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3 - Quasi-bialgebras and Quasi-Hopf Algebras
Daniel Bulacu, Universitatea din Bucureşti, Romania, Stefaan Caenepeel, Vrije Universiteit Brussel, Florin Panaite, Freddy Van Oystaeyen, Universiteit Antwerpen, Belgium
Book:

Quasi-Hopf Algebras

Published online:

21 February 2019

Print publication:

21 February 2019, pp 103-146
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Summary

We introduce the concepts of quasi-bialgebra and quasi-Hopf algebra by using a categorical point of view.We present the basic properties of these objects and study their invariance under a twist. We also introduce the dual notions, called dual quasi-bialgebra and dual quasi-Hopf algebra.

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
D. Bulacu, S. Caenepeel, B. Torrecillas
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 54 / Issue 3 / October 2011

Published online by Cambridge University Press:

14 June 2011, pp. 613-641
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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].

The Representation Ring of the Twisted Quantum Double of a Finite Group
S. J. Witherspoon
Journal:

Canadian Journal of Mathematics / Volume 48 / Issue 6 / 01 December 1996

Published online by Cambridge University Press:

20 November 2018, pp. 1324-1338

Print publication:

01 December 1996
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We provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.