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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: [email protected], [email protected]
ALASTAIR KING
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom e-mails: [email protected], [email protected]
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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

REFERENCES

Bakalov, B. and Kirillov, A., Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21 (American Mathematical Society, Providence, RI, 2001).Google Scholar
Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V., Tensor categories, Mathematical Surveys and Monographs, vol. 205, (American Mathematical Society, Providence, RI, 2015).CrossRefGoogle Scholar
Evans, D. E. and Kawahigashi, Y., Orbifold subfactors from Hecke Algebras II quantum doubles and braiding, Comm. Math. Phys. 196(2) (Aug 1998), 331–361.10.1007/s002200050424CrossRefGoogle Scholar
Kong, L., Cardy condition for open-closed field algebras, Comm. Math. Phys. 283(1) (2008), 25–92.10.1007/s00220-008-0555-9CrossRefGoogle Scholar
Mac Lane, S., Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, 2nd edition, (Springer-Verlag, New York, NY, 1998).Google Scholar
Müger, M., On the structure of modular categories, Proc. London Math. Soc. (3), 87(2) (2003), 291–308.CrossRefGoogle Scholar
Ocneanu, A., Chirality for operator algebras, in Subfactors (Kyuzeso, 1993), (World Sci. Publ., River Edge, NJ, 1994), 39–63.Google Scholar
Popa, S., Shlyakhtenko, D. and Vaes, S., Cohomology and L2-Betti numbers for subfactors and quasi-regular inclusions. Int. Math. Res. Not. IMRN. 2018(8) (2018), 2241–2331.Google Scholar