Definition of PU(2,1)?

Question

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of the two dimensional complex ball. What is the definition of $PU(2,1)$, how does it differ from $PU(2)$, and how does it act on the complex ball?

Answer 1

$U(p,q)$ is the symmetry group of the Hermitian form of type $(p,q),$ defined by $$|| (z_1,\ldots,z_n)||^2 = |z_1|^2 + \cdots + |z_p|^2 - |z_{p+1}|^2 - \cdots - |z_{p+q}|^2.$$

Let's consider the case $p = n, q = 1$. Any vector in $\mathbb C^{n+1}$ can have positive, zero, or negative norm, and pretty clearly this depends only on the complex line spanned by $z$. We can thus speak of lines of positive, negative, or zero norm.

The lines of negative norm then form a copy of the complex $n$-ball, sitting inside $\mathbb C P^n$. The property of having negative norm is invariant under $U(n,1)$, and so we get an action of $PU(n,1)$ on the $n$-ball, compatible with the usual action of $PGL(n+1, \mathbb C)$ on $\mathbb C P^n$.

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This link seems helpful. (Not sure if it's freely available.)