### Abstract

Original language | English |
---|---|

Number of pages | 18 |

Journal | Glasgow Mathematical Journal |

Early online date | 17 Jun 2019 |

DOIs | |

Publication status | E-pub ahead of print - 17 Jun 2019 |

### Keywords

- monoidal categories
- modular tensor categories

### Cite this

**Decomposing the Tube Category.** / Hardiman, Leonard; King, Alastair.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Decomposing the Tube Category

AU - Hardiman, Leonard

AU - King, Alastair

PY - 2019/6/17

Y1 - 2019/6/17

N2 - 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.

AB - 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.

KW - monoidal categories

KW - modular tensor categories

U2 - 10.1017/S001708951900020X

DO - 10.1017/S001708951900020X

M3 - Article

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 0017-0895

ER -