Geometries over Universal Left Conjugacy Closed Quasifields

A class of examples of loops with universal left conjugacy property defined in the Canadian Journal of Mathematics (Vol. 46 (1994), pp. 1027--1056) is provided by proving that finite André systems have universal left conjugacy closed multiplicative loops. Then quasifields, in which the multiplicative loop has this property are characterized geometrically.     

Semi-simple groups of compact 16-dimensional planes

We determine all semi-simple automorphism groups of dimension at least 17 of a locally compact 16-dimensional affine plane without fixed elements.     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

Three-Dimensional Topological Loops with Solvable Multiplication Groups

We prove that each 3-dimensional connected topological loop L having a solvable Lie group of dimension ≤5 as the multiplication group of L is centrally nilpotent of class 2. Moreover, we classify the solvable non-nilpotent Lie groups G which are multiplication groups for 3-dimensional simply connected topological loops L and dim G ≤ 5. These groups are direct products of proper connected Lie groups and have dimension 5. We find also the inner mapping groups of L.     

Γ-amenability of locally compact groups

Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to F-amenable groups and give various characterizations for F-amenable groups. A sequence of characterizations of F-amenable groups is given here by developing the well-known Flner＇s conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.     

Isotopy and geotopy for ternary rings of projective planes

We investigate the question of when two ternary rings coordinatize the same finite projective plane. Necessary and sufficient conditions are obtained for right quasifields and for right distributive linear rings whose multiplicative loop is a group.     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified. Received 10 February 1997; in final form 19 December 1997     

Almost disjoint families of connected sets

We investigate the question which (separable metrizable) spaces have a ‘large’ almost disjoint family of connected (and locally connected) sets. Every compact space of dimension at least 2 as well as all compact spaces containing an ‘uncountable star’ have such a family. Our results show that the situation for 1-dimensional compacta is unclear.     

g-Quasikörper

A generalization of a construction for finite quasifields by Foulser is given, which yields all finite Dickson-nearfields. This constuction is then used to obtain locally finite quasifields as unions of chains of finite g-quasifields.     

Compact projective planes with homogeneous ovals

Closed ovals exist only in 2-or 4-dimensional compact projective planes. We show that a plane of dimension 2 or 4 contains ahomogeneous closed oval iff the automorphism group contains SO₂ or SO₃, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.     

On some d-dimensional dual hyperovals in

In [H. Taniguchi, On d-dimensional dual hyperovals in PG(2d,2), Innov. Incidence Geom., in press], we construct d-dimensional dual hyperovals in PG(2d,2) from quasifields of characteristic 2. In this note, we show that, if d-dimensional dual hyperovals in PG(2d,2) constructed from nearfields are isomorphic, then those nearfields are isomorphic. Some results on dual hyperovals constructed from quasifields are also proved.     

Henstock type integral in harmonic analysis on zero-dimensional groups

A Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform.     

Loops which are semidirect products of groups

We construct loops which are semidirect products of groups of affinities. As their elements in many cases one may take transversal subspaces of an affine space. In particular we obtain in this manner smooth loops having Lie groups of affine real transformations as the groups generated by left translations, whereas the groups generated by right translations are smooth groups of infinite dimension. We also determine the Akivis algebras of these loops. This paper was supported by DAAD.     

Linear groups as right multiplication groups of quasifields

For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of finite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the isomorphism of the corresponding translation planes.     

A simple definition for locally compact quantum groups

In this Note we propose a simple definition of a locally compact quantum group in reduced form. By the word “reduced” we mean that we suppose the Haar weight to be faithful, and hence we define in fact arbitrary locally compact quantum groups represented on the L²-space of the Haar weight. We construct the multiplicative unitary associated with our quantum group. We construct the antipode with its polar decomposition, and the modular element. We prove the unicity of the Haar weights, define the dual and prove a Pontryagin duality theorem.     

Unitaires multiplicatifs commutatifs

In a previous article, we proved that every commutative multiplicative unitary is a multiple of the multiplicative unitary associated to a locally compact group. In the present Note we give a simpler proof of this generalization of a theorem of Weil. To cite this article: S. Baaj, G. Skandalis, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Some Minkowski planes with 3-dimensional automorphism group

A new rather large family of 2-dimensional locally compact topological Minkowski planes with an at least 3-dimensional automorphism group is introduced here. Isomorphism classes and automorphisms of these planes are determined.     

The spaces of meromorphic Prym differentials on a finite Riemann surface

In the previous articles the second author started constructing a general theory of multiplicative functions and Prym differentials on a compact Riemann surface for arbitrary characters. Function theory on compact Riemann surfaces differs substantially from that on finite Riemann surfaces. In this article we start constructing a general function theory on variable finite Riemann surfaces for multiplicative meromorphic functions and differentials. We construct the forms of all elementary Prym differentials for arbitrary characters and find the dimensions of, and also construct explicit bases for, two important quotient spaces of Prym differentials. This yields the dimension of and a basis for the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters.     

Kurzweil-Henstock type integral on zero-dimensional group and some of its applications

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case. Supported by RFFI-08-01-00669.     

On Discrete Zariski-Dense Subgroups of Algebraic Groups

We investigate under which conditions an algebraic group G defined over a locally compact field k admitr a subgroup Γ? G(k) which is dense in the Zariski topology, but discerte in the topology induced by the locally compact topology on k. For non—solvable groups we provide a complete answer.