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Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$ algebras
- Source :
- Algebras and Representation Theory. 10:77-95
- Publication Year :
- 2006
- Publisher :
- Springer Science and Business Media LLC, 2006.
-
Abstract
- Let G be a group and assume that (Ap)p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δp,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 Δp,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) 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.
- Subjects :
- Discrete mathematics
Pure mathematics
Quantum group
Direct sum
General Mathematics
Mathematics::Rings and Algebras
Quantum algebra
Representation theory of Hopf algebras
Quasitriangular Hopf algebra
Hopf algebra
Multiplier (Fourier analysis)
Mathematics::Quantum Algebra
Hopf lemma
Mathematics
Subjects
Details
- ISSN :
- 15729079 and 1386923X
- Volume :
- 10
- Database :
- OpenAIRE
- Journal :
- Algebras and Representation Theory
- Accession number :
- edsair.doi...........4472658b44e13ea938f763624f7d3144