Lie groups and Lie algebras

Recall that the vector fields on a manifold \({\textrm{vect}(M)}\) form an infinite-dimensional Lie algebra. The group structure of a Lie group \({G}\) permits the definition of special vector fields that form a Lie subalgebra of \({\textrm{vect}(G)}\) with many useful properties. In particular, this special Lie algebra describes the infinitesimal behavior of \({G}\), i.e. the behavior near the identity. In physics, Lie groups are used to describe many transformations, with their infinitesimal generators thus described by Lie algebras.

An Illustrated Handbook