The PBW Theorem, also known as the Poincaré-Birkhoff-Witt theorem, establishes a significant connection between a Lie algebra and its universal enveloping algebra. This theorem asserts that the universal enveloping algebra of a Lie algebra has a basis that consists of certain ordered monomials formed from the elements of the Lie algebra, which reflects the structure of the Lie algebra itself. This result is fundamental in understanding how representations of Lie algebras can be studied through their enveloping algebras.
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