Intuition/Significance of adjoint representation of a Lie group

Question

I understand the definition of the adjoint representation of a Lie group. But why is that important? In particular, why is it a natural choice of group representation?

Answer 1

In case you prefer math intuitive and non-technical like me, I have summarized my thoughts on your question. I also very recommend to look at the links in @Dietrich Burde´s comment.

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With adjoint representation, we can look at the elements of the Lie group (let´s call it $G$) as at linear transformations of its Lie algebra (let´s call it $\mathfrak{g}$). But why and how?

For me, this is a problem that is common in math. We have a group $G$ with some abstract elements. Recall that any Lie group $G$ is also a manifold, so the elements live in some wiggly curved space. Of course, we would like to move everything into some more pleasant space where we can perform more magic. Ideally, a vector space, because working on vector spaces is just easier.

So, to "move" the elements of $G$ into the world of vectors, we want to represent them as matrices that act on a vector space.

But how do we even start with this? We need to decide how to represent the elements of $G$, ideally in an easy and natural way.

The fact that every Lie group $G$ is a manifold whose elements satisfy the group axioms tells us that $G$ looks in the neighborhood of any point (= any group element) like $\mathbb{R}^n$.

Now recall that the Lie algebra of a group is defined as the tangent space at the identity element $e$ of $G$. Denote this tangent space $T_eG$. Lie algebras are important, because a curved object which is the Lie group $G$, a Lie group G, can be quite nicely described by a flat object, which is the Tangent space $T_eG$ of $G$ at the identity $e$.”

So, we got from $G$ to a vector space $T_eG$.

Are there more possible vector spaces which can be used for representation of $G$? Of course. We can look at representations of a given group on any vector space. (There is even a theorem saying that every Lie algebra is isomorphic to a matrix Lie algebra. This means that the knowledge of ordinary linear algebra is enough to study Lie algebras because every Lie algebra can be viewed as a set of matrices.)

However, there is exactly one distinguished vector space that comes automatically with each group: Its own Lie algebra. And this representation on the Lie algebra is called the adjoint representation.

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I hope this helps to understand, why adjoint representation is precisely the natural choice.