The Composition Method in the Theory of Lie Algebras

Abstract

The composition method was developed by A. I. Shirshov. To begin with, in [372], he proved the theorem on embedding a Lie algebra of finite or countable dimension into a two-generated Lie algebra, applying this method. In [278], the notion of a composition was introduced and a lemma, the first variant of a composition lemma, was proven (see [278], Lemma 4). The composition lemma in its modern form appeared in the work of L. A. Bokut’ [53]. Therein A. I. Shirshov also presented applications of the method in deciding the word problem for Lie algebras given by generators and defining relations and to studying one-relator Lie algebras. Next, in [373] he employed the composition method in his constructing a basis of the amalgamated free product of Lie algebras.