Publisher Summary
The focus of this chapter is on chain geometries. In a projective geometry over a ring R, it makes no sense to intrinsically consider a projective line. Nevertheless, if R is a K-algebra, then on the projective line P1(R) there is an interesting incidence structure, the blocks of which are the K-sublines embedded into P1(R) in a canonical way, and this gives the chain geometry over R. The notion of chain geometries arose from efforts to unify the treatment of such different geometries as the geometry of Mobius (lines and circles of the Euclidean plane), of Laguerre–Lie (spears and cycles), and of Minkowski (the pseudo-Euclidean plane with its hyperbolas as circles). The chain geometry Σ(K, R) is called Laguerre geometry provided the parallel relation is an equivalence relation on P (R) and every chain meets every parallel class of points. A weak chain space is an incidence structure Σ = (P, C). The blocks of Σ are called chains and two different points are called distant provided they are incident with a common chain.

Chapter 14 – Chain Geometries

In this paper, we bring together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them are rare and are scattered over the literature. Our main goal is to present the basics from either side, to develop, or at least sketch, the principal connections between them, and to make recommendations for further reading.

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Divisible designs, Laguerre geometry, and beyond

The main result of the present paper is that the projective line over a ring R is con- nected with respect to the relation ''distant'' if, and only if, R is a GE2-ring.

The connected components of the projective line over a ring

Jordan homomorphisms connect directly with the structure of certain nonassociative rings including the classical Jordan algebras. The concept of a Jordan homomorphism generalizes in a natural way the ones of homomorphism and antihomomorphism of rings: in fact, by a well-known theorem of HUA'S [7], Jordan homomorphisms of (non-necessarily commutative) fields, and more generally Jordan homomorphisms onto prime rings [5], have to be homomorphisms or anti-homomorphisms of rings. Jordan homomorphisms also have an interest in Projective Geometry, where they can describe homomorphisms of geometric structures: to be exact, any Jordan homomorphism of an arbitrary field F, hence any automorphism or anti-automorphism of F, extends to a permutation of points of the underlying projective line by a direct action on the coordinates of any point. It is well-known that the induced permutation preserves the intrinsic geometric structure of the line. This structure is either the linear one inherited from an embedding into a projective space of higher dimension, in characteristic =~2 we then have harmonic quadruples (see [1], pp. 71--93), or is the non-linear one determined by a suitable subfield K, and we have a geometry of chains, a M6bius plane if [F : K] = 2 and F is commutative (see [4]). In recent years many authors, following a suggestion of R. BAER, have tried to verify whether this situation also occurs or not when passing to the more general case of an arbitrary (associative) ring, or of an arbitrary K-algebra for chain geometries. To verify that a Jordan homomorphism induces in the underlying projective line a well-defined mapping is then very difficult. Nevertheless for the case of a local ring, where the points have a very simple representation, the result has been achieved in [8], for the linear case, and in [6] for the non-linear case (here only for finite dimensional local K-algebras, K in the centre of the algebras). However, it is possible that, in the general case, there exist examples of Jordan homomorphisms which do not preserve the linear structure of the projective line: in [2], p. 367, we can infer examples which do not preserve a structure stronger than the linear one, defined by generalized harmonic quadruples. On the other hand, a Jordan homomorphism of finite dimensional K-algebras, containing K in their centre, has to be K-semilinear if it induces an homomorphism of the underlying chain geometries [6]. This is not true, of course, if K is not central,

Jordan homomorphisms, chain geometries and the fundamental theorem

This paper is devoted to the study of translation planes of order q2 and kernel GF(q) that admit a collineation group of order q2 in the linear translation complement We give a representation of this group by a suitable set of matrices depending on some functions over GF(q) Using this representation we obtain several results concerning the existence and the collineation group of the plane

A structure theory for two-dimensional translation planes of order q 2 that admit collineation groups of order q 2

E. ARTIN (1957), R. BAER (1952) and J.DIEUDONNE (1951) emphasized many times the structural identity between classical projective geometry and linear algebra over a division ring. Then Baer pointed out a possible extension of this structural identity to the case of a ring, generating intense research activity in the area of geometric algebra over rings. In this direction the most significant results appear: Ojanguren and Sridharan’s article on the fundamental theorem of projective geometry over a commutative ring; some theorems due to Klingenberg, Bass and Suslin on the structure of the general linear group over appropriate rings and some valuable notes of O’Meara’s on the automorphisms of linear groups over an integral domain.

Topics in Geometric Algebra over Rings

Nilpotent Lie algebras with 2-dimensional commutator ideals

A remark on the projectivities of the projective line over a commutative ring

Solvable Extensions of Nilpotent Complex Lie Algebras of Type 2n,1,1

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