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A Few Remarks on the Tube Algebra of a Monoidal Category

Part of: Selfadjoint operator algebras Categories with structure Groups and algebras in quantum theory

Published online by Cambridge University Press:  08 May 2018

Sergey Neshveyev  and
Makoto Yamashita
Sergey Neshveyev
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway (sergeyn@math.uio.no)
Makoto Yamashita*
Affiliation:
Department of Mathematics, Ochanomizu University, Otsuka 2-1-1, Bunkyo, 112-8610 Tokyo, Japan (yamashita.makoto@ocha.ac.jp)
*
*Corresponding author.
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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.

Keywords

C*-tensor categorytube algebraMorita equivalence

MSC classification

Primary: 81R15: Operator algebra methods
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 735 - 758
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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