Poincaré–Birkhoff–Witt theorem
In mathematics, more specifically in abstract algebra, 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.
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. 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
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