Structure of commutative hypergroups

交换超群的结构

• 批准号: 234833369
• 负责人: Professor Dr. Herbert Heyer (†)
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/234833369?language=en
• 关键词: Structure; commutative; hypergroups

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.

超群是局部紧空间，其上的有界测度卷积类似于局部紧群上的有界测度卷积。自1975年以来，存在一种公理化的方法来研究超群理论。超群的例子是由Gelfand对导出的双陪集空间，更具体的例子是卷积分别由正交多项式或特殊函数定义的非负整数或非负真实的数。超群结构有着广泛的应用，从二阶微分方程理论（Sturm-Liouville问题）到概率论（随机游动的性质）。关于该理论的全面论述，参见申请人的专著[1]和[2]。这里要描述的研究计划的中心主题是超群的扩展，即从已知的超群中产生新的超群的野心。这样人们就希望得到一个超群的结构理论。有条理地成功的第一个想法是将群的上同调理论推广到超群，特别是推广了G. W。Mackey的上循环理论在申请人与他的同事S.来自奈良教育大学的Kawakami已经成功地处理了扩展问题（参考列表I和II）。对于庞特里亚金超群的特殊类，已经获得了令人惊讶的结果（参考文献列表II的[HK 1]）。在交换超群的上同调理论的方向上的第一步已经成功（参考文献列表II的[HK 24]）。本申请所涉及的项目涉及交换超群的不可约性定理。有人猜想，这个定理至少对由群在超群上的作用定义的半直积成立，对具有超正规稳定性超群的一般超群也成立。在半直积的情形下，与S. Kawakami几乎完成了[3]。为了丰富有趣的例子的剧目，它是打算研究字符超群，即对偶，离散Mautner群[4]。显然，这样的研究是狭义相关的理论诱导表示和对偶的nonbelief局部紧群。