What is the main use of Lie brackets in the Lie algebra of a Lie group?

I am beginner in Lie group theory, and I can't find the answer a question I am asking myself : I know that the Lie algebra $\mathfrak g$ of a Lie group $G$ is more or less the tangent vector of $G$ at the identity, so that $\mathfrak g$ have a very interesting property : linearity.

However $\mathfrak g$ has another property : it is stable under Lie brackets $[.,.]$.

For me when I study Lie groups I always find linearity of Lie algebras really important, and I don't see and I didn't find why the stability under Lie brackets is important. What is the main result/property of Lie groups using this property?

That would be great if you could light me!

Solution 1:

I think about that this way.

In some sence geometry is "difficult" and algebra is "easy". So you want to obtain as much information as possible from studying Lie algebras instead of Lie groups, and then transering your results from algebras back to groups. So your bracket is the most natural operation on the tangent space that sort of allows you to do that. You can reinterpret a lot of properties of your Lie group (commutativity, solvability, (semi-)simplicity et.c.) into properties of the bracket on the Lie algebra. For example, simplicity for groups encodes into the property that Lie algebra does not have non-trivial ideals, and so on. You also have analogs between subgroups of different kind of your Lie group $G$ and subalgebras of $g$. For example, tangent space to the center of Lie group $G$ is the center of Lie algebra $g$, i.e. $Z(g)=\{x\in g|[x,y]=0,\forall y\in g\}$.

I hope I've helped a bit.

Solution 2:

As a take-home message, I would say that the Lie bracket encodes the structure of group multiplication. The linear structure alone is very uninformative; it really just tells you about the dimension of the manifold.

More precisely, the Baker-Campbell-Hausdorff formula tells you that the group multiplication (at least for elements near the identity, and hence in the image of the exponential map $\exp$) can be entirely obtained from Lie brackets. Therefore, as pointed out in Sasha Patotski's post, (almost) all group properties have Lie bracket analogies.

In fact, the Lie bracket is particularly nice, giving very concise expressions of Lie group properties. Thinking about the adjoint representation, Killing form and so on is a good way in.

Remark: The caveats above reflect the problems like $\exp$ mapping the Lie algebra of $O(3)$ back to just $SO(3)$, and a more subtle problem where the Lie algebra of $SL_2(\mathbb R)$ is mapped to a strict subset of that group (even though the group is connected).

Solution 3:

A good question. There are many aspects of the situation... At least one fundamental structure can be understood in the following way. First, imagining that $t$ is an "infinitesimal", so that $t^3=0$ (not $t^2=0$!) (or equivalent...), and imagining that elements of the Lie group near the identity are $g=1+tx$ and $h=1+ty$ (with $x,y$ in the Lie algebra) observe that $(1+tx)(1+ty)(1-tx)(1-ty)=1+t^2(xy-yx)$. Thus, we care about $xy-yx=[x,y]$.

E.g., for matrix Lie groups, so that $x,y$ are matrices, this makes sense, where $xy-yx$ is in the matrix algebra.

Yes, several issues are left hanging after this walk-through, but the symbol-pattern proves to be excellent, in essentially all incarnations.