Journal of Lie Theory 20 (2010), No. 2, 253--282
Copyright Heldermann Verlag 2010
Associative Geometries. II: Involutions, the Classical Torsors, and their Homotopes
Wolfgang Bertram
Institut Élie Cartan, Université de Nancy, Boulevard des Aiguillettes, B. P. 239, 54506 Vandoeuvre-lès-Nancy, France
bertram@iecn.u-nancy.fr
Michael Kinyon
Department of Mathematics, University of Denver, 2360 S Gaylord Street, Denver, CO 80208, U.S.A.
mkinyon@math.du.edu
For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called "homotopes". The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion.
Keywords: Classical groups, homotope, associative triple systems, semigroup completion, involution, linear relation, adjoint relation, complemented lattice, orthocomplementation, generalized projection, torsor.
MSC: 20N10, 17C37, 16W10
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