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The Haar measure on finite quantum groups

Author: 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.

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A. Van Daele
Affiliation: Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium

Received by editor(s): March 18, 1996
Received by editor(s) in revised form: July 8, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society