Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.This paper was supported by DAAD.     

Loops, their cores and symmetric spaces

This paper is devoted to the relations among affine symmetric spaces, smooth Bol and Moufang loops, smooth left distributive quasigroups and differentiable 3-nets. The results are used to prove the analyticity of smooth Moufang loops and left distributive quasigroups with involutive left translations as well as to show the Lie nature of transformation groups naturally related to some classes of smooth binary systems and 3-nets. In the last section we establish power series expansion for local loops with weak associativity conditions and apply the methods of the previous sections in order to describe geodesic loops having euclidean lines either as their geodesic lines or as geodesic lines of their core. The first author was partly supported by the Deutsche Forschungsgemeinschaft and by OTKA Grant no. T020545.     

Characterizing Translations on Groups by Cosets of Their Subgroups

On a group, constant functions and left translations by group elements map left cosets into left cosets for every subgroup. We determine classes of groups for which this property of preserving cosets characterizes constants and translations, e.g., finite non-abelian groups that are perfect, partitioned, primitive, or generated by elements of prime order p. For certain classes of groups we construct other coset-preserving functions, in particular, power endomorphisms and functions defined in terms of the subgroup lattice.     

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.     

Coxeter–Chein Loops

In 1974, Orin Chein discovered a new family of Moufang loops which are now called Chein loops. Such a loop can be created from any group W together with ?₂ by a variation on a semidirect product. We first settle an open problem, originally proposed by Petr Vojtěchovský in 2003, by finding a minimal presentation for the Chein loop with respect to a presentation for W. We then study these loops in the case where W is a Coxeter group and show that it has what we call a Chein-Coxeter system, a small set of generators of order 2, together with a set of relations closely related to the Coxeter relations and Chein relations. In particular, even if the Moufang loop is infinite, it is finitely presented. Viewing these presentations as amalgams of loops, we then apply methods due to Blok and Hoffman to describe a family of twisted Coxeter–Chein loops.     

Schreier loops

We study systematically the natural generalization of Schreier’s extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes.     

Hall triple systems and commutative Moufang exponent 3 loops: The case of nilpotence class 2

We study here the structure of the coordinatizing loops of Hall Triple Systems which are Steiner Triple Systems, other than affine geometries, in which every plane is an affine plane. These loops are necessarily non-associative commutative Moufang loops of exponent 3 and we concentrate on those which involve the least complicated computations, namely, those having central nilpotence class 2. After presenting introductory material from Bruck's theory of loops we determine precisely the structure of finitely generated commutative Moufang exponent 3 loops of nilpotence class 2 in terms of a very useful representation. We then give a general method of constructing such loops; and since the triple systems can always be recovered from the loops, this enables us to construct many new Hall Triple Systems. These designs are particularly interesting since they are the only known “non-classical” perfect matroid designs.     

Orbit Equivalence and Topological Conjugacy of Affine Actionson Compact Abelian Groups

 We study topological rigidity of affine actions on compact connected metrizable abelian groups. We also classify one parameter flows of translations up to orbit equivalence and discrete group actions by translations up to topological conjugacy. (Received 21 December 1998; in revised form 2 June 1999)     

Left invariant special Kähler structures

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.     

Topological loops with three-dimensional solvable left translation group

We classify all connected topological loops having a three-dimensional solvable Lie group G as the group topologically generated by their left translations. It is surprising that to the non-nilpotent Lie group G having precisely one one-dimensional normal subgroup there are topological but no differentiable strongly left alternative loops.     

Some recent applications of functional equations to geometry

The results of applications of solutions of functional equations or of methods used in the theory of functional equations to the following subjects are discussed in this paper. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations. One-parameter subsemigroups of affine transformations and their homomorphisms. Extensions of homomorphisms from sub-semigroups to groups generated by them. Determination of all collineations on subsets of general projective planes and their extensions to the entire plane.     

On left conjugacy closed loops in which the left multiplication group is normal

A loopQ is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. This papers investigates those LCC loops where the group generated by left translations is normal in the group generated by both left and right translations.     

Simple Bol Loops

In this article we investigate the Bol loops and connected with them groups. We prove an analog of the Doro's theorem for Moufang loops and find a criterion for simplicity of Bol loops. One of the main results obtained is the following: If the right multiplication group of a connected finite Bol loop S is a simple group, then S is a Moufang loop.     

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.     

Kollineationen von Translationsstrukturen

Translationstructures are generalized affine spaces. They can be described algebraically by partitions of groups. For desarguesian affine spaces the group is a vectorspace and the partition is the set of all onedimensional subspaces. In this case each collineation fixing 0 is a regular semilinear mapping, i.e. an automorphism of the vectorspace. In the general case it is a mapping called equivalence. Each equivalence of a partition is an automorphism iff the set of translations of the group is a normal subgroup of the collineationgroup. The translations form a normal subgroup, if the group is finite or abelian. We prove some theorems for the infinite non abelian case.     

On loop extensions satisfying one single identity and cohomology of loops

In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3 Nishigori, N. (1963). On loop extensions of groups and M-cohomology groups I. J. Sci. Hiroshima Univ. Ser. A-I 27:151–165. [Google Scholar]] and of Kenneth and Leedham-Green [2 Kenneth, W. J., Leedham-Green, Ch. R. (1990). Loop cohomology. Czec. Math. J. 40(2):182–194.[Web of Science ®] , [Google Scholar]]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.     

Semiautomorphic inverse property loops

We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and if a is in the commutant, then a² is a Moufang element, a³ is a c-element and a⁶ is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein’s and de Barros and Juriaans’ doubling constructions.     

Polynomial structures for nilpotent groups

If a polycyclic-by-finite rank- group admits a faithful affine representation making it acting on properly discontinuously and with compact quotient, we say that admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups . Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups , affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Malcev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the ``affine defect number'. We prove that the known counterexamples to Milnor's question have the smallest possible affine defect, i.e. affine defect number equal to one.

     

Translations and parallel classes of lines in affine designs

The structure of affine designs admitting all possible translations in one direction is investigated. A method is given for constructing such designs using Cartesian groups, analogous to the known method for affine planes.     

Oversampling and preservation of tightness in affine frames

The problem of how an oversampling of translations affects the bounds of an affine frame has been proposed by Chui and Shi. In particular, they proved that tightness is preserved if the oversampling factor is coprime with the dilation factor. In this paper we study, in the dyadic dilation case, oversampling of translation by factors which do not satisfy the above condition, and prove that tightness is preserved only in the case of affine frames generated by wavelets having frequency support with very particular properties.