nLab tangent Lie algebra

Given an algebraic group $G$ in characteristic zero, or a finite dimensional Lie group, one associates to it a Lie algebra, its tangent Lie algebra which is the Lie subalgebra $\mathcal{X}^{linv}(G)$ of the Lie algebra $\mathcal{X}^(G)$ left invariant vector fields on $G$ with respect to the usual Lie bracket of vector fields.

The value of a left-invariant vector field $X$ at the unit element $e$ is a tangent vector $X_e$ at $e$. It appears that the specialization/evaluation at the unit element map $\mathcal{X}^{linv}(G)\to T_e G$ is an isomorphism of vector spaces, which is often considered as an identification. However, one needs to look into vector fields in order to find the bracket, hence defining the tangent Lie algebra as the tangent vector space at $e$ misses the bracket (which come from consideration of infintesimals of second order). One can instead work with right invariant vector fields $\mathcal{X}^{rinv}(G)$ and obtain an isomorphic Lie algebra; the isomorphism is of course, by comparing the specialization at $e$.

Within $\mathcal{X}^(G)$, all right invariant vector fields commute with all left invariant vector fields.

The correspondence $G\mapsto\mathcal{X}^{linv}(G)$ is functorial.

A generalization of a tangent Lie algebra of a Lie group is the tangent Lie algebroid.

The first idea of the tangent Lie algebra was explained in a letter of Sophus Lie to his friend Meyer in 1874 (See the historical appendix to Bourbaki, Lie groups and Lie algebras vol. 3).