Representation of a Lie group

Summary

A representation of a Lie group $G$ is a continuous homomorphism from $G$ to the group of invertible linear maps of a vector space. It lets the group act on vectors by linear maps, so that group theory can be studied with linear algebra. Each finite-dimensional representation has a derivative, a representation of the Lie algebra, which often determines it. Representations of Lie groups are central in harmonic analysis, number theory and quantum physics.