What is PSU(1,1) PSU(2,1)? Is there a general group of the form PSU(n,m)?

Question

I have seen it written about groups PSU(1,1), PSU(2,1). But what exactly are these? The definitions were not given, and I can't seem to find a definition online.

Moreover is there a general class of these groups of the form PSU(n,m)?

Answer 1

The unitary group of signature $(n,m)$ is the group of matrices preserving a Hermitian form of signature $(n,m)$, and is isomorphic to this group preserving a particularly simple such form (a diagonal one): $$\operatorname{U}(n,m) = \left\{ \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \operatorname{GL}_{n+m}(\mathbb{C}) : AA^* - BB^* = I_n,~ DD^*-CC^* = I_m,~ AC^* = BD^* \right\}$$

The special unitary group $\operatorname{SU}(n,m) = \left\{ M \in \operatorname{U}(n,m) : \det(M) = 1 \right\}$ is restricted to those matrices of determinant $1$.

The projective special unitary group $\operatorname{PSU}(n,m) = SU(n,m)/Z$ is the quotient of this group be its subgroup of scalar matrices $Z=\left\{\begin{bmatrix} zI_n & 0 \\ 0 & zI_m \end{bmatrix} : |z|=1, z^{n+m} = 1 \right\}$.

See the wikipedia page for more information and references.