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

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 G₂. 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 G₂ 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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