of Y⊗X. The authors introduce the notion of “moderate growth’’ for an object: there exists an integer n such that dim End (X⊗ⁿ) < n!. This condition is equivalent to the usual growth constraints known from Deligne’s work and from Schur–Weyl duality.

The central result (Theorem 1.1) states that any non‑negligible object of moderate growth is automatically rigid, i.e. it possesses a left and right dual. The proof proceeds by choosing a retraction r: X⊗Y→1 and a splitting s:1→X⊗Y (with Y witnessing non‑negligibility) and arranging that the key morphisms built from the braiding are nilpotent. The authors then consider the braid group Bₙ and its symmetric subgroup Sₙ. For each permutation t∈Sₙ they construct a morphism f_t∈End (X⊗ⁿ) by interpreting t as a braid diagram using the chosen retraction and splitting. They define a bilinear form Φ on End (X⊗ⁿ) by closing diagrams to obtain scalars in k. Using a combinatorial analysis of braid relations (Lemma 2.3) they show that Φ(f_s⁻¹,f_t)=δ_{s,t}. Consequently the set {f_t | t∈Sₙ} is linearly independent, forcing dim End (X⊗ⁿ)≥n!. If dim End (X⊗ⁿ) is strictly smaller than n! for some n, the only way to avoid this contradiction is that the nilpotent morphisms vanish, which forces the retraction and splitting to be invertible and hence X to be rigid.

From this theorem several corollaries follow. Corollary 1.2 asserts that if the whole category has moderate growth, every non‑negligible object is rigid. In the language of r‑categories (categories where each object has a representing object for Hom(–⊗X,1)), this yields Corollary 1.3: any semisimple braided r‑category of moderate growth is rigid. This immediately simplifies Huang’s proof that representation categories of many vertex operator algebras (VOAs) are rigid, because those categories are known to be finite semisimple braided r‑categories.

A further application concerns modular functors. For a finite split semisimple monoidal category C, the data of a C‑modular functor (in the sense of Bakalov–Kirillov) is shown to be equivalent to giving C a modular fusion category structure (Corollary 1.4). This answers a question raised in