Analogues of $SU(2)$ and $SO(3)$

Question

The groups $SU_2(\mathbb{C})$ and $SO_3(\mathbb{R})$ are interesting in geometry, and there is a $2$-to-$1$ map from $SU_2(\mathbb{C})$ to $SO_3(\mathbb{R})$. There are finitely many finite groups in $SO_3(\mathbb{R})$ (up to isomorphism), and correspondingly, there are finitely many finite groups in $SU_2(\mathbb{C})$, which are generally called as double groups.

What are the higher dimensional analogues of $SU_2(\mathbb{C})$, $SO_3(\mathbb{R})$, with similar map (as $2$-to-$1$ above), and where we can realize interesting families of finite groups as "subgroups" (as above).

Answer 1

Regarding generalizing the $2$-to-$1$ map, you want covering space theory. The covering space theory of Lie groups is well-understood; every connected Lie group $G$ has a universal cover $\tilde{G}$ which is a simply connected Lie group, completely determined by its Lie algebra $\mathfrak{g}$, and the covering map $\tilde{G} \to G$ is the quotient by a discrete central subgroup of $\tilde{G}$. So to classify such covering maps it suffices to

• classify simply connected Lie groups (equivalently, Lie algebras) and
• compute their centers.

The classification of Lie algebras is hopeless in general, but it can in some sense (see Levi decomposition) be reduced to the semisimple case, where amazingly there is a complete classification.

Example. If $G = \text{SO}(n)$ is the special orthogonal group of rotations in $\mathbb{R}^n$, then for $n \ge 3$ the corresponding universal cover is the spin group $\text{Spin}(n)$ (so $\text{Spin}(3) \cong \text{SU}(2)$; this is an exceptional isomorphism). The spin groups are important in geometry and physics.

Subexample. If $G = \text{SO}(4)$ then its universal cover is $\text{Spin}(4)$, which has an exceptional isomorphism with $\text{SU}(2) \times \text{SU}(2)$. As in the case $n = 3$ this exceptional isomorphism can be explained using quaternions.

The simplest introduction to Lie theory I know, which will at least acquaint you with the basic examples, is Stillwell's Naive Lie theory.

Regarding finite subgroups, one keyword here is "McKay correspondence." You may also be interested in learning about the representation theory of finite groups.