Actions of the unit circle on finite complex matrices Let $M_2(\mathbb{C})$ be the algebra of $2\times 2$ complex matrices and $\mathbb{S}^1$ the unit circle.

How many actions of $\mathbb{S}^1$ on $M_2(\mathbb{C})$ exist (up to isomorphism)? And on $M_n(\mathbb{C})?$

 A: As Qiaochu mentioned in a comment, every automorphism of $M_n(\mathbb{C})$ is inner (this is easy to show by considering what happens to matrices with only one nonzero entry), so you are asking to classify continuous homomorphisms $S^1\to PGL_n(\mathbb{C})$.  These are easier to think about by passing to the universal covers, so let's think about homomorphisms $f:\mathbb{R}\to SL_n(\mathbb{C})$.  Every such homomorphism is of the form $f(t)=\exp(2\pi i tA/n)$ for a unique $A\in \mathfrak{sl}_n(\mathbb{C})$ (the factor of $2\pi i/n$ is included because it will make the final answer nicer).  When does such a homomorphism descend to a map $S^1\to PGL_n(\mathbb{C})$ (thinking of $S^1$ as $\mathbb{R}/\mathbb{Z})$)?  Well, this happens iff $f(1)=\exp(2\pi i A/n)$ is a scalar matrix that is an $n$th root of unity.  Explicitly, this happens iff $A$ is a diagonalizable matrix whose eigenvalues are integers that are all equal mod $n$.
So to sum up, actions of $S^1$ on $M_n(\mathbb{C})$ are naturally in bijection with traceless diagonalizable matrices $A$ whose eigenvalues are integers that are all equal mod $n$.  Given such a matrix, the action of $e^{2\pi i t}\in S^1$ on $M_n(\mathbb{C})$ is given by conjugation by $\exp(2\pi i tA/n)$.  For $n=2$, such matrices $A$ have a particularly simple description: they are matrices with eigenvalues $k$ and $-k$ for some integer $k$.
