Representations On A Complex Finite-dimensional Vector Space
Let us first discuss representations acting on finite-dimensional complex vector spaces. A representation of a Lie group G on a finite-dimensional complex vector space V is a smooth group homomorphism Ψ:G→Aut(V) from G to the automorphism group of V.
For n-dimensional V, the automorphism group of V is identified with a subset of complex square-matrices of order n. The automorphism group of V is given the structure of a smooth manifold using this identification. The condition that Ψ is smooth, in the definition above, means that Ψ is a smooth map from the smooth manifold G to the smooth manifold Aut(V).
If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into GL(n,C). This is known as a matrix representation.
Read more about this topic: Representation Of A Lie Group
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