A rigidity result for extensions of braided tensor C*-categories derived from compact matrix quantum groups

TL;DR

This paper proves a rigidity property for braided tensor categories derived from compact matrix quantum groups, showing certain functors are full unless the deformation parameter has magnitude 1, with implications for subcategory embeddings.

Contribution

It establishes a rigidity result for braided tensor *-functors from quantum group representation categories, highlighting restrictions on embeddings and subcategory structures.

Findings

Any braided tensor *-functor from Rep(G_) is full if || 1.

Restriction functors to proper quantum subgroups cannot be braided.

Temperley--Lieb categories with dimension >2 cannot embed into larger braided tensor categories.

Abstract

Let G be a classical compact Lie group and G_\mu the associated compact matrix quantum group deformed by a positive parameter \mu (or a nonzero and real \mu in the type A case). It is well known that the category Rep(G_\mu) of unitary f.d. representations of G_\mu is a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_\mu) to another braided tensor C*-category with irreducible tensor unit is full if |\mu|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows that the Temperley--Lieb category generated by an object of dimension >2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.