Vol. 18, No. 3, 1966

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Über eine Beziehung zwischen Malcev-Algebren und Lietripelsystemen

Ottmar Loos

Vol. 18 (1966), No. 3, 553–562

A Malcev algebra is an anticommutative algebra which satisfies the identity

xy⋅xz = (xy ⋅z)x + (yz ⋅x)x+ (zx⋅x)y.

In this paper we construct to every Malcev algebra A a Lie triple system TA and study the relations between them. A number of properties hold for a Malcev algebra if and only if they hold for the associated Lie triple system. E.g. the algebra A is solvable (semisimple, simple) if and only if TA is. Moreover the radicals of A and TA coincide. We shall prove:

Theorem A. A finite dimensional Malcev algebra A over a field of characteristics zero is semisimple if and only if the Killing form of A is nondegenerate.

Let C be the Cayley algebra over an algebraically closed field of characteristic zero. It has been shown by Sagle, that a simple 7-dimensional Malcev algebra A can be obtained from C. Using a further theorem of Sagle we prove

Theorem B. Every simple finite dimensional non-Lie Malcev algebra over an algebraically closed field of characteristic zero is isomorphic to A.

Mathematical Subject Classification
Primary: 17.60
Secondary: 17.30
Received: 1 May 1965
Published: 1 September 1966
Ottmar Loos
Fakultät für Mathematik und Informatik
FernUniversität in Hagen
Lützowstr. 125
D-55084 Hagen