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 , , . #m/thm/lie It follows that is injective.

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 .

#state/develop | #lang/en | #SemBr