Point-line spaces related to Jordan pairs

A point-line space is an abstract geometric object that consists of a set of points and a set of lines such that on each line there are at least two points. A large class of point-line spaces with high symmetry comes along with buildings, combinatorial objects that are introduced by Jacques Tits and help to study algebraic objects with geometric methods. To formulate quantum mechanics as abstract and general as possible, the physicist Pascual Jordan invented a non-associative algebraic structure which is now called Jordan algebra. A generalisation of Jordan algebras are the so-called Jordan pairs. In 1975, Ottmar Loos classified all Jordan pairs with finite dimension. As a matter of fact, the list of Ottmar Loos matches in a certain way a part of the list of types of buildings given by Jacques Tits. The buildings of the types that correspond to the types of the Jordan pairs provide a class of point-line spaces. This class consists of two exceptional types and an infinite number of types that occur in four series of increasing dimension. There is a natural way to enlarge these series to the cases of infinite dimension. Together with the two exceptional types this is the class of point-line spaces that we consider to be the point-line spaces related to Jordan pairs of arbitrary dimension. The present work gives a characterisation of point-line spaces that determines exactly this class and uses four rather simple axioms. Moreover, we give a full classification of the point-line spaces satisfying these axioms and prove that it is the class mentioned above.

Jaar van publicatie:2009

Toegankelijkheid:Open