The Haar measure on finite quantum groups

Daele, A.

doi:10.1090/S0002-9939-97-04037-9

The Haar measure on finite quantum groups
HTML articles powered by AMS MathViewer

by A. Van Daele PDF

Proc. Amer. Math. Soc. 125 (1997), 3489-3500 Request permission

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

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
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
Edward G. Effros and Zhong-Jin Ruan, Discrete quantum groups. I. The Haar measure, Internat. J. Math. 5 (1994), no. 5, 681–723. MR 1297413, DOI 10.1142/S0129167X94000358
T. H. Koornwinder, Representations of the twisted $\textrm {SU}(2)$ quantum group and some $q$-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 1, 97–117. MR 993682
Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
A. Van Daele, Dual pairs of Hopf $*$-algebras, Bull. London Math. Soc. 25 (1993), no. 3, 209–230. MR 1209245, DOI 10.1112/blms/25.3.209
A. Van Daele, Discrete quantum groups, J. Algebra 180 (1996), no. 2, 431–444. MR 1378538, DOI 10.1006/jabr.1996.0075
S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157
S.L. Woronowicz, Compact quantum groups, Preprint University of Warsaw (1992).