{c}

The author seems to have uploaded the entire book by chapters at: https://www.physics.drexel.edu/~bob/LieGroups.html

And the author is the cutest: https://www.physics.drexel.edu/~bob/Personal.html[].

Overview:
* Chapter 3: gives a bunch of examples of important <matrix Lie groups>. These are done by imposing certain types of constraints on the <general linear group>, to obtain <subgroups> of the general linear group. Feels like the start of a <classification (mathematics)>
* Chapter 4: defines <Lie algebra>. Does some basic examples with them, but not much of deep interest, that is mostl left for Chapter 7
* Chapter 5: calculates the <Lie algebra> for all examples from chapter 3
* Chapter 6: don't know
* Chapter 7: describes how the <exponential map> links <Lie algebras> to <Lie groups>


Lie Groups, Physics, and Geometry by Robert Gilmore (2008)

The one parameter subgroup of a <Lie group> for a given element $M$ of its <Lie algebra> is a <subgroup> of $G$ given by:
$$
{ e^{tM} \in G | t \in \R }
$$

Intuitively, $M$ is a direction, and $t$ is how far we move along a given direction. This intuition is especially vivid in for example in the case of the <Lie algebra of SO(3)>, the <rotation group>.

One parameter subgroups can be seen as the continuous analogue to the <cycle of an element of a group>.


One parameter subgroup

This is a good and simple first example of <Lie algebra> to look into.


Translation group

The most important example is perhaps <SO(3)> and <SU(2)>, both of which have the same <Lie algebra>, but are not isomorphic.


Two different Lie groups can have the same Lie algebra

{c}
{wiki}

Like everything else in <Lie groups>, first start with the <matrix> as discussed at <Lie algebra of a matrix Lie group>{full}.

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 multilinear map>[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:
* the dimension
* the <Lie bracket>
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.

Bibliography:
* https://physicstravelguide.com/advanced_tools/group_theory/lie_algebras#tab__concrete on <Physics Travel Guide>
* http://jakobschwichtenberg.com/lie-algebra-able-describe-group/ by <Jakob Schwichtenberg>


Ciro Santilli @cirosantilli 37

 Incoming links: Lie algebra