Representations

0.00 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.

1.40 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.
2.25 There are applications to other areas of mathematics, like invariant theory.
2.50 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

4.30 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:

5.30 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

$R_{*}^{\mathbf{C}}\colon\mathfrak{g}\otimes\mathbf{C}\to\mathfrak{gl}(n,% \mathbf{C})$ .

6.15 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:

7.55 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.

9.20 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

11.00 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.