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Title:  The Drinfel'd double for groupcograded multiplier Hopf algebras 
Authors:  DELVAUX, Lydia Van Daele, A. 
Issue Date:  2007 
Publisher:  SPRINGER 
Citation:  ALGEBRAS AND REPRESENTATION THEORY, 10(3). p. 197221 
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 nondegenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called Gcograded multiplier Hopf algebras. They are a generalization of the Hopf groupcoalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action pi of the group G on a Gcograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel'd double Dpi 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 finitetype Hopf groupcoalgebra, is an example of the construction above. In this case, the action is nontrivial but related with the adjoint action of the group on itself. Now, the double is again a Gcograded multiplier Hopf algebra. 
Notes:  Katholieke Univ Leuven, Dept Math, B3001 Heverlee, Belgium. Limburgs Univ Ctr, Dept Math, B3590 Diepenbeek, Belgium.VAN DAELE, A, Katholieke Univ Leuven, Dept Math, Celestijnenlaan 200B, B3001 Heverlee, Belgium.lydia.delvaux@luc.ac.be alfons.vandaele@wis.kuleuven.ac.be 
URI:  http://hdl.handle.net/1942/3981 
DOI:  10.1007/s1046800690421 
ISI #:  000246177600001 
ISSN:  1386923X 
Category:  A1 
Type:  Journal Contribution 
Validation:  ecoom, 2008

Appears in Collections:  Research publications

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