Study of noncommutative real Jordan division algebras of dimension 8 whose Lie algebra of derivations is nontrivial
Milnor and Bott and, independently, Kervaire proved that every finite dimensional division algebra over the reals must have dimension 1, 2, 4 or 8. The algebras of the reals, complex, quaternions and octonions are the classical examples. All these algebras are alternative and exhibit many symmetries, which results for the quaternions and octonions in ``large'' Lie algebras of derivations: \(su (2)\) and compact \(G_2\) respectively. \textit{G. M. Benkart} and \textit{J. M. Osborn} [Am. J. Math. 103, No. 6, 1135-1150 (1981; Zbl 0474.17002)] and [Pac. J. Math. 96, 265-300 (1981; Zbl 0474.17003)] studied what the different possibilities for the Lie algebra of derivations of a finite dimensional real division algebra are, and also gave a list of the possibilities for the structure of the algebra as a module for its Lie algebra of derivations. The existence of examples in some cases was left open. In a subsequent paper, \textit{G. M. Benkart}, \textit{D. Britten} and \textit{J. M. Osborn} [Can. J. Math. 34, No. 3, 550-588 (1982; Zbl 0487.17002)] studied the possibilities for the finite dimensional flexible (that is, \((xy)x = x(yx)\;\forall x,y)\) real division algebras. In dimension 8, these algebras were shown to be either ``scalar isotopes'' of quadratic and flexible real division algebras or ``generalized pseudo-octonion algebras''. From these results, it follows that the noncommutative real Jordan division algebras of dimension 8 are exactly the real quadratic flexible division algebras. The paper under review is devoted to studying this latter class of eight dimensional real division algebras under the hypothesis of the existence of a nonzero derivation or a nontrivial automorphism. The algebras with a nonzero derivation are described in terms of seven scalar parameters and this is drastically improved if large derivation algebras are imposed. Also examples are provided for one of the possibilities left open by Benkart and Osborn \((su (2)\) as derivation algebra and decomposition ``1 + 1 + 3 + 3'' as a module for \(su (2))\). The description of quadratic algebras is decomposed in a scalar and a vector part and a construction of quadratic flexible algebras which generalizes the classical Cayley-Dickson doubling process plays a key role in the paper. It must be remarked that all the possibilities left open by Benkart and Osborn do appear with just one exception, and that in all cases there are examples which are nonunital composition algebras, as has been recently shown by \textit{José María Pérez} [Composition algebras (Spanish); Doctoral dissertation, University of Zaragoza, 1996].
scientific article; zbMATH DE number 840733