Source: /cirosantilli/lie-algebra-of-a-matrix-lie-group

= Lie algebra of a matrix Lie group

For this sub-case, we can define the <Lie algebra> of a Lie group $G$ as the set of all matrices $M \in G$ such that for all $t \in \R$:
e^{tM} \in G
If we fix a given $M$ and vary $t$, we obtain a <subgroup> of $G$. This type of subgroup is known as a <one parameter subgroup>.

The immediate question is then if every element of $G$ can be reached in a unique way (i.e. is the exponential map a <bijection>). By looking at the <matrix logarithm> however we conclude that this is not the case for <real> matrices, but it is for <complex> matrices.

* <Lie algebra of GL(n)>
* <Lie algebra of SL(2)>
* <Lie algebra of SO(3)>
* <Lie algebra of SU(2)>

TODO example it can be seen that the Lie algebra is not closed <matrix multiplication>, even though the corresponding group is by definition. But it is closed under the <Lie bracket> operation.