# Poincaré–Birkhoff–Witt theorem

Poincaré–Birkhoff–Witt theorem For the Poincaré–Birkhoff fixed-point theorem, see Poincaré–Birkhoff theorem.

In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt.

The terms PBW type theorem and PBW theorem may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups.

Contents 1 Statement of the theorem 2 More general contexts 3 History of the theorem 4 Notes 5 References 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 of which the elements are totally ordered by some relation which we denote ≤.

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

Theorem.[1] 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 ..., xn) of elements of X which is non-decreasing in the order ≤, that is, x1 ≤x2 ≤ ... ≤ xn. Extend h to all canonical monomials as follows: if (x1, x2, ..., xn) is a canonical monomial, let {displaystyle h(x_{1},x_{2},ldots ,x_{n})=h(x_{1})cdot h(x_{2})cdots h(x_{n}).} Then h is injective on the set of canonical monomials and the image of this set {displaystyle {h(x_{1},ldots ,x_{n})|x_{1}leq ...leq x_{n}}} forms a basis for U(L) as a K-vector space.

Stated somewhat differently, consider Y = h(X). Y is totally ordered by the induced ordering from X. The set of monomials {displaystyle y_{1}^{k_{1}}y_{2}^{k_{2}}cdots y_{ell }^{k_{ell }}} where y1

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