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     

Moufang loops with commuting inner mappings

We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most 2. The 6-divisible Moufang loops with commuting inner mappings have nilpotency class at most 2. There is a Moufang loop of order 2¹⁴ with commuting inner mappings and of nilpotency class 3.     

A note on Bol loops of order 2 n k

It is shown that, for eachn 2 and k 3, there exist at least 2 ⁿ -3 non-isomorphic loops of order 2 ⁿ k which are Bol but not Moufang. In most cases this bound can be improved.     

A note on the solvability of finite Moufang loops

In this short note we show that finite Moufang loops with nilpotent inner mapping groups are solvable. Received: 24 May 2006     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.     

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.     

A class of loops with right alternative loop rings

The purpose of this paper is to exhibit a class of loops which have strongly right alternative loop rings that are not alternative..     

The existence of free products in existence varieties of regular semigroups

It is shown that an existence varietyV of regular semigroups contains (all) free products if and only ifV consists solely of locally inverse orE-solid semigroups.Presented by B. M. Schein.The author is indebted to the Australian Research Council for financial support (ARC grant A69231516).     

The construction of orthogonalk-skeins and latink-cubes

Aequationes mathematicae -     

Secant varieties of toric varieties

Let X_P be a smooth projective toric variety of dimension n embedded in P^r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety . We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties X_A embedded using a set of lattice points A⊂P∩Zⁿ containing the vertices of P and their nearest neighbors.     

Some results about the projective normality of abelian varieties

We reduce the problem of the projective normality of polarized abelian varieties to check the rank of very explicit matrices. This allows us to prove some results on normal generation of primitive line bundles on abelian threefolds and fourfolds. We also give two situations where the projective normality always fails. Finally we make some conjecture. Received: 1 September 2004; revised: 10 March 2005     

On syzygies of projective bundles over abelian varieties

Syzygies or $N_p$-property of an ample line bundles on abelian varieties are well known. In this paper, we study defining equations and syzygies among them of projective bundles over abelian varieties. We prove an analogue of Pareschi's theorem (or Lazarsfeld's conjecture) on abelian varieties, extended to projective bundles over an abelian variety.     

Special inner projections of projective varieties

LetX⊃P ^r be a non-degenerate variety and Γ(X) the closure inX of the set of all pointsP∈X such that the projection ofX fromP is not birational. Here we study the irreducible components of Γ(X), using proofs and ideas contained in a paper by Calabri and Ciliberto concerning the outer non birational projections. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Permutation representations of left quasigroups

The concept of a permutation representation has recently been extended from groups to quasigroups. Following a suggestion of Walter Taylor, the concept is now further extended to left quasigroups. The paper surveys the current state of the theory, giving new proofs where necessary to cover the general case of left quasigroups. Both the Burnside Lemma and the Burnside algebra appear in this new context. This paper is dedicated to Walter Taylor. Received August 9, 2005; accepted in final form March 7, 2006.     

Quasigroups,right quasigroups and category coverings

The category of modules over a fixed quasigroup in the category of all quasigroups is equivalent to the category of representations of the fundamental groupoid of the Cayley diagram of the quasigroup in the category of abelian groups. The corresponding equivalent category of coverings, and the generalization to the right quasigroup case, are also described.Dedicated to the memory of Alan DayPresented by J. Sichler.     

Symmetric incidence groupoids

In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.This paper was sponsored by Vigoni Program 1999.     

The representation type of Hecke algebras of type B

This paper determines the representation type of the Iwahori-Hecke algebras of type B when q≠±1. In particular, we show that a single parameter non-semisimple Iwahori-Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno's (J. Algebra 149 (1992) 287).     

Semidirect Product of Loops and Fibrations

Starting from two loops (H, +) and (K, ·), a new loop L can be defined by means of a suitable map Θ : K → Sym H (cf. [3]). Such a loop is called semidirect product of H and K with respect to Θ and denoted by H ×_Θ K =: L. Here we consider the class of those semidirect products in which Θ : K → Aut(H, +) is a homomorphism, this situation being quite akin to the group case. Some relevant algebraic properties of the loop L (Bol condition, Moufang etc.) can be inherited from H and K. In the case that K is a group we investigate the possibility of describing L by a partition (or fibration). In this way we propose a generalization of [8] for the non associative case. Received: September 20, 2007. Revised: November 8, 2007.