The Haar measure on finite quantum groups
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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
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Additional Information
- A. Van Daele
- Affiliation: Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium
- Email: alfons.vandaele@wis.kuleuven.ac.be
- Received by editor(s): March 18, 1996
- Received by editor(s) in revised form: July 8, 1996
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 3489-3500
- MSC (1991): Primary 16W30
- DOI: https://doi.org/10.1090/S0002-9939-97-04037-9
- MathSciNet review: 1415374