# Jordan- and Lie geometries

## Wolfgang Bertram

Address: Institut Élie Cartan de Lorraine (IECL), Université de Lorraine at Nancy, CNRS, INRIA, Boulevard des Aiguillettes, B.P. 239, F-54506 Vandœuvre-lès-Nancy, France

E-mail: wolfgang.bertram@univ-lorraine.fr

Abstract: 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 knowledge of Lie theory – we give complete definitions and explain the results by presenting examples, such as Grassmannian geometries.

AMSclassification: primary 16W10; secondary 17C37, 20N10, 22A30, 51B25, 51P05, 81P05.

Keywords: Jordan algebra (triple system, pair), associative algebra (triple systems, pair), Lie algebra (triple system), graded Lie algebra, symmetric space, torsor (heap, groud, principal homogeneous space), homotopy and isotopy, Grassmannian, generalized projective geometry.

DOI: 10.5817/AM2013-5-275