A VERSION OF THE POINCARE-BIRKHOFF-WITT THEOREM

In this note I shall prove that if L is a finite-dimensional Lie algeb ra over a field F of characteristic zero which is generated as an alge bra by a set of elements {e(1), e(2),...,e(k)}, then the universal env eloping algebra U(L) of L is linearly generated by monomials spanned b y the elements {e(i)} of an a priori bounded width. As an application, a criterion of Kostant for a left ideal of U(L) to be of finite codim ension is proved by purely algebraic means.