Chemistry - Poincaré-Birkhoff-Witt theorem

In the theory of Lie algebras, the Poincaré-Birkhoff-Witt theorem is a fundamental result characterizing the universal enveloping algebra of a Lie algebra.

Recall that any vector space V over a field has a Hamel 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 which are totally ordered by some relation which we denote ≤.

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

Theorem. Let L be a Lie algebra over K and X a totally ordered Hamel basis for 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

$h(x_1, x_2, \ldots, x_n) = h(x_1) \cdot h(x_2) \cdots h(x_n).$

Then h is injective and its range is a Hamel basis for 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

$y_1^{k_1} y_2^{k_2} \cdots y_\ell^{k_\ell}$

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

Note that the monomials in Y form a basis as a vector space. The multiplicative structure of U(L) is determined by the structure constants of the Lie algebra; that is the coefficients c_u,v,x such that

$[u,v] = \sum_{x \in X} c_{u,v,x}\; x$

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. 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.

References

G. Hochschild, The Theory of Lie Groups, Holden-Day, 1965.