Affine Reductive Spaces of Small Dimension and Left A-Loops

In this paper we determine the at least 4-dimensional affine reductive homogeneous manifolds for an at most 9-dimensional simple Lie group or an at most 6-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable left A-loops. Using this we classify all global almost differentiable left A-loops L having either a 6-dimensional semi-simple Lie group or the group as the group topologically generated by their left translations. Moreover, we determine all at most 5-dimensional left A-loops L with as the group topologically generated by their left translations.     

Bol loops with non-normal nuclei

The purpose of this paper is to expand the collection of Bol loops that have nuclei that are not normal. In doing so, we will show that for each of the Bol loops given by Daniel and Karl Robinson in [2] there are uncountably many more. Clearly this will require us to give examples of infinite Bol loops with non-normal nuclei, which until now have apparently been absent from the literature.Received: 20 August 2001     

Subloop incompatible Bol loops

We give a necessary modification of Proposition 1.18 in Nagy and Strambach (Loops in Group Theory and Lie Theory. de Gruyter Expositions in Mathematics Berlin, New York, 2002) and close the gap in the classification of differentiable Bol loops given in Figula (Manuscrp Math 121:367–385, 2006). Moreover, using the factorization of Lie groups we determine the simple differentiable proper Bol loops L having the direct product G ₁ × G ₂ of two groups with simple Lie algebras as the group topologically generated by their left translations such that the stabilizer of the identity element of L is the direct product H ₁ × H ₂ with H _i < G _i . Also if G ₁ = G ₂ = G is a simple permutation group containing a sharply transitive subgroup A, then an analogous construction yields a simple proper Bol loop. If A is cyclic and G is finite and primitive, then all such loops are classified.     

The varieties of loops of Bol-Moufang type

A loop identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, viz. ((xy)x)z = x(y(xz)). Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varieties, and determine all inclusions between them, providing all necessary counterexamples, too. This extends and completes the programme of Fenyves [Fe69]. Received October 23, 2003; accepted in final form April 12, 2005.     

A class of simple proper Bol loops

The existence of finite simple non-Moufang Bol loops has long been considered to be one of the main open problems in the theory of loops and quasigroups. In this paper, we present a class of simple proper Bol loops. This class contains finite and new infinite simple proper Bol loops. This paper was written during the author’s Marie Curie Fellowship MEIF-CT-2006-041105 at the University of Würzburg (Germany).     

Loops on spheres having a compact-free inner mapping group

We prove that any topological loop homeomorphic to a sphere or to a real projective space and having a compact-free Lie group as the inner mapping group is homeomorphic to the circle. Moreover, we classify the differentiable 1-dimensional compact loops explicitly using the theory of Fourier series. Authors’ addresses: ágota Figula, Mathematisches Institut der Universit?t Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany and Institute of Mathematics, University of Debrecen, P.O.B. 12, H-4010 Debrecen, Hungary; Karl Strambach, Mathematisches Institut der Universit?t Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany     

On the extension of involutorial Bol loops

The group theoretical problem of the existence of a system of representativesT of the subgroup H of G such that T consists of conjugacy classes of involutions leads to the theory of Bol loops of exponent 2. In this paper, we develop a theory of extensions of such loops and give two applications of the theory. First, we classify all (left) Bol loops of exponent 2 of order 16; second, we classify all Bol loops of exponent 2 whose right nucleus has index 2. In particular, we give a class of examples of non-nilpotent such Bol loops. The second author was supported by the “János Bolyai” Fellowship, the Blaschke Stiftung and the OTKA grants F030737, T029849.     

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.     

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.     

The Campbell- Hausdorff Series of Local Analytic Bruck Loops

The class of local analyitic Bruck loops (or equivalently K-loops) is strongly related to locally symmetric spaces. In particular, both have Lie triple systems as their tangent algebra. In this paper, we consider the existence and some properties of the Campbell-Hausdorff series of local analytic Bruck loops (K-loops). This formula can be used to determine the local symmetries of the associated symmetric space.     

Classification of homogeneous almost cosymplectic three-manifolds

The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R×N, where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map.     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

Homogeneous Geodesics of Three-Dimensional Unimodular Lorentzian Lie Groups

We consider three-dimensional unimodular Lie groups equipped with a Lorentzian metric and we determine, for all of them, their sets of homogeneous geodesics through a point. Dedicated to the memory of Professor Aldo Cossu Authors supported by funds of M.U.R.S.T., G.N.S.A.G.A. and the University of Lecce.     

Finite Bruck loops

Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops , showing that is essentially the direct product of a Bruck loop of odd order with a -element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite -element Bruck loops are -loops, leaving open the question of whether such obstructions actually exist.

     

On Ricci negative solvmanifolds and their nilradicals

In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some applications.     

A remark on left invariant metrics on compact Lie groups

We show that a left invariant metric on a compact Lie group G with Lie algebra has some negative sectional curvature if it is obtained by enlarging a biinvariant metric on a subalgebra , unless the semi-simple part of is an ideal of This answers a question raised in [8]. Received: 7 May 2007     

Harmonic morphisms from the compact semisimple Lie groups and their non-compact duals

In this paper we prove the local existence of complex-valued harmonic morphisms from any compact semisimple Lie group and their non-compact duals. These include all Riemannian symmetric spaces of types II and IV. We produce a variety of concrete harmonic morphisms from the classical compact simple Lie groups SO(n), SU(n), Sp(n) and globally defined solutions on their non-compact duals SO(n,C)/SO(n), SLn(C)/SU(n) and Sp(n,C)/Sp(n).     

On the connection theory of extensions of transitive Lie groupoids

Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for the classification of their respective Lie algebroids.