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DECOMPOSING THE TUBE CATEGORY

Part of: Categories with structure

Published online by Cambridge University Press:  17 June 2019

LEONARD HARDIMAN Open the ORCID record for LEONARD HARDIMAN [Opens in a new window]  and
ALASTAIR KING Open the ORCID record for ALASTAIR KING [Opens in a new window]
LEONARD HARDIMAN
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom e-mails: leonard.p.a.hardiman@bath.edu, a.d.king@bath.ac.uk
ALASTAIR KING
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom e-mails: leonard.p.a.hardiman@bath.edu, a.d.king@bath.ac.uk
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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 through 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.

MSC classification

Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 2 , May 2020 , pp. 441 - 458
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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References

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