Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

The Haar measure on finite quantum groups

Author(s): A. Van Daele
Journal: Proc. Amer. Math. Soc. 125 (1997), 3489-3500.
MSC (1991): Primary 16W30
MathSciNet review: 1415374
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Abstract: By a finite quantum group, we will mean in this paper a finite-dimensional Hopf algebra. A left Haar measure on such a quantum group is a linear functional satisfying a certain invariance property. In the theory of Hopf algebras, this is usually called an integral. It is well-known that, for a finite quantum group, there always exists a unique left Haar measure. This result can be found in standard works on Hopf algebras. In this paper we give a direct proof of the existence and uniqueness of the left Haar measure on a finite quantum group. We introduce the notion of a faithful functional and we show that the Haar measure is faithful. We consider the special case where the underlying algebra is a $^{*}$ -algebra with a faithful positive linear functional. Then the left and right Haar measures coincide. Finally, we treat an example of a root of unity algebra. It is an example of a finite quantum group where the left and right Haar measures are different. This note does not contain many new results but the treatment of the finite-dimensional case is very concise and instructive.

References:

1.: E. Abe, Hopf Algebras, Cambridge University Press, 1977. MR 83a:16010
2.: S. Baaj & G. Skandalis, Unitaires Multiplicatifs et dualité pour les Produits Croisés de C-Algèbres. Ann. Scient. Ec. Norm. Sup., 4ème srie, 26 (1993), 425-488. MR 94e:46127
3.: E.G. Effros & Z.-J. Ruan, Discrete Quantum Groups I. The Haar Measure, Int. J. Math. 5 (1994), 681-723. MR 95j:46089
4.: T. Koornwinder, Representations of the twisted quantum group and some -hypergeometric orthogonal polynomials, Proc. Koninklijke Nederlandse Akademie Wetenschappen, Series A 92 (1989), 97-117. MR 90h:17016
5.: S. Sakai, C-algebras and W-algebras, Springer Verlag (Berlin), 1971. MR 56:1082
6.: M.E. Sweedler, Hopf Algebras, Mathematical Lecture Notes Series, Benjamin (New York), 1969. MR 40:5705
7.: A. Van Daele, Dual Pairs of Hopf -algebras, Lecture notes (second version), Bull. London. Math. Soc. 25 (1993), 209-230. MR 94c:16053
8.: A. Van Daele, Discrete Quantum Groups, J. of Algebra 180 (1996), 431-444. MR 97a:16076
9.: S.L. Woronowicz, Compact Matrix Pseudogroups, Comm. Math. Physics 111 (1987), 613-665. MR 88m:46079
10.: S.L. Woronowicz, Compact quantum groups, Preprint University of Warsaw (1992).

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