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Journal of Lie Theory 20 (2010), No. 2, 215--252
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 Vandoeuvre-lès-Nancy, France

Michael Kinyon
Department of Mathematics, University of Denver, 2360 S Gaylord Street, Denver, CO 80208, U.S.A.

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

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