Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/3981
Title: The Drinfel'd double for group-cograded multiplier Hopf algebras
Authors: DELVAUX, Lydia 
Van Daele, A.
Issue Date: 2007
Publisher: SPRINGER
Source: ALGEBRAS AND REPRESENTATION THEORY, 10(3). p. 197-221
Abstract: Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G) circle times K(G )) by the formula (Delta(f))(p,q) = f(pq) for all f is an element of K (G) and p,q is an element of G. In this paper we consider multiplier Hopf algebras B (over C) such that there is an embedding I : K(G) -> M(B). This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called G-cograded multiplier Hopf algebras. They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action pi of the group G on a G-cograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel'd double D-pi where the coproduct and the product depend on the action pi. We also treat the *-algebra case. If pi is the trivial action, we recover the usual Drinfel'd double associated with the pair [A, B]. On the other hand, also the Drinfel'd double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a G-cograded multiplier Hopf algebra.
Notes: Katholieke Univ Leuven, Dept Math, B-3001 Heverlee, Belgium. Limburgs Univ Ctr, Dept Math, B-3590 Diepenbeek, Belgium.VAN DAELE, A, Katholieke Univ Leuven, Dept Math, Celestijnenlaan 200B, B-3001 Heverlee, Belgium.lydia.delvaux@luc.ac.be alfons.vandaele@wis.kuleuven.ac.be
Keywords: G-cograded multiplier Hopf algebra; comultiplication; multiplier Hopf algebra; Hopf group-coalgebra; Drinfel'd double
Document URI: http://hdl.handle.net/1942/3981
ISSN: 1386-923X
e-ISSN: 1572-9079
DOI: 10.1007/s10468-006-9042-1
ISI #: 000246177600001
Category: A1
Type: Journal Contribution
Validations: ecoom 2008
Appears in Collections:Research publications

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