03 - Lie groups and homomorphisms
We expand on our discussion of groups to include the maps between groups. We also present the idea behind Lie groups. Finally we give a formal definition of a group representation.
Lie Groups
A Lie group is a group with one or more continuous parameters that may also have a geometric interpretation. The abelian group $\mathsf{U}(1)$ has a single, continuous parameter that served as a coordinate on the circle, $S^{1}$.
The translations associated to the vector space $\mathbb{R}^{n}$ also form an abelian Lie group. Actually, this idea holds for any vector space. Essentially, translations model the group of vector addition.
The general linear group of invertible matrices is - unsurprisingly - also a group. It is a group whose operation is matrix multiplication. Given that each matrix element is a continuous parameter, $\mathsf{GL}(V)$ is also a Lie group.
Note that the full vector space of endomorphisms $\mathsf{End}(V)$ is not a group under matrix multiplication, as some matrices are not invertible.
Homomorphisms
Let $G$ and $H$ be groups. A map $f$ between these groups is called a group homomorphism if it respects the group multiplication,
$$f(gh) = f(g)f(h),\quad g,h \in G.$$
In this language, a linear map might be thought of as a vector space homomorphism.
We are now in a position to give our first precise definition of a representation. A group representation of a group $G$ on a vector space $V$ is a homomorphism $\pi$ from $G$ to a subgroup of $\mathsf{GL}(V)$:
$$\pi : G \rightarrow \mathsf{GL}(V).$$
In this context $V$ is called a module for $G$ or a $G$-module.
A Confounding Point
One confusing thing about groups like $\mathsf{GL}(V)$, $\mathsf{U}(n)$, and $\mathsf{O}(n)$ is that they are also defined in terms of matrices that are supposed to represent them.
The homomorphism $\pi$ is a trivial in these so-called defining modules. As we shall see in detail, groups like $\mathsf{O}(n)$ have their own identity, independent of their definition in terms of space like $\mathsf{End}(\mathbb{R}^{n}$). In particular, there is an infinite tower of modules for $\mathsf{O}(n)$, each with a different dimension.
