Projective Planes, Graphs, and Simple Algebras Original Research Article

Let X be a projective plane of order n. A flag x0 set membership, variant l0 in X determines a graph F which is used to construct a central simple Jordan algebra A of dimension n3 − 1 over a field k of characteristic 2 with x2 = 0 for all x set membership, variant A. When n is even, the group of all algebra automorphisms of A is isomorphic to the symplectic group Sp(n3, k). We also imbed every finite connected graph G whose adjacency matrix is non-singular mod 2 in a strongly regular graph G*, and we construct a central simple Lie algebra over image/2image from G*; in particular, F gives a central simple Lie algebra.