A few remarks on the tube algebra of a monoidal category

TL;DR

This paper investigates the structure of tube algebras in rigid C*-tensor categories, revealing their relation to Drinfeld doubles and Morita equivalences, with implications for quantum groups and Temperley-Lieb categories.

Contribution

It establishes that the tube algebra of a quantum group's representation category is a corner of its Drinfeld double and shows Morita equivalence of tube algebras under certain conditions.

Findings

Tube algebra of a quantum group's representation category is a full corner of the Drinfeld double.

Tube algebras of weakly Morita equivalent categories are strongly Morita equivalent.

Structural insights into Temperley-Lieb categories for d>2.

Abstract

We prove two results on the tube algebras of rigid C$^*$-tensor categories. The first is that the tube algebra of the representation category of a compact quantum group $G$ is a full corner of the Drinfeld double of $G$. As an application we obtain some information on the structure of the tube algebras of the Temperley-Lieb categories $TL(d)$ for $d>2$. The second result is that the tube algebras of weakly Morita equivalent C$^*$-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the $2$-category defining the Morita context.