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Multiplier Hopf Algebras

A. Van Daele
Transactions of the American Mathematical Society
Vol. 342, No. 2 (Apr., 1994), pp. 917-932
DOI: 10.2307/2154659
Stable URL: http://www.jstor.org/stable/2154659
Page Count: 16
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Multiplier Hopf Algebras
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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 Δ from A to the multiplier algebra M(A ⊗ A) of A ⊗ A. We impose certain conditions on Δ (such as coassociativity). Then we call the pair (A, Δ) 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 (Δ f)(s, t) = f(st) with s, t ∈ G and f ∈ 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 *-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a *-algebra.

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