Ciro Santilli @cirosantilli 34

Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".

Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.

Another important way to think about Lie algebras, is as infinitesimal generators.

Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.

To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.

As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.

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We can almost reach the Lie algebra of any isometry group in a single go. For every $X$ in the Lie algebra we must have:because $e^{tX}$ has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".

Then:so we reach:With this relation, we can easily determine the Lie algebra of common isometries:

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

Because $GL(n)$ is a Lie group we can use Section "Lie algebra of a matrix Lie group".

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 $M_n$.

Is the solution to a system of linear ordinary differential equations, the exponential function is just a 1-dimensional subcase.

Note that more generally, the matrix exponential can be defined on any ring.

The matrix exponential is of particular interest in the study of Lie groups, because in the case of the Lie algebra of a matrix Lie group, it provides the correct exponential map.