The first Whitehead lemma for Malcev algebras

Abstract

In the present paper the first Whitehead lemma for separable Malcev algebras is proved, the dimensions being finite and the characteristic of the base field necessarily equal to zero. A consequence is the theorem of Malcev-Harish-Chandra for Malcev algebras. To get the lemma a structure theorem for modules over semisimple split Malcev algebras is proved. 1. Introduction. In this paper Whitehead's first lemma is proved for separable Malcev algebras. It says that every derivation of a separable Malcev algebra in a Malcev module is inner, the characteristic of the base field being zero. A consequence is the theorem of Malcev-Harish-Chandra for Malcev algebras. These theorems are known for Lie algebras which are special Malcev algebras, for alternative, and for Jordan algebras. The lemma is proved by means of the classification of irreducible Malcev modules over simple split Malcev algebras. This classification is done in (1). 2. Definitions. Let A be an algebra over a field k, the characteristic of k not