Derivations and Cayley derivations of generalized Cayley-Dickson algebras (Q1073160)

The author studies the derivation algebras derived from the infinite series of algebras derived from the Cayley-Dickson process. The author has extended the work of R. D. Schafer which restricted the scalars to be a field of characteristic \(\neq 2,3\). This paper allows arbitrary rings of scalars. Let A be an algebra with involution *. A *-module is a bimodule M consisting of *-elements m such that \(am=ma*\) for all a in A. A Cayley derivation of A into a *-module M is a linear map C such that \(C(xy)=C(x)y*+C(y)x\). The author shows how the derivations of C(A,\(\mu)\), (an algebra generated by the Cayley-Dickson construction), are built out of derivations, Cayley derivations, and skew nuclear elements of A. The author then shows how the Cayley derivations of C(A,\(\mu)\) are built out of the Cayley derivations and anti-derivations of A. In most cases of dimension \(\geq 8\) there are no Cayley derivations at all.