{c}
{tag=Lie algebra of a matrix Lie group}

Is the <set of all n-by-y square matrices>.

Because <GL(n)> is a <Lie group> we can use <Lie algebra of a matrix Lie group>{full}.

For every matrix $x$ in the <set of all n-by-y square matrices> $M_n$, $e^x$ has inverse $e^-x$.

Note that this works even if $x$ is not <invertible>, and therefore not in <GL(n)>!

Therefore, the Lie algebra of <GL(n)> is the entire <M_n>.

Lie algebra of $GL(n)$

Lie algebra of GL(n)

{c}
{tag=Lie algebra of a matrix Lie group}

From <Lie algebra of a isometry group>{full} we reach:

Lie algebra of $O(n)$

Lie algebra of O(n)

{c}
{tag=Lie algebra of a matrix Lie group}

= Lie algebra of the special linear group
{c}
{synonym}


Lie algebra of $SL(n)$

Lie algebra of SL(n)

{c}
{tag=Lie algebra of a matrix Lie group}

Bibliography:
* <Physics from Symmetry by Jakob Schwichtenberg (2015)> page 54.

Lie algebra of $SU(2)$

Lie algebra of SU(2)

{c}

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.

Examples:
* <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.

Ciro Santilli @cirosantilli 37

 Tagged: Lie algebra of a matrix Lie group