Decomposing the Tube Category

TL;DR

This paper explores the decomposition of the tube category in a modular tensor category, detailing how Hom spaces and objects can be broken down into irreducible components using diagram calculus.

Contribution

It provides a detailed description of the two types of decompositions of the tube category and identifies primitive idempotents within the category.

Findings

Hom spaces decompose into summands via the Yoneda embedding.

Objects decompose into irreducibles corresponding to primitive idempotents.

Diagram calculus is used extensively for proofs and descriptions.

Abstract

The tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces decompose into summands factoring though irreducible functors, in a manner analogous to decomposing an algebra as a sum of matrix algebras. We describe these summands. Secondly, under the Yoneda embedding, each object decomposes into irreducibles, which correspond to primitive idempotents in the category itself. We identify these idempotents. We make extensive use of diagram calculus in the description and proof of these decompositions.