Representations of graded Lie algebras

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Introduction. Let L be a graded Lie algebra (see ?2) over a commutative ring R with unit. We shall deal with graded representations of L. In ?2 we define the universal enveloping algebra U of L and prove a PoincareBirkhoff-Witt theorem for U. In ?3 we prove that if L is a finite dimensional graded Lie algebra over a field of characteristic : 2, then L has a faithful finite dimensional representation. A key lemma for this proof is a result on (nongraded) Lie algebras which states that if L is a finite dimensional Lie algebra over a field of characteristic 0 then L has a faithful, finite dimensional representation f such that f (x) is nilpotent whenever ad x is nilpotent. This work is the first section of a doctoral thesis written under the direction of Professor Gerhard Hochschild. The author takes this opportunity to thank him for his generous advice and instruction.