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 certaines 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)