Fusion operators and cocycloids in monoidal categories
type: Item
instance of: scholarly article
title: Fusion operators and cocycloids in monoidal categories
zbMATH Open document ID: 0912.16020
DOI: 10.1023/A:1008655911796
published in: Applied Categorical Structures
publication date: 1999-05-25T00:00:00Z
review text: The Yang-Baxter equation has been studied extensively in the context of monoidal categories. The fusion equation, which appears to be the Yang-Baxter equation with a term missing, has been studied mainly in the context of Hilbert spaces. This paper endeavours to place the fusion equation in an appropriate categorical setting. The author begins by establishing the bijection between fusion operators and 3-cocycles, and shows how these arise from bialgebras. Some constructions with 3-cocycles are described showing, in particular, the relation to the Stasheff-MacLane pentagon. An object is called a tricocycloid if it is equipped with an invertible 3-cocycle. It should be regarded as a generalized Hopf algebra, and hence, as a generalized (quantum) group. By considering representations and using techniques of Tannaka duality, Hopf algebras are associated with tricocycloids. And, the author makes some speculations on higher cocycloids. Finally, the generic tricocycloid is constructed.
reviewed by: Fang Li
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 53A45
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1195511
zbMATH Keywords: braidings
zbMATH Keywords: monoidal categories
zbMATH Keywords: fusion equation
zbMATH Keywords: fusion operators
zbMATH Keywords: 3-cocycles
zbMATH Keywords: bialgebras
zbMATH Keywords: generalized Hopf algebras
zbMATH Keywords: Tannaka duality
author: Ross H. Street
MaRDI profile type: Publication
full work available at URL: a:1008655911796
OpenAlex ID: W86190141
Wikidata QID: Q56687271
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description: scientific article; zbMATH DE number 1195511