Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 111, 17 pages      arXiv:1605.07010      https://doi.org/10.3842/SIGMA.2016.111

Hypergroups Related to a Pair of Compact Hypergroups

Herbert Heyer a, Satoshi Kawakami b, Tatsuya Tsurii c and Satoe Yamanaka d
a) Universität Tübingen, Mathematisches Institut, Auf der Morgenstelle 10, 72076, Tübingen, Germany
b) Nara University of Education, Department of Mathematics, Takabatake-cho Nara, 630-8528, Japan
c) Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai Osaka, 599-8531, Japan
d) Nara Women's University, Faculty of Science, Kitauoya-higashimachi, Nara, 630-8506, Japan

Received June 02, 2016, in final form November 10, 2016; Published online November 18, 2016

Abstract
The purpose of the present paper is to investigate a hypergroup associated with irreducible characters of a compact hypergroup $H$ and a closed subhypergroup $H_0$ of $H$ with $|H/H_0|$<$+ \infty$. The convolution of this hypergroup is introduced by inducing irreducible characters of $H_0$ to $H$ and by restricting irreducible characters of $H$ to $H_0$. The method of proof relies on the notion of an induced character and an admissible hypergroup pair.

Key words: hypergroup; induced character; semi-direct product hypergroup; admissible hypergroup pair.

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References

1. Bloom W.R., Heyer H., Harmonic analysis of probability measures on hypergroups, de Gruyter Studies in Mathematics, Vol. 20, Walter de Gruyter & Co., Berlin, 1995.
2. Dunkl C.F., The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331-348.
3. Dunkl C.F., Structure hypergroups for measure algebras, Pacific J. Math. 47 (1973), 413-425.
4. Heyer H., Katayama Y., Kawakami S., Kawasaki K., Extensions of finite commutative hypergroups, Sci. Math. Jpn. 65 (2007), 373-385.
5. Heyer H., Kawakami S., Hypergroup structures arising from certain dual objects of a hypergroup, J. Math. Soc. Japan, to appear.
6. Heyer H., Kawakami S., Extensions of Pontryagin hypergroups, Probab. Math. Statist. 26 (2006), 245-260.
7. Heyer H., Kawakami S., A cohomology approach to the extension problem for commutative hypergroups, Semigroup Forum 83 (2011), 371-394.
8. Heyer H., Kawakami S., An imprimitivity theorem for representations of a semi-direct product hypergroup, J. Lie Theory 24 (2014), 159-178.
9. Heyer H., Kawakami S., Tsurii T., Yamanaka S., A commutative hypergroup associated with a hyperfield, arXiv:1604.04361.
10. Heyer H., Kawakami S., Tsurii T., Yamanaka S., Hypergroups arising from characters of a compact group and its subgroup, arXiv:1605.03744.
11. Heyer H., Kawakami S., Yamanaka S., Characters of induced representations of a compact hypergroup, Monatsh. Math. 179 (2016), 421-440.
12. Hirai T., Classical method of constructing a complete set of irreducible representations of semidirect product of a compact group with a finite group, Probab. Math. Statist. 33 (2013), 353-362.
13. Jewett R.I., Spaces with an abstract convolution of measures, Adv. Math. 18 (1975), 1-101.
14. Kawakami S., Tsurii T., Yamanaka S., Deformations of finite hypergroups, Sci. Math. Japan e-2015 (2015), 2015-21, 11 pages.
15. Spector R., Aperçu de la théorie des hypergroupes, in Analyse Harmonique sur les groupes de Lie (Sém. Nancy-Strasbourg, 1973-1975), Lecture Notes in Math., Vol. 497, Springer, Berlin, 1975, 643-673.
16. Sunder V.S., Wildberger N.J., Fusion rule algebras and walks on graphs, in The Proceedings of the Fifth Ramanujan Symposium on Harmonic Analysis, Editor K.R. Parthasarathy, Ramanujan Institute, 1999, 53-80.
17. Vrem R.C., Harmonic analysis on compact hypergroups, Pacific J. Math. 85 (1979), 239-251.
18. Willson B., Configurations and invariant nets for amenable hypergroups and related algebras, Trans. Amer. Math. Soc. 366 (2014), 5087-5112.