Universal enveloping algebra

Poincaré-Birkhoff-Witt theorem

Let be a Lie algebra over and its universal enveloping algebra with the canonical Lie algebra homomorphism . For some ordered basis of , the universal enveloping algebra has a basis consisting of ordered products for , , . lie It follows that is injective.

Proof

proof

Corollaries

• Let and . Then the following defines a -linear isomorphism:

  Therewithal if is an -module then the following defines a -linear isomorphism

  where the codomain is the induced module .¹

• The associated graded algebra of is the symmetric algebra .

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.5, p. 16 ↩