The non-orthogonal Cayley-Dickson construction and the octonionic structure of the E8-lattice

Using a conic (= degree 2) algebra B over an arbitrary commutative ring, a scalar in the base ring, and a linear form on B as input, the non-orthogonal Cayley-Dickson construction produces a conic algebra C as output and collapses to the standard (orthogonal) Cayley-Dickson construction if the linear form is zero. Conditions on the input parameters that are necessary and sufficient for C to satisfy various algebraic properties (like associativity or alternativity) are derived. Sufficient conditions guaranteeing non-singularity of C even if B is singular are also given. As an application we show how the algebras of Hurwitz quaternions and of Dickson or Coxeter octonions over the rational integers can be obtained from the non-orthogonal Cayley-Dickson construction.

(To appear in J. of Algebra and its Applications)


H. P. Petersson < holger.petersson@fernuni-hagen.de >