The Poincaré-Birkhoff-Witt Theorem is a fundamental result in the theory of Lie algebras that establishes a relationship between the universal enveloping algebra of a Lie algebra and the algebra of polynomials. It asserts that the universal enveloping algebra has a basis formed by the ordered monomials in terms of a chosen basis of the Lie algebra, leading to a structure that reflects both the combinatorial properties of these monomials and the algebraic structure of the Lie algebra itself.
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