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 from big G to big G L n C, that is an assignment of a matrix R of g to each group element such that of g_1 g_2 equals R of g_1 times R of g_2 and R of the identity equals the identity matrix. The image of this representation is a group of matrices which is a quotient of big 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.
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 from big G to big G L n C we get a Lie algebra representation R star from little g to little g l n C, that is a linear map such that R star of X bracket Y equals R star X bracket R star Y.
Here, little g l n C is a complex vector space, but little g is a vector space over the real numbers. When I say that R star from little g to little g l n C is linear, the only thing that makes sense is for it to be real linear, i.e. R star lambda X equals lambda R star X for all real numbers lambda We will later complexify little g to obtain a complex vector space little g tensor C and get an associated complex linear map R star superscript C from little g to little g l n C.
The key property of R star was the equation R of exp X equals exp of R star X This tells us that R determines R star by differentiation: R star X equals d by d t at t = 0 of R of exp t X Using the formula R of exp X equals exp of R star X, we see that R star determines R of g for all g in exp of little g. Does that mean R star determines R of g for all g in big G?
If big G is a path-connected group (i.e. any two matrices in big G are connected by a smooth path of matrices in big G) then R is determined by R star.
This is because, in this case, big G is generated as a group by the subset exp of little g inside big G. The proof of this lemma is an exercise.
Given R star from little g to little g l n C, does R of exp X equals exp of R star X give a well-defined representation R from big G to big G L n C? Lie's theorem told us that this is true if big G is simply-connected. If big 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.
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.
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,\CC), 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).
Suppose we have a Lie algebra little g inside little g l n R consisting of real matrices. Consider the subspace little g tensor C inside little g l n C consisting of matrices of the form M + i N with M and N in little g. Show that:
this is a Lie subalgebra, i.e. that it is preserved by Lie bracket
if f from little g to little g l m C is a real-linear Lie algebra homomorphism then f superscript C from little g tensor C to little g l m C defined by f superscript C of M + i N equals f of M plus i f of N for M and N in little g is also a Lie algebra homomorphism.