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|Title: ||Group-Cograded Multiplier Hopf (*-)Algebras|
|Authors: ||Abd El-Hafez, A.T.|
Van Daele, A.
|Issue Date: ||2006|
|Citation: ||ALGEBRAS AND REPRESENTATION THEORY, 10. p. 77-95|
|Abstract: ||Let G be a group and assume that (Ap)p2G is a family of algebras with identity. We have
a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p; q 2 G, there is given a unital
homomorphism ¢p;q : Apq ! Ap Aq satisfying certain properties.
Consider now the direct sum A of these algebras. It is an algebra, without identity, except
when G is a finite group, but the product is non-degenerate. The maps ¢p;q can be used
to define a coproduct ¢ on A and the conditions imposed on these maps give that (A; ¢)
is a multiplier Hopf algebra. It is G-cograded as explained in this paper.
We study these so-called group-cograded multiplier Hopf algebras. They are, as explained
above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover,
our point of view makes it possible to use results and techniques from the theory of multiplier
Hopf algebras in the study of Hopf group-coalgebras (and generalizations).
In a separate paper, we treat the quantum double in this context and we recover, in a
simple and natural way (and generalize) results obtained by Zunino. In this paper, we
study integrals, in general and in the case where the components are finite-dimensional.
Using these ideas, we obtain most of the results of Virelizier on this subject and consider
them in the framework of multiplier Hopf algebras.|
|ISI #: ||000243320700004|
|Type: ||Journal Contribution|
|Validation: ||ecoom, 2008|
|Appears in Collections: ||Research publications|
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