# Representations

## Representations

In the next part of the course, we're going to focus on representations of Lie groups. Remember that a (complex) representation of a group is a homomorphism $R\colon G\to GL(n,\mathbf{C})$ , that is an assignment of a matrix $R(g)$ to each group element such that $R(g_{1}g_{2})=R(g_{1})R(g_{2})$ and $R(1)=I$ . The image of this representation is a group of matrices which is a quotient of $G$ . In this course, we'll focus on smooth representations of matrix groups.

Why are we focusing on representations? There are many fantastic applications:

• There are internal applications to the theory of Lie groups: by studying at the adjoint representation, you can completely classify the Lie algebras of compact Lie groups.

• There are applications to other areas of mathematics, like invariant theory.

• There are applications to particle physics. For example, by comparing tables of hadrons found in particle accelerators with weight diagrams of representations of the Lie group $SU(3)$ , Gell-Mann (and independently Ne'eman) was able to guess at the underlying internal structure of these particles and so develop the quark model of matter. This led him to predict a particle with strangeness equal to minus 3, which was discovered in 1964. We will discuss this application in detail.

## Recap

I just want to recap the punchline of the first half of the course, as this will be the basis for everything that comes after.

Given a smooth representation $R\colon G\to GL(n,\mathbf{C})$ we get a Lie algebra representation $R_{*}\colon\mathfrak{g}\to\mathfrak{gl}(n,\mathbf{C})$ , that is a linear map such that $R_{*}[X,Y]=[R_{*}X,R_{*}Y]$ .

Remark:

Here, $\mathfrak{gl}(n,\mathbf{C})$ is a complex vector space, but $\mathfrak{g}$ is a vector space over the real numbers. When I say that $R_{*}\colon\mathfrak{g}\to\mathfrak{gl}(n,\mathbf{C})$ is linear, the only thing that makes sense is for it to be real linear, i.e. $R_{*}(\lambda X)=\lambda R_{*}(X)\ \forall\lambda\in\mathbf{R}.$ We will later complexify $\mathfrak{g}$ to obtain a complex vector space $\mathfrak{g}\otimes\mathbf{C}$ and get an associated complex linear map $R_{*}^{\mathbf{C}}\colon\mathfrak{g}\otimes\mathbf{C}\to\mathfrak{gl}(n,% \mathbf{C})$ .

The key property of $R_{*}$ was the equation $R(\exp X)=\exp(R_{*}X).$ This tells us that $R$ determines $R_{*}$ by differentiation: $R_{*}(X)=\left.\frac{d}{dt}\right|_{t=0}R(\exp(tX)).$ Using the formula $R(\exp X)=\exp(R_{*}X)$ , we see that $R_{*}$ determines $R(g)$ for all $g\in\exp(\mathfrak{g})$ . Does that mean $R_{*}$ determines $R(g)$ for all $g\in G$ ?

Lemma:

If $G$ is a path-connected group (i.e. any two matrices in $G$ are connected by a smooth path of matrices in $G$ ) then $R$ is determined by $R_{*}$ .

This is because, in this case, $G$ is generated as a group by $\exp(\mathfrak{g})\subset G$ . The proof of this lemma is an exercise.

Given $R_{*}\colon\mathfrak{g}\to\mathfrak{gl}(n,\mathbf{C})$ , does $R(\exp X)=\exp(R_{*}X)$ give a well-defined representation $R\colon G\to GL(n,\mathbf{C})$ ? Lie's theorem told us that this is true if $G$ is simply-connected. If $G$ is not simply-connected, we need to think. The first example we'll consider is $U(1)$ , which is not simply-connected, but all the other examples we will consider are simply-connected.

## Plan

Our plan for the rest of the course is:

• study the representations of $U(1)$

• study the representations of $SU(2)$

• study the representations of $SU(3)$

• study the general theory.

## Pre-class exercise

Exercise:

In an earlier video, we constructed a map $R\colon SU(2)\to SO(3)$ . Check that $R(M_{1})=R(M_{2})$ if and only if $M_{1}=\pm M_{2}$ . Given a representation $S\colon SO(3)\to GL(n,\mathbf{C})$ , we get a representation $R\circ S\colon SU(2)\to SO(3)$ . Show that a representation $T\colon SU(2)\to SO(3)$ has this form if and only if $T(-M)=T(M)$ for all $M\in SU(2)$ .

Exercise:

Suppose we have a Lie algebra $\mathfrak{g}\subset\mathfrak{gl}(n,\mathbf{R})$ consisting of real matrices. Consider the subspace $\mathfrak{g}\otimes\mathbf{C}\subset\mathfrak{gl}(n,\mathbf{C})$ consisting of matrices of the form $M+iN$ with $M$ and $N$ in $\mathfrak{g}$ . Show that:

• this is a Lie subalgebra, i.e. that it is preserved by Lie bracket

• if $f\colon\mathfrak{g}\to\mathfrak{gl}(m,\mathbf{C})$ is a real-linear Lie algebra homomorphism then $f^{\mathbf{C}}\colon\mathfrak{g}\otimes\mathbf{C}\to\mathfrak{gl}(m,\mathbf{C})$ defined by $f^{\mathbf{C}}(M+iN)=f(M)+if(N)$ for $M$ and $N$ in $\mathfrak{g}$ is also a Lie algebra homomorphism.