Vol. 42, No. 1, 1972

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On Malcev algebras

T. S. Ravisankar

Vol. 42 (1972), No. 1, 227–234

A Malcev algebra is an anticommutative algebra satisfying the identity (xy)(xz) = ((xy)z)x+((yz)x)x+((zx)x)y. Various notions of solvability have been introduced for a Malcev algebra mainly with the aim of proving the following result or one of its corollaries for the related notion of semisimplicity.

Theorem A. A semisimple Malcev algebra over a field of characteristio zero is a direct sum of ideals which are simple as algebras.

In the present paper all these notions of solvability are shown to be equivalent. This fact enables simplifications of proofs of some known results. The Killing form of a Malcev algebra is considered and is shown to possess some of the nice properties of the classical Cartan-Killing form of a Lie algebra. Finally, the structure of the Malcev algebra is studied in relation to that of the derivation algebra of an associated Lie triple system.

Mathematical Subject Classification 2000
Primary: 17E05
Secondary: 17A30
Received: 19 September 1969
Revised: 23 March 1972
Published: 1 July 1972
T. S. Ravisankar