It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt.
The set of monomials where y1 Note that the unit element 1 corresponds to the empty canonical monomial. The theorem then asserts that these monomials form a basis for U(L) as a vector space. It is easy to see that these monomials span U(L); the content of the theorem is that they are linearly independent. such that This relation allows one to reduce any product of y's to a linear combination of canonical monomials: The structure constants determine yiyj – yjyi, i.e. what to do in order to change the order of two elements of Y in a product. This fact, modulo an inductive argument on the degree of (non-canonical) monomials, shows one can always achieve products where the factors are ordered in a non-decreasing fashion. The Poincaré–Birkhoff–Witt theorem can be interpreted as saying that the end result of this reduction is unique and does not depend on the order in which one swaps adjacent elements. If L is a Lie algebra over a field, the canonical map L → U(L) is injective. Already at its earliest stages, it was known that K could be replaced by any commutative ring, provided that L is a free K-module, i.e., has a basis as above. To extend to the case when L is no longer a free K-module, one needs to make a reformulation that does not use bases. This involves replacing the space of monomials in some basis with the symmetric algebra, S(L), on L. In the case that K contains the field of rational numbers, one can consider the natural map from S(L) to U(L), sending a monomial Then, passing to the associated graded, one gets a canonical morphism T(L) → grU(L), which kills the elements vw - wv for v, w ∈ L, and hence descends to a canonical morphism S(L) → grU(L). Then, the (graded) PBW theorem can be reformulated as the statement that, under certain hypotheses, this final morphism is an isomorphism of commutative algebras. More generally, the PBW theorem as formulated above extends to cases such as where (1) L is a flat K-module, (2) L is torsion-free as an abelian group, (3) L is a direct sum of cyclic modules (or all its localizations at prime ideals of K have this property), or (4) K is a Dedekind domain. Finally, it is worth noting that, in some of these cases, one also obtains the stronger statement that the canonical morphism S(L) → grU(L) lifts to a K-module isomorphism S(L) → U(L), without taking associated graded. This is true in the first cases mentioned, where L is a free K-module, or K contains the field of rational numbers, using the construction outlined here (in fact, the result is a coalgebra isomorphism, and not merely a K-module isomorphism, equipping both S(L) and U(L) with their natural coalgebra structures such that In four papers from the 1880s Alfredo Capelli proved, in different terminology, what is now known as the Poincaré–Birkhoff–Witt theorem in the case of They have found out that the majority of the sources before Bourbaki's 1960 book call it Birkhoff-Witt theorem. Following this old tradition, Fofanova[4] in her encyclopaedic entry says that Poincaré obtained the first variant of the theorem. Ton-That and Tran[3] conclude that "Poincaré had discovered and completely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff". On the other hand, they point out that "Poincaré makes several statements without bothering to prove them".