Structure of commutative hypergroups

• Applicant: Professor Dr. Herbert Heyer (†)
• Subject Area: Mathematics
• Term: from 2013 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 234833369

Hypergroups are locally compact spaces on which the bounded measures convolve similar to those on a locally compact group. There exists an axiomatic approach to the theory of hypergroups since around 1975. Examples of hyper groups are double coset spaces derived from Gelfand pairs, more concrete examples are the nonnegative integers or the nonnegative real numbers whose convolutions are defined by orthogonal polynomials of by special functions respectively. Hyper group structures enjoy widespread applications ranging from the theory of differential equations of second order (Sturm-Liouville problems) to the theory of probability (properties of random walks). For comprehensive expositions on the theory see the applicant’s monographs [1] and [2]. The central topic of the research program to be described here is the extension of hyper groups, i.e. the ambition to produce new hyper groups from already known ones. This way one hopes to arrive at a structural theory of hyper groups. Methodically a first idea to succeed is to extend the cohomology theory for groups to hyper groups, in particular to generalize G.W. Mackey’s theory of cocycles. In earlier papers published by the applicant together with his colleague S. Kawakami from the Nara University of Education the extension problem has been successfully dealt with (Reference lists I and II). For the special class of Pontryagin hyper groups surprising results have been obtained ([HK1] of Reference list II). A first step in the direction to a cohomology theory for commutative hyper groups has been successful ([HK 24] of Reference List II). The project which the present application refers to concerns the imprimitivity theorem for commutative hypger groups. There is the conjecture that this theorem can be established at least for semidirect products which are defined by an action of a group on a hyper group, also for general hyper groups with supernormal stability hyper groups. In the case of semi direct products a paper jointly written with S. Kawakami is almost complete [3]. In order to enrich the repertoire of interesting examples it is intended to study the character Hyper group, i.e. the dual, of the discrete Mautner group [4]. Clearly, such a study is narrowly related to the theory of induced representation and duality of nonbelief locally compact groups.

• DFG Programme: Research Grants
• International Connection: Japan
• Participating Person: Professor Dr. Satoshi Kawakami