Representation theory of Lie groups

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras. The physics literature sometimes passes over the distinction between Lie groups and Lie algebras.

Let us first discuss representations acting on finite-dimensional vector spaces over a field ${\displaystyle \mathbb {C} }$. (Occasionally representations over the field of real numbers are also considered.) A representation of a Lie group G on a finite-dimensional vector space V over ${\displaystyle \mathbb {C} }$ is a smooth group homomorphism

where ${\displaystyle \mathrm {GL} (V)}$ is the group of all invertible linear transformations of ${\displaystyle V}$. For n-dimensional V, the group ${\displaystyle \mathrm {GL} (V)}$ is identified with the ${\displaystyle \mathrm {GL} (n;\mathbb {C} )}$, the group of ${\displaystyle n\times n}$ invertible matrices. Smoothness of the map should be regarded as a technicality, in that any homomorphism ${\displaystyle \psi }$ that is continuous will automatically be smooth.

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