Association schemes and hypergroups

In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two interesting classes of hypergroups (partition hypergroups and linearly ordered hypergroups) are realizable. Along the way, we prove that a certain class of projective geometries is equipped with a canonical association scheme structure which allows us to link three objects; association schemes, hypergroups, and projective geometries (see, Section 1.2 for details).     

Sine functions on hypergroups

In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.     

Amenability notions of hypergroups and some applications to locally compact groups

Different notions of amenability on hypergroups and their relations are studied. Developing Leptin's theorem for discrete hypergroups, we characterize the existence of a bounded approximate identity for hypergroup Fourier algebras. We study the Leptin condition for discrete hypergroups derived from the representation theory of some classes of compact groups. Studying amenability of the hypergroup algebras for discrete commutative hypergroups, we obtain some results on amenability properties of some central Banach algebras on compact and discrete groups.     

Ramsey theory for hypergroups

Semigroup Forum - In this paper, Ramsey theory for discrete hypergroups is introduced with emphasis on polynomial hypergroups, discrete orbit hypergroups and hypergroup deformations of semigroups....     

Hypergroupes de typeC

Some results on the right hypergroups of typeC are given. This class of hypergroups containD-hypergroups and cogroups (Eaton and Utumi). Connections between groups orD-hypergroups and right hypergroups of typeC are studied. Lastly all right hypergroups of typeC of cordinality smaller than five are determined (they are allD-hypergroups). Travail accompli avec l’aide du M.P.I.     

Harmonic functions on hypergroups

We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.     

On Sperner (semi)hypergroups

In this paper using Sperner family we introduce and study a new class of (semi)hypergroups that we call the class of Sperner (semi)hypergroups. We characterize strongly Sperner semihypergroups of order 3 and the Sperner Rosenberg hypergroups of order 3.     

Laws of large numbers for polynomial hypergroups and some applications

There are introduced moments on polynomial hypergroups. These moments are used to prove strong laws of large number (SSLLNs) for random walks on the nonnegative integers that are homogeneous with respect to a polynomial hypergroup where SLLNs of different kind appear for polynomial hypergroups thth different properties. Furthermore, we discuss polynomial hypergroups that are associated with some discrete semigroups in a canonical way, and, using SLLNs for polynomial hypergroups, we get SLLNs for isotropic random walks on some discrete semigroups.     

On the construction of hypergroups

We give a method for construction of finite, abelian hypergroups of matrices which, while including most previously known cases, also gives rise to a new family of hypergroups.     

On the construction of hypergroups

We give a method for construction of finite, abelian hypergroups of matrices which, while including most previously known cases, also gives rise to a new family of hypergroups.     

On the γ-Cyclic Hypergroups

The class of γ-complete hypergroups and γ-cyclic hypergroups is introduced. Several properties and examples are found.     

Hypergroups and polygroups in diffeology

Abstract

We introduce diffeological algebraic hyperstructures such as diffeological hypergroups and diffeological polygroups, which are generalizations of diffeological groups. After providing some examples of these notions, we investigate the relationships between diffeological hypergroups and diffeological groups. It is proved that there is an equivalence of categories between subductions over diffeological groups and diffeological complete hypergroups. We finally show how every diffeological polygroup is a subpolygroup of a diffeological poly-monoid of hyper-diffeomorphisms.

Communicated by J. L. Gomez Pardo     

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ?,?,? were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.     

On the dual of a commutative signed hypergroup

Signed hypergroups are convolution structures similar to hypergroups, though being not necessarily positivity-preserving. We prove a generalized Plancherel theorem for positive definite measures on a commutative signed hypergroup, with an analogue of the classical Plancherel theorem as a special case. Moreover, signed hypergroups with subexponential growth are studied. As an application, the dual of the Laguerre convolution structure on ℝ₊ is determined.     

Fourier algebra of a hypergroup – II. Spherical hypergroups

We in this article, introduce a class of hypergroups called ultraspherical hypergroups and show that the Fourier space of an ultraspherical hypergroup forms a Banach algebra under pointwise product. These hypergroups need not be commutative and include for example double coset hypergroups. We also show that the structure space of this algebra equals the underlying hypergroup. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sine functions on compact commutative hypergroups

Recently, Fechner and Székelyhidi introduced sine functions on hypergroups. They conjectured that on a compact hypergroup, all sine functions are constant zero. We prove this conjecture for compact commutative hypergroups by Fourier analysis.     

Duality and Entropy for Finite Commutative Hypergroups and Fusion Rule Algebras

We develop a theory of harmonic analysis and duality for finitecommutative hypergroups by considering somewhat more generalobjects called signed hypergroups. A notion of entropy is defined,and a Second Law of Thermodynamics is established. Applicationsto group theory and to the fusion rule algebras of conformalfield theory are given.     

On the γ n -complete hypergroups and K H hypergroups

The class of γn-complete hypergroups is introduced. Several properties and examples are found both of γn-complete hypergroups and of KH hypergroups.     

Algebraic quantum hypergroups

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman.We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.     

Continuous Gabor Transform for Strong Hypergroups

We define a continuous Gabor transform for strong hypergroups and prove a Plancherel formula, an L ² inversion formula and an uncertainty principle for it. As an example, we show how these techniques apply to the Bessel–Kingman hypergroups and to the dual Jacobi polynomial hypergroups. These examples have an interpretation in the setting of radial functions on R ^d and zonal functions on compact two-point homogeneous spaces, where they provide a new transform which possesses many properties of the classical Gabor transform.