Orthomodular Lattices of Subspaces Obtained from Quadratic Forms |
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Authors: | J. C. Carréga R. Mayet |
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Affiliation: | 1. Institut Girard Desargues, UMR 5028 du CNRS, Université Lyon 1, 69622, Villeurbanne Cedex, France
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Abstract: | Being given a field K of characteristic different from 2 and 3, a 3-dimensional vector space E over K, and a nonsingular symmetric bilinear form φ over E, we define a structure of orthomodular lattice T(E,φ) on the set of all nonisotropic subspaces of E. We give a structure Theorem about the irreducible and 3-homogeneous subalgebras of T(E,φ). In particular, these subalgebras are all of the form T(E',φ ') where E' is a 3-dimensional subspace of E, if E is regarded as a vector space over a subfield K' of K, and φ ' is induced by φ. This structure Theorem allows us to achieve an old project, concerning minimal orthomodular lattices (an orthomodular lattice L is called minimal if it is nonmodular and if all its proper subalgebras are either modular, or isomorphic to L). |
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