In these lecture notes we report on research aiming at understanding the relation beween algebras and geometries, by focusing on the classes of Jordan algebraic and of associative structures and comparing them with Lie structures. The geometric object sought for, called a generalized projective, resp. an associative geometry, can be seen as a combination of the structure of a symmetric space, resp. of a Lie group, with the one of a projective geometry. The text is designed for readers having basic...

Dans cet article, en utilisant les algèbres de Jordan euclidiennes, nous étudions l’espace de Hardy $H^2\left(\Xi \right)$ d’un espace symétrique de type Cayley $ℳ=G/H$. Nous montrons que le noyau de Cauchy-Szegö de $H^2\left(\Xi \right)$ s’exprime comme somme d’une série faisant intervenir la fonction $c$ de Harish-Chandra de l’espace symétrique riemannien $D=G/K$, la fonction $c$ de l’espace symétrique $c$-dual $𝒩$ de $ℳ$ et les fonctions sphériques de l’espace symétrique ordonné $𝒩$. Nous établissons, dans le cas où la dimension de l’algèbre de Jordan associée...

We characterize an important class of generalized projective geometries $(X,X^{\text{'}})$ by the following essentially equivalent properties: (1) $(X,X^{\text{'}})$ admits a central null-system; (2) $(X,X^{\text{'}})$ admits inner polarities: (3) $(X,X^{\text{'}})$ is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s...