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The representation theory of SU(2)

Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of $\text{SU}(2)$ (which you’ll recall we assumed way back in this previous post). As a first step, it turns out that the finite-dimensional representation theory of compact groups looks a lot like the finite-dimensional representation theory of finite groups, and this will be a major boon to three of the proofs. The last proof will instead proceed by classifying irreducible representations of the Lie algebra $\mathfrak{su}(2)$.

At the end of the post we’ll briefly describe what we can conclude from all this about electrons orbiting a hydrogen atom.

The quaternions and Lie algebras II

We now know what a Lie algebra is and we know they are abstractions of infinitesimal symmetries, which are given by derivations. Today we will see what we can say about associating infinitesimal symmetries to continuous symmetries: that is, given a matrix Lie group $G$, we will describe its associated Lie algebra $\mathfrak{g}$ of infinitesimal elements and the exponential map $\mathfrak{g} \to G$ which promotes infinitesimal symmetries to real ones.

As in the other post, I will be ignoring some technical details for the sake of exposition. For example, I am generally not specifying how I’m topologizing various objects, and this is because of the general fact that a finite-dimensional real vector space has a unique Hausdorff topology compatible with addition and scalar multiplication. Whenever I talk about limits in such a vector space, I therefore don’t need to specify how I’m imposing a topology, although it will generally be convenient to induce it via a norm (which I am also not specifying).

Euler characteristic as homotopy cardinality

Let $X$ be a finite CW complex with $c_0$ vertices, $c_1$ edges, and in general $c_i$ different $i$-cells. The Euler characteristic

$\displaystyle \chi(X) = \sum_{i \ge 0} (-1)^i c_i$

is a fundamental invariant of $X$, and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula $V - E + F = 2 = \chi(S^2)$ for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups $H_i(X, \mathbb{Q})$, but actually one can work on a more intuitive level characterized by the following slogan:

The Euler characteristic is a homotopy-invariant generalization of cardinality.

More precisely, the above expression for Euler characteristic can be deduced from three simple axioms:

1. Cardinality: $\chi(\text{pt}) = 1$.
2. Homotopy invariance: If $X \sim Y$, then $\chi(X) = \chi(Y)$.
3. Inclusion-exclusion: Suppose $X$ is the union of two subcomplexes $A, B$ whose intersection $A \cap B$ is a subcomplex of both $A$ and $B$. Then $\chi(X) = \chi(A) + \chi(B) - \chi(A \cap B)$.

Of course, this isn’t enough to conclude that there actually exists an invariant with these properties. Nevertheless, it’s enough to motivate the search for a proof that such an invariant exists.