Birkhoff-Witt theorem

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 [1]; the theorem was subsequently completely demonstrated by E. Witt [2] and G.D. Birkhoff [3]. The theorem remains valid if $k$ is a principal ideal domain [4], 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 [5].

References

[1]	H. Poincaré, "Sur les groupes continus" Trans. Cambr. Philos. Soc. , 18 (1900) pp. 220–225
[2]	E. Witt, "Treue Darstellung Liescher Ringe" J. Reine Angew. Math. , 177 (1937) pp. 152–160
[3]	G.D. Birkhoff, "Representability of Lie algebras and Lie groups by matrices" Ann. of Math. (2) , 38 : 2 (1937) pp. 526–532
[4]	M. Lazard, "Sur les algèbres enveloppantes universelles de certain algèbres de Lie" C.R. Acad. Sci. Paris Sér. I Math. , 234 (1952) pp. 788–791
[5]	A.I. Shirshov, "On representations of Lie rings in associative rings" Uspekhi Mat. Nauk , 8 : 5 (1953) pp. 173–175 (In Russian)
[6]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[7]	A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[8]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[9]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)