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


[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)
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