# The coordination of a hexagonal-barbilian plane by a quadratic jordan algebra

Torrence, Eve Alexandra Littig, Department of Mathematics, University of Virginia

Faulkner, John, Department of Mathematics, University of Virginia

Ward, Harold, Department of Mathematics, University of Virginia

Keller, Gordon E., Department of Mathematics, University of Virginia

In this paper we define a Hexagonal-Barbilian plane which generalizes the notion of both a hexagonal geometry and a Faulkner plane. We define transvections on the plane and show that they form a group with Steinberg relations of type G_{2}. The group of transvections is indexed by a module *J* over a ring with some of the properties of a quadratic Jordan algebra. We also show a purely algebraic result; we use *J* to construct an algebra *G*, then *J* is a quadratic Jordan algebra if and only if *G* is a Lie algebra with certain maps acting as Lie automorphisms.

If *J* is a quadratic Jordan algebra, the Lie automorphisms of *G* form a group with Steinberg relations of type G_{2} and from such a group we can build a Hexagonal-Barbilian plane. The tangent bundle of a Hexagonal-Barbilian plane is defined and we make progress towards showing that the sections of the tangent bundle form a Lie algebra similar to *G*.

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PHD (Doctor of Philosophy)

Jordan algebras, Lie algebras, Geometry, Algebraic

English

All rights reserved (no additional license for public reuse)

1991