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7 - Finite-Dimensional Quasi-Hopf Algebras

Published online by Cambridge University Press:  21 February 2019

Daniel Bulacu ,
Stefaan Caenepeel ,
Florin Panaite  and
Freddy Van Oystaeyen
Daniel Bulacu
Affiliation:
Universitatea din Bucureşti, Romania
Stefaan Caenepeel
Affiliation:
Vrije Universiteit Brussel
Florin Panaite
Affiliation:
Institute of Mathematics of the Romanian Academy
Freddy Van Oystaeyen
Affiliation:
Universiteit Antwerpen, Belgium
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Summary

We show that for a finite-dimensional quasi-Hopf algebra H the space of integrals in H, and the space of cointegrals on H, have dimension 1. We characterize semisimple and symmetric quasi-Hopf algebras with the help of integrals, and prove a formula for the fourth power of the antipode in terms of the modular elements by using the machinery provided by Frobenius algebras. The chapter ends with a freeness theorem stating that any finite-dimenisonal quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

Keywords

(co)integralFrobenius algebrasymmetric algebrasemisimple quasi-Hopf algebramodular elements
Type
Chapter
Information
Quasi-Hopf Algebras
A Categorical Approach
 , pp. 253 - 304
Publisher: Cambridge University Press
Print publication year: 2019

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