Poincaré–Birkhoff–Witt theorem

The theorem states that ordered monomials in a basis of a Lie algebra form a basis of its universal enveloping algebra. It identifies this associative algebra’s size and structure with those of the corresponding symmetric algebra.

Universal enveloping algebra: The associative algebra generated by a Lie algebra subject to relations encoding its bracket. The theorem describes a basis for this algebra using ordered products of Lie algebra elements.
Tensor algebra: The associative algebra formed from all tensor powers of a vector space, with concatenation as multiplication. The enveloping algebra is constructed as a quotient of the tensor algebra.
Henri Poincaré: French mathematician whose work helped establish topology, celestial mechanics, and mathematical physics. His work on Lie groups and algebras is one source of the theorem’s name.
Poincaré–Birkhoff–Witt basis: The ordered-monomial basis of a universal enveloping algebra obtained from an ordered basis of its Lie algebra. It is the explicit basis whose existence the theorem guarantees.
Diamond lemma: A criterion ensuring that rewriting rules yield unique normal forms when all critical ambiguities resolve. It generalizes the reduction argument often used to prove ordered-monomial bases.
Symmetric algebra: The commutative algebra generated by a vector space, equivalently its tensor algebra modulo commutativity relations. The theorem gives its graded dimensions as the dimensions of the enveloping algebra’s associated graded algebra.
Lie bracket: A bilinear, alternating operation on a Lie algebra satisfying the Jacobi identity. The relations identifying commutators with brackets are precisely the quotient relations used in PBW.
Garrett Birkhoff: American mathematician known for work in algebra, lattice theory, and differential equations. His 1937 treatment helped establish the theorem in its modern algebraic form.
Verma module: A highest-weight module for a Lie algebra induced from a one-dimensional representation of a Borel subalgebra. PBW monomials in lowering operators provide a standard spanning basis for it.
Lyndon basis: A basis of a free Lie algebra constructed from Lyndon words and their bracketings. It provides a specialized Lie algebra basis before PBW turns basis elements into enveloping-algebra monomials.

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Capelli identityRelated: Ordered monomials make precise the algebraic setting behind the identity’s noncommuting products.