Narrow Results

There is a long-standing belief that the modular tensor categories ${𝒞}(𝔤,k)$, for $k\in {\mathbb{Z}}_{\ge 1}$ and finite-dimensional simple complex Lie algebras $𝔤$, contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels $k$. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when $𝔤$ has rank 2, contributing proofs and explicit bounds when $𝔤$ is of type $B_2$ or $G_2$, adding to the previously known positive results for types $A_1$ and $A_2$.

Generalized Temperley–Lieb–Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong ∗-pseudofunctors into the ∗-2-category of row-finite separable bigraded Hilbert spaces. We classify these modules up to ∗-equivalence in terms of weighted bi-directed fair and balanced graphs in the spirit of Yamagami’s classification of fiber functors on TLJ categories and DeCommer and Yamashita’s classification of unitary modules for ${\mathrm{\text{Rep(SU}}}_q(2))$.