# Birkhoff-Witt theorem

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Poincaré–Birkhoff–Witt theorem

A theorem about the representability of Lie algebras in associative algebras. Let $G$ be a Lie algebra over a field $k$, let $U(G)$ be its universal enveloping algebra, and let $B = \{ {b _ {i} } : {i \in I } \}$ be a basis of the algebra $G$ which is totally ordered in some way. All the possible finite products $b _ {\alpha _ {1} } \dots b _ {\alpha _ {r} }$, where $\alpha _ {1} \leq \dots \leq \alpha _ {r}$, then form a basis of the algebra $U(G)$, and it thus follows that the canonical homomorphism $G \rightarrow U(G)$ is a monomorphism.

It is possible to construct a Lie algebra $L(R)$ for any associative algebra $R$ by replacing the operation of multiplication in $R$ with the commutator operation

$$[xy] = xy - yx .$$

The Birkhoff–Witt theorem is sometimes formulated as follows: For any Lie algebra $G$ over any field $k$ there exists an associative algebra $R$ over this field such that $G$ is isomorphically imbeddable in $L(R)$.

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 $k$ is a principal ideal domain , in particular for Lie rings without operators, i.e. over $\mathbf Z$, but in the general case of Lie algebras over an arbitrary domain of operators the theorem is not valid .

How to Cite This Entry:
Birkhoff–Witt theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff%E2%80%93Witt_theorem&oldid=22128