Semigroup Forum

, Volume 83, Issue 3, pp 371–394 | Cite as

A cohomology approach to the extension problem for commutative hypergroups

RESEARCH ARTICLE
First Online:
Received:
Accepted:

Abstract

The purpose of this paper is to determine all commutative hypergroup extensions of a countable discrete commutative hypergroup by a locally compact Abelian group, in terms of second order cohomology of hypergroups, a notion which generalizes the cohomology of groups.

Keywords

Hypergroups Extensions Second order cohomology 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bloom, W.R., Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter Studies in Mathematics, vol. 20. de Gruyter, Berlin (1995) MATHGoogle Scholar
  2. 2.
    Dunkl, C.F., Ramirez, D.E.: A family of compact P ∗-hypergroups. Trans. Am. Math. Soc. 202, 339–356 (1975) MathSciNetMATHGoogle Scholar
  3. 3.
    Fournier, J.J.F., Ross, K.A.: Random Fourier series on compact abelian hypergroups. J. Aust. Math. Soc. A 37(1), 45–81 (1984) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Heyer, H., Jimbo, T., Kawakami, S., Kawasaki, K.: Finite commutative hypergroups associated with actions of finite abelian groups. Bull. Nara Univ. Ed. 54(2), 23–29 (2005) MathSciNetGoogle Scholar
  5. 5.
    Heyer, H., Kawakami, S.: Extensions of Pontryagin hypergroups. Probab. Math. Stat., 26(2), 245–260 (2006) MathSciNetMATHGoogle Scholar
  6. 6.
    Heyer, H., Katayama, Y., Kawakami, S., Kawasaki, K.: Extensions of finite commutative hypergroups. Sci. Math. Jpn., 65(3), 373–385 (2008) MathSciNetGoogle Scholar
  7. 7.
    Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18(1), 1–101 (1975) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kawakami, S., Ito, W.: Crossed products of commutative finite hypergroups. Bull. Nara Univ. Ed. 48(2), 1–6 (1999) MathSciNetGoogle Scholar
  9. 9.
    Kawakami, S., Kawasaki, K., Yamanaka, S.: Extensions of the Golden hypergroup by finite abelian groups. Bull. Nara Univ. Ed. 57(2), 1–10 (2008) MathSciNetGoogle Scholar
  10. 10.
    Kawakami, S., Sakao, M., Yamanaka, S.: Extensions of hypergroups of order two by locally compact abelian groups. Bull. Nara Univ. Ed. 59(2), 1–6 (2010) MathSciNetGoogle Scholar
  11. 11.
    Kawakami, S., Yamanaka, S.: Extensions of the Golden hypergroup by locally compact abelian groups. Preprint (2010) Google Scholar
  12. 12.
    Kawakami, S., Sakao, M., Takeuchi, T.: Non-splitting extensions of hypergroups of order two. Bull. Nara Univ. Ed. 56(2), 7–13 (2007) MathSciNetGoogle Scholar
  13. 13.
    Mackey, G.W.: Unitary representations of group extensions I. Acta Math. 99, 265–311 (1958) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ross, K.A.: Centers of hypergroups. Trans. Am. Math. Soc. 243, 251–269 (1978) MATHCrossRefGoogle Scholar
  15. 15.
    Voit, M.: Substitution of open subhypergroups. Hokkaido Math. J. 23, 143–183 (1994) MathSciNetMATHGoogle Scholar
  16. 16.
    Voit, M.: Hypergroups on two tori. Semigroup Forum 76, 192–203 (2008) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Vrem, R.C.: Hypergroup joins and their dual objects. Pac. J. Math. 111(2), 483–495 (1984) MathSciNetMATHGoogle Scholar
  18. 18.
    Vrem, R.C.: Connectivity and supernormality results for hypergroups. Math. Z. 195(3), 419–428 (1987) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Wildberger, N.J.: Strong hypergroups of order three. J. Pure Appl. Algebra 174, 95–115 (2002) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Zeuner, H.: Duality of commutative hypergroups. In: Heyer, H. (ed.) Probability Measures on Groups, X, pp. 467–488. Plenum, New York (1991) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany
  2. 2.Department of MathematicsNara University of EducationNaraJapan

Personalised recommendations