I am reading some papers in physics. I don't know some notations in those papers. For example, $SU(1, 1)$, $U(1)$. I think these are Lie groups which consist of matrices. But I don't know what kind of matrices are in these groups.
What are elements in $SU(1, 1)$ and how to show that $U(1)$ is the maximal subgroup of $SU(1, 1)$?
Thank you very much.
The group $\rm{SU}(1,1)$ has a faithful representation as the group of complex matrices $$\left\{ \begin{bmatrix} a & b \\ \bar{b} & \bar{a}\end{bmatrix} \text{ st. } |a|^2 - |b|^2 = 1 \right\} .$$
The group $\rm U(1)$ is a natural subgroup of $\rm{SU}(1,1)$, in the representation above $$ \mathrm{U}(1) \simeq \left\{\begin{bmatrix} a & 0 \\ 0 & \bar{a}\end{bmatrix}, a \in \mathbb U\right\}.$$
Any subgroup of $\rm{SU}(1,1)$ containing strictly $\rm U(1)$ contains a matrix $\mathrm{M} = \begin{bmatrix} a & b \\ \bar{b} & \bar{a}\end{bmatrix}$ with $|a|^2 - |b|^2 = 1$ and $a$ and $b$ both not zero.
This implies that $|a| > 1$ and so $\rm M$ has an eigenvalue strictly greater than $1$. Hence $||\mathrm M||^n \to \infty$ and the group is not compact.