Poincaré–Birkhoff–Witt theorem

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In the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (Poincare (1900) , Birkhoff (1937) , Witt (1937) ; frequently contracted to PBW theorem) is a fundamental result giving an explicit description of the universal enveloping algebra of a Lie algebra. The term 'PBW type theorem' or even 'PBW theorem' may also refer to various analogues of the original theorem, comparing a noncommutative algebra and its associated graded algebra, in particular, in the area of quantum groups.

[edit] Statement of the theorem

Recall that any vector space V over a field has a basis; this is a set S such that any element of V is a unique (finite) linear combination of elements of S. In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases the elements of which are totally ordered by some relation which we denote ≤.

If L is a Lie algebra over a field K, there is a canonical K-linear map h from L into the universal enveloping algebra U(L).

Theorem. Let L be a Lie algebra over K and X a totally ordered basis of L. A canonical monomial over X is a finite sequence (x₁, x₂ ..., x_n) of elements of X which is non-decreasing in the order ≤, that is, x₁ ≤x₂ ≤ ... ≤ x_n. Extend h to all canonical monomials as follows: If (x₁, x₂, ..., x_n) is a canonical monomial, let

Then h is injective on the set of canonical monomials and its range is a basis of the K-vector space U(L).

Stated somewhat differently, consider Y = h(X). Y is totally ordered by the induced ordering from X. The set of monomials

where y₁ <y₂ < ... < y_n are elements of Y, and the exponents are non-negative, together with the multiplicative unit 1, form a basis for U(L). Note that the unit element 1 corresponds to the empty canonical monomial.

The multiplicative structure of U(L) is determined by the structure constants in the basis Y, that is, the coefficients c_u,v,x such that

The Poincaré–Birkhoff–Witt theorem can be interpreted as saying that the product of canonical monomials in Y can be reduced uniquely to a linear combination of canonical monomials by repeatedly using the structure equations. The reduction part of this is clear: the structure constants determine uv − vu, i.e. what to do in order to change the order of two elements of X in a product. This fact, modulo an inductive argument on the degree of sums of monomials, shows one can always achieve products where the factors are ordered in a non-decreasing fashion.

Corollary. If L is a Lie algebra over a field, the canonical map L → U(L) is injective. In particular, any Lie algebra over a field is isomorphic to a Lie subalgebra of an associative algebra.

[edit] History of the theorem

Ton-That and Tran have investigated the history of the theorem. 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 in her encyclopaedic entry says that H. Poincaré obtained the first variant of the theorem. She further says that the theorem was subsequently completely demonstrated by E. Witt and G.D. Birkhoff. It appears that pre-Bourbaki sources were not familiar with Poincare's paper.

Birkhoff and Witt do not mention Poincare's work in their 1937 papers. Cartan and Eilenberg in their 1956 book call the theorem Poincare-Witt Theorem and attribute the complete proof to Witt. Bourbaki were the first to use all three names in their 1960 book. Knapp presents a clear illustration of the shifting tradition. In his 1986 book he calls it Birkhoff-Witt Theorem while in his later 1996 book he switches to Poincare-Birkhoff-Witt Theorem.

It is not clear whether Poincare's result was complete. Ton-That and Tran conclude that Poincare had discovered and completely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff. On the other hand, they point out that Poincare makes several statements without bothering to prove them. Their own proofs of all the steps are rather long according to their admission.

[edit] References

• G.D. Birkhoff, Representability of Lie algebras and Lie groups by matrices Ann. of Math. (2) , 38 : 2 (1937) pp. 526–532
• T.S. Fofanova (2001), "Birkhoff–Witt theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• G. Hochschild, The Theory of Lie Groups, Holden-Day, 1965.
• H. Poincaré, Sur les groupes continus Trans. Cambr. Philos. Soc. , 18 (1900) pp. 220–225
• E. Witt, Treue Darstellung Liescher Ringe J. Reine Angew. Math. , 177 (1937) pp. 152–160
• T. Ton-That, T.-D. Tran, Poincaré's proof of the so-called Birkhoff-Witt theorem Rev. Histoire Math., 5 (1999), pp. 249-284.
• N. Bourbaki, Éléments de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie. Hermann, Paris, 1960.
• A. W. Knapp, Representation theory of semisimple groups. An overview based on examples. Princeton University Press, 1986.
• A. W. Knapp, Lie groups beyond an introduction. Birkhäuser Boston, 1996.