The universal enveloping algebra is the associative algebra generated by subject to . Its modules are the same as Lie algebra representations.
For an ordered basis of a Lie algebra , the ordered monomials form a basis of . In particular, a triangular decomposition gives as vector spaces.
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The universal enveloping algebra is a fundamental concept in the theory of Lie algebras and representation theory. Given a Lie algebra \(\mathfrak{g}\), its universal enveloping algebra, denoted as \(U(\mathfrak{g})\), is an associative algebra that encodes the structure of the Lie algebra in such a way that representation theory can be applied to it using methods of associative algebras.