Poincaré-Birkhoff-Witt theorem

Let 𝔤 be a Lie algebraMathworldPlanetmath over a field k, and let B be a k-basis of 𝔤 equipped with a linear order . The Poincaré-Birkhoff-Witt-theorem (often abbreviated to PBW-theorem) states that the monomials

x1x2xn with x1x2xn elements of B

constitute a k-basis of the universal enveloping algebra U(𝔤) of 𝔤. Such monomials are often called ordered monomials or PBW-monomials.

It is easy to see that they span U(𝔤): for all n, let Mn denote the set


and denote by π:n=0BnU(𝔤) the multiplicationPlanetmathPlanetmath map. Clearly it suffices to prove that


for all n; to this end, we proceed by inductionMathworldPlanetmath. For n=0 the statement is clear. Assume that it holds for n-10, and consider a list (x1,,xn)Bn. If it is an element of Mn, then we are done. Otherwise, there exists an index i such that xi>xi+1. Now we have

π(x1,,xn) =π(x1,,xi-1,xi+1,xi,xi+2,,xn)

As B is a basis of 𝔨, [xi,xi+1] is a linear combinationMathworldPlanetmath of B. Using this to expand the second term above, we find that it is in i=0n-1π(Mi) by the induction hypothesis. The argument of π in the first term, on the other hand, is lexicographically smaller than (x1,,xn), but contains the same entries. Clearly this rewriting proces must end, and this concludes the induction step.

The proof of linear independence of the PBW-monomials is slightly more difficult, but can be found in most introductory texts on Lie algebras, such as the classic below.


  • 1 N. Jacobson. . Dover Publications, New York, 1979
Title Poincaré-Birkhoff-Witt theorem
Canonical name PoincareBirkhoffWittTheorem
Date of creation 2013-03-22 13:03:38
Last modified on 2013-03-22 13:03:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type TheoremMathworldPlanetmath
Classification msc 17B35
Synonym PBW-theorem
Related topic LieAlgebra
Related topic UniversalEnvelopingAlgebra
Related topic FreeLieAlgebra