Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Eiichi Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. MR 594432
  • [2] Saad Baaj and Georges Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de 𝐶*-algèbres, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 425–488 (French, with English summary). MR 1235438
  • [3] M. E. Sweedler, Hopf algebras, Math. Lecture Notes Ser., Benjamin, New York, 1969.
  • [4] A. Van Daele, Dual pairs of Hopf *-algebras, Bull. London Math. Soc. 25 (1993), no. 3, 209–230. MR 1209245,
  • [5] -, Discrete quantum groups, preprint K. U. Leuven (August 1993).
  • [6] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16W30

Retrieve articles in all journals with MSC: 16W30

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society