Composition algebras and PG(m, 2)

Abstract

Neil A. Gordon, Trevor M. Jarvis, Johannes G. Maks, Ron Shaw The multiplication law for the non-associative algebra of Cayley numbers can be expressed (in a suitable basis) in the form exey -- (-1)f(x'Y)ex+y, x, y E V, where V = V(3,2) denotes the 3-dimensional vector space over GF(2) = $~ = {0,1}. All i=2-valued functions f which give rise to a Cayley algebra are determined, as are those which give rise to the algebra of split octonions. In the case of Cayley numbers the invariance group of f is a flag-transitive subgroup of GL(3,2) which is isomorphic to ~7 >~ ~3. In the course of some more general considerations a new proof of Hurwitz's theorem is obtained. 1. INTRODUCTION Recall that a real octonion algebra is a real composition algebra of dimension eight, that is, a real algebra ~4 with unit and equipped with a non-degenerate quadratic form Q such that Q(ala2) = Q(al)Q(a2) for all ai E Jg. Up to isomorphism there are only two such algebras: the Cayley octonion algebra of signature (8,0) and the split octonion algebra of neutral signature (4,4). Let V(3,2) denote a vector space of dimension three over the Galois field GF(2) = I= 2. It is well-known that V(3,2) may be used to label an orthonormal basis {ex} of the Cayley algebra such that exey = ~- ex§ for all x,y e V(3,2): see Freudenthal [1] and Bourbaki [2], Ch.III, Appendix, Prop.4, p.616. We adopt a projective geometry point of view by considering the seven non-zero elements of V(3,2) to be the seven points in the Fano plane PG(2,2). Three points x,y,z E PG(2,2) are collinear if and only if x + y + z = 0. Each line contains three points and, dually, through each point pass exactly three lines.