# Birkhoff-Witt theorem

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Poincaré–Birkhoff–Witt theorem

A theorem about the representability of Lie algebras in associative algebras. Let be a Lie algebra over a field , let be its universal enveloping algebra, and let be a basis of the algebra which is totally ordered in some way. All the possible finite products , where , then form a basis of the algebra , and it thus follows that the canonical homomorphism is a monomorphism.

It is possible to construct a Lie algebra for any associative algebra by replacing the operation of multiplication in with the commutator operation The Birkhoff–Witt theorem is sometimes formulated as follows: For any Lie algebra over any field there exists an associative algebra over this field such that is isomorphically imbeddable in .

The first variant of this theorem was obtained by H. Poincaré ; the theorem was subsequently completely demonstrated by E. Witt  and G.D. Birkhoff . The theorem remains valid if is a principal ideal domain , in particular for Lie rings without operators, i.e. over , but in the general case of Lie algebras over an arbitrary domain of operators the theorem is not valid .