Lie Theory Through Examples 4
Posted by John Baez
This week in our seminar we’ll do some examples illustrating how a representation of a simply-connected complex simple Lie group $G$ gives rise to a function $d : L^* \to \mathbb{N}$ where $L^*$ is the ‘weight lattice’ of $G$. Wonderfully, this function completely determines the representation (up to equivalence).
In physics, the most famous example is the meson octet, corresponding to the obvious representation of $SU(3)$ on $sl(3,\mathbb{C})$. It looks like this…
Here we see 8 mesons classified according to their charge $q$ and strangeness $s$ — which are eigenvalues of two elements of $su(3)$ generating a maximal torus in $SU(3) \subset SL(3,\mathbb{C})$.
It takes a bit of practice to turn this chart into a function from the hexagonal $A_3$ lattice to the natural numbers! Each particle counts for one. So, the 6 corners of the hexagon have $d = 1$, while the point in the middle, drawn as two points in this chart, has $d = 2$, since there are two mesons here with vanishing charge and strangeness: the neutral pion $\pi^0$ and the eta $\eta$. The total dimension of this representation is thus $6 + 2 = 8$ — the dimension of $sl(3,\mathbb{C})$.
Here are the lecture notes:
- Lecture 4 (Oct. 28) - Classifying representations using weights. The example of A_{1}, which corresponds to the group SU(2). The relation between SU(2) and SO(3), and their representations. The example of A_{2}, which corresponds to the group SU(3).
By the way, I rewrote last week’s notes.
Re: Lie Theory Through Examples 4
Nice lecture, but there’s a typo in the U(1) matrix on page 1.