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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Multiplier Hopf algebras

Author: A. Van Daele
Journal: Trans. Amer. Math. Soc. 342 (1994), 917-932
MSC: Primary 16W30
MathSciNet review: 1220906
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Abstract: In this paper we generalize the notion of Hopf algebra. We consider an algebra A, with or without identity, and a homomorphism $ \Delta $ from A to the multiplier algebra $ M(A \otimes A)$ of $ A \otimes A$. We impose certain conditions on $ \Delta $ (such as coassociativity). Then we call the pair $ (A,\Delta )$ a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where $ (\Delta f)(s,t) = f(st)$ with $ s,t \in G$ and $ f \in A$. We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a $ \ast$-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a $ \ast$-algebra.

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Article copyright: © Copyright 1994 American Mathematical Society