Example

Let G be a real Lie group, g its Lie algebra and θ an analytic representation of G on a finite-dimensional real vector space E. Then the corresponding homomorphism of g into gl(E) is a representation of g on E.*

Let U be the enveloping algebra of g. Proposition 1 of § 2, no. 1 defines a one-to-one correspondence between the set of representations of g on M and the set of representations of U on M. On the other hand we know (Algebra, Chapter VIII, § 13, no. 1) that there is an equivalence between the notion of representation of the associative algebra U and that of left U-module.