Howe-Moore type theorems for quantum groups and rigid C*-tensor categories

TL;DR

This paper extends Howe-Moore type theorems to quantum groups and rigid C*-tensor categories, showing that certain categories exhibit convergence properties of multipliers, with applications to quantum groups and subfactor theory.

Contribution

It introduces Howe-Moore properties for quantum groups and rigid C*-tensor categories, proving convergence of multipliers and characterizing central states in specific quantum groups.

Findings

Representation categories of q-deformed Lie groups satisfy Howe-Moore property.

Temperley-Lieb-Jones categories with principal graph A_infinity satisfy Howe-Moore property.

Explicit characterization of central states on quantum SU_q(N).

Abstract

We formulate and study Howe-Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast}$-tensor categories. We say that a rigid $C^{\ast}$-tensor category $\mathcal{C}$ has the Howe-Moore property if every completely positive multiplier on $\mathcal{C}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe-Moore property. As an immediate consequence, we deduce the Howe-Moore property for Temperley-Lieb-Jones standard invariants with principal graph $A_{\infty}$. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan-Wenzl categories. Additionally, in the specific case of the quantum groups $\mathrm{SU}_q(N)$, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of $\mathrm{SU}_q(N)$, which coincide with the completely positive multipliers on the representation category of $\mathrm{SU}_q(N)$.