Faithful Representations of Lie Algebras

Abstract

The object of this paper is to give a new proof of the theorem that every Lie algebra over a field K of characteristic zero, has a faithful representation. The first proof of this result, at least when K is algebraically closed, is due to Ado (1). Later Cartan (2) gave a simpler and entirely different proof for the case when K is the field of either real or complex numbers. Cartan's proof depends on the integration of the Maurer-Cartan equations and therefore is of a non-algebraic character.! The present proof is of course algebraic and seems to differ from the earlier ones in approaching the problem quite directly. Also the result established is slightly sharper than the usual one in so far as we assert the existence of a faithful representation in which every element of the maximal nilpotent ideal of the given Lie algebra is mapped on a nilpotent matrix. I am very much indebted to Professor C. Chevalley for his advice and help in improving the presentation of the proof. Also I should like to thank Dr. G. D. Mostow for many interesting and valuable discussions. All algebras (whether Lie algebras or associative algebras) and vector spaces appearing in this paper are to be understood over the basic field K. A linear Lie algebra 2 is a Lie algebra whose elements are endomorphisms of some given vector space, the bracket operation in 2 being defined by [X,Y] = XY YX. As far as possible we follow the notation and terminology of Chevalley's book (3) and his papers (4). In particular, if 2 is a Lie algebra and X e 2 we denote by ad X the derivation of 2 defined by (ad X)Y = [XY](Y ).2 The following notion of the semidirect sum of a Lie algebra and its algebra of derivations' is important for our purpose. DEFINITION. Let S be a Lie algebra and i) the algebra of its derivations. By the semidirect sum of V and Z is meant a Lie algebra 2 + ) defined as follows. Considered as a vector space 2 + Z is the direct sum of 2 and Z so that an element of 2 + i) is a pair (X, D) with X e and D e Z. The bracket operation in S + Z is defined by