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The non-orthogonal Cayley–Dickson construction and the octonionic structure of the $E_{8}$ $E_{8}$ -lattice

Holger P. Petersson

Fakultät für Mathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany

E-mail Address: holger.petersson@fernuni-hagen.de

https://doi.org/10.1142/S0219498817502309Cited by:0 (Source: Crossref)

Abstract

Using a conic $(= degree - 2)$ $(= degree - 2)$ algebra $B$ $B$ over an arbitrary commutative ring, a scalar $μ$ $μ$ and a linear form $s$ $s$ on $B$ $B$ as input, the non-orthogonal Cayley–Dickson construction produces a conic algebra $C : = Cay (B; μ, s)$ $C : = Cay (B; μ, s)$ and collapses to the standard (orthogonal) Cayley–Dickson construction for $s = 0$ $s = 0$ . Conditions on $B, μ, s$ $B, μ, s$ that are necessary and sufficient for $C$ $C$ to satisfy various algebraic properties (like associativity or alternativity) are derived. Sufficient conditions guaranteeing non-singularity of $C$ $C$ even if $B$ $B$ 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.

Communicated by I. Shestakov

Meirem Bruder, Jörn P. Petersson, zur Vollendung des 80. Lebensjahres gewidmet

Keywords:

AMSC: 17A45, 17A75, 11E12

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