In this section, we will discuss the representation theory of Lie groups and Lie algebras – as far as it can be discussed without using the structure theory of semisimple Lie algebras. Unless specified otherwise, all Lie groups, algebras, and representations are finite-dimensional, and all representations are complex. Lie groups and Lie algebras can be either real or complex; unless specified otherwise, all results are valid both for the real and complex case.

Basic definitions

Let us start by recalling the basic definitions.

Definition 4.1. A representation of a Lie group G is a vector space V together with a morphism ρ:G → GL(V).

A representation of a Lie algebra g is a vector space V together with a morphism ρ:g → gl(V).

A morphism between two representations V, W of the same group G is a linear map f:V → W which commutes with the action of G:f ρ(g) = ρ(g)f. In a similar way one defines a morphism of representations of a Lie algebra. The space of all G-morphisms (respectively, g-morphisms) between V and W will be denoted by HomG(V, W) (respectively, Homg(V, W)).

Remark 4.2. Morphisms between representations are also frequently called intertwining operators because they “intertwine” action of G in V and W.

The notion of a representation is completely parallel to the notion of module over an associative ring or algebra; the difference of terminology is due to historical reasons.