
Journal of Lie Theory 20 (2010), No. 2, 215252 Copyright Heldermann Verlag 2010 Associative Geometries. I: Torsors, Linear Relations and Grassmannians Wolfgang Bertram Institut Élie Cartan, Université de Nancy, Boulevard des Aiguillettes, B. P. 239, 54506 VandoeuvrelèsNancy, France bertram@iecn.unancy.fr Michael Kinyon Department of Mathematics, University of Denver, 2360 S Gaylord Street, Denver, CO 80208, U.S.A. mkinyon@math.du.edu We define and investigate a geometric object, called an "associative geometry", corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in Part II of this work. Keywords: Associative algebras and pairs, torsor, heap, groud, principal homogeneous space, semitorsor, linear relations, homotopy, isotopy, Grassmannian, generalized projective geometry. MSC: 20N10, 17C37, 16W10 [ Fulltextpdf (314 KB)] for subscribers only. 