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 >