Representation of a Lie algebra

Summary

A representation of a Lie algebra $\mathfrak{g}$ realises the elements of $\mathfrak{g}$ as linear maps of a vector space. In it, the bracket becomes the commutator of linear maps. Equivalently, it is a $\mathfrak{g}$-module, or a module over the universal enveloping algebra of $\mathfrak{g}$. The derivative of a representation of a Lie group is a representation of its Lie algebra. Representations of Lie algebras are central in Lie theory and in quantum physics.