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The braided monoidal structures on the category of vector spaces graded by the Klein group

Part of: Hopf algebras, quantum groups and related topics Categories with structure

Published online by Cambridge University Press:  14 June 2011

D. Bulacu ,
S. Caenepeel  and
B. Torrecillas
D. Bulacu
Affiliation:
Faculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14, 010014 Bucharest 1, Romania (daniel.bulacu@fmi.unibuc.ro) Faculty of Engineering, Vrije Universiteit Brussel, 1050 Brussels, Belgium (scaenepe@vub.ac.be)
S. Caenepeel
Affiliation:
Faculty of Engineering, Vrije Universiteit Brussel, 1050 Brussels, Belgium (scaenepe@vub.ac.be)
B. Torrecillas
Affiliation:
Departamento Álgebra y Análisis Matemático, Universidad de Almería, Ctra. Sacramento S/N, La Cañada de San Urbano, 04120 Almería, Spain (btorreci@ual.es)
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Abstract

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Let k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

Keywords

braided monoidal categoryKlein groupgroup cohomologyquasi-Hopf algebra

MSC classification

Secondary: 16T05: Hopf algebras and their applications 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 613 - 641
Copyright
Copyright © Edinburgh Mathematical Society 2011

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