Existence Theorem

Abstract

In this section F may be an arbitrary field (except where otherwise noted). We shall associate to each Lie algebra L over F an associative algebra with 1 (infinite dimensional, in general), which is generated as “freely” as possible by L subject to the commutation relations in L. This “universal enveloping algebra” is a basic tool in representation theory. Although it could have been introduced right away in Chapter I, we deferred it until now in order to avoid the unpleasant task of proving the Poincaré-Birkhoff-Witt Theorem before it was really needed. The reader is advised to forget temporarily all the specialized theory of semisimple Lie algebras.