TITLE: Why study Lie algebras?
QUESTION [137 upvotes]: I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics and physics. I visited a course on Lie groups, and an elementary one on Lie algebras. But I don't fully understand how those theories are being applied. I actually don't even understand the importance of Lie groups in differential geometry.
I know, among others, of the following facts:

If $G$ and $H$ are two Lie groups, with $G$ simply connected, and $\mathfrak{g,h}$ are their respective Lie algebras, then there is a one to one correspondance between Lie algebra homomorphisms $\mathfrak{g}\rightarrow\mathfrak{h}$ and group homomorphisms $G\rightarrow H$.
The same remains true if we replace $H$ with any manifold $M$: any Lie algebra homomorphism from $\mathfrak{g}$ to the Lie algebra $\Gamma(TM)$ of smooth vector fields on $M$ gives rise to a local action of $G$ on $M$.
Under some conditions like (I think) compactness, the cohomology of $\mathfrak{g}$ is isomorphic to the real cohomology of the group $G$. I know that calculating the cohomology of $\mathfrak{g}$ is tractable in some cases.
There is a whole lot to be said of the representation theory of Lie algebras.
Compact connected centerless Lie groups $\leftrightarrow$ complex semisimple Lie algebras

How do people use Lie groups and Lie algebras? What questions do they ask for which Lie groups or algebras will be of any help? And if a geometer reads this, how, if at all, do you use Lie theory? How is the representation theory of Lie algebras useful in differential geometry?

REPLY [3 votes]: I hope you may be interested in your questions restricted to Dynamical Systems (DS):

How do people use Lie groups and Lie algebras in DS?
What questions do they ask for which Lie groups or algebras will be
of any help in DS?

In DS people are interested in particular in: dense orbits and invariant measures for actions of groups on a manifold. A tool to study this is Ergodic Theory (ET).
There is an abstract ET of amenable groups actions (Ornstein-Weiss), I have seen that Lie groups provide: concrete examples, counter-examples and extensions of this abstract theory. I have also seen that one important reason why Lie groups allow to prove deep ET results is because they are well adapted to the theory of harmonic analysis, and harmonic analysis is one of the main machinery of ET.
The ET of Lie groups is a very broad theory, just a few examples:
Example 1.
Let $G\subset SL(\mathbb{Z},d)$ (such that $G$ acts irreducibly on $\mathbb{R}^d$ and $G$ does
not contain an abelian subgroup of finite index). Then the $G$-action on the d-torus is strongly ergodic (''the G-invariatn measure is the unique G-invariant mean'').  Moreover, there is a characterization of the $G$-action with $G\subset SL(\mathbb{Z},d)$ on the d-torus that are strongly ergodic. And more in abstract, if $G$ is a connected non-compact simple Lie group with finite center, there is a characterization for the action to be strongly ergodic.
These are theorems of A. FURMAN and Y. SHALOM that generalize Rosenblatt:

A. Furman and Y. Shalom. Sharp ergodic theorems for group actions and
strong ergodicity.
J. Rosenblatt. Uniqueness of invariant means for measure preserving transformations. Trans. Amer.
Math. Soc. 265 (1981), 623–636

Example 2.
Orbit-equivalence rigidity (Zimmer):
Let $G_1$ and $G_2$ be noncompact, simple Lie groups, with $R$-$rank(G_1)\geq 2.$ If $(G_1,X_1)$ is orbit equivalent to $(G_2,X_2)$, then $G_1$ is locally isomorphic to $G_2$, and, up to a group automorphism, the actions are isomorphic.
Example 3.
A. Avila, B. Fayad, and A. Kocsard provide in (On manifolds supporting distributionally uniquely ergodic diffeomorphisms) some counterexamples to a conjecture proposed by Forni in “On the Greenfield-Wallach and Katok conjectures in dimension three.
A Good references for ET of actions of Lie group is R. Zimmer. Ergodic Theory and Semisimple Lie Groups. Birkh¨auser, Boston, 1984.
You can find deep results by: A. Avila, V. Bergelson,  G. Forni, J. Rosenblatt, A. Furman, H. Furstenberg, D. Y. Kleinbock, G. A. Margulis, Y. Shalom,  A. Katok and Zimmer, between many others.