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Group-cograded multiplier Hopf ($$*$$-)algebras. (English) Zbl 1129.16027
Let $$G$$ be a group and assume that $$(A_p)_{p\in G}$$ is a family of algebras with identity. For each pair $$p,q\in G$$, there is given a unital homomorphism $$\Delta_{p,q}\colon A_{pq}\to A_p\otimes A_q$$ satisfying certain properties. The authors consider the direct sum $$A=\bigoplus_{p\in G}A_p$$. $$A$$ is an algebra in a natural way and the product is non-degenerate. The maps $$\Delta_{p, q}$$ can be used to define a coproduct $$\Delta$$ on $$A$$ such that $$(A,\Delta)$$ is a multiplier Hopf algebra.
In the paper under review, the authors show that $$A$$ is a group-cograded multiplier Hopf algebra, which is more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, their 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 addition, the authors study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, the authors obtain most of the results of A. Virelizier [J. Pure Appl. Algebra 171, No. 1, 75-122 (2002; Zbl 1011.16023)] on this subject and consider them in the framework of multiplier Hopf algebras.
Reviewer: Li Fang (Hangzhou)

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W50 Graded rings and modules (associative rings and algebras) 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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