Manifold properties of matrix groups

As real manifolds, we can list various properties of the matrix groups for \({n>1}\) and \({rs\neq0}\).

Group Dimension Compact Connectedness
\({GL\left(n,\mathbb{R}\right)}\) \({n^{2}}\) No 2 components
\({SL\left(n,\mathbb{R}\right)}\) \({n^{2}-1}\) No Connected
\({O\left(n\right)}\) \({n\left(n-1\right)/2}\) Yes 2 components
\({SO\left(n\right)}\) \({n\left(n-1\right)/2}\) Yes Connected
\({O\left(r,s\right)}\) \({n\left(n-1\right)/2}\) No 4 components
\({SO\left(r,s\right)}\) \({n\left(n-1\right)/2}\) No 2 components
\({Sp\left(2n,\mathbb{R}\right)}\) \({\left(2n\right)\left(2n+1\right)/2}\) No Connected
\({GL\left(n,\mathbb{C}\right)}\) \({2n^{2}}\) No Connected
\({SL\left(n,\mathbb{C}\right)}\) \({2(n^{2}-1)}\) No Simply connected
\({U\left(n\right)}\) \({n^{2}}\) Yes Connected
\({SU\left(n\right)}\) \({n^{2}-1}\) Yes Simply connected
\({Sp\left(2n,\mathbb{C}\right)}\) \({\left(2n\right)\left(2n+1\right)}\) No Simply connected
\({Sp\left(n\right)}\) \({n\left(2n+1\right)}\) Yes Simply connected

Notes: In particular, since \({U(n)}\) is compact and connected, any unitary matrix \({U}\) can be written as \({U=e^{iH}}\) for some hermitian matrix \({H}\).

Since the exponential map is surjective for any compact connected Lie group, it is surjective for \({SO(n)}\), \({U(n)}\) and \({SU(n)}\). As it turns out, it is also surjective for \({SO(3,1)^{e}}\) and \({GL(n,\mathbb{C})}\).

We can also characterize the topology of matrix groups by noting some diffeomorphisms of their manifolds:

  • \({O(n+1)/O(n)\cong SO(n+1)/SO(n)\cong S^{n}}\)
  • \({U(n+1)/U(n)\cong SU(n+1)/SU(n)\cong S^{2n+1}}\)
  • In particular, we then have \({U(1)\cong SO(2)\cong S^{1},\: SU(2)\cong S^{3}}\)
  • \({O(n)\cong S^{0}\times SO(n);\: U(n)\cong S^{1}\times SU(n)}\)
  • Thus \({SO\left(n+1\right)/O(n)\cong\mathbb{R}\textrm{P}^{n}}\); \({SU\left(n+1\right)/U(n)\cong\mathbb{C}\textrm{P}^{n}}\)
  • In particular, we then have \({SO(3)\cong\mathbb{R}\textrm{P}^{3}}\)
  • \({SO(4)\cong S^{3}\times SO(3)}\); \({SO(8)\cong S^{7}\times SO(7)}\)
  • \({Sp(2,\mathbb{R})\cong SL(2,\mathbb{R})}\); \({Sp(2,\mathbb{C})\cong SL(2,\mathbb{C})}\); \({Sp(1)\cong SU(2)\cong S^{3}}\)

With regard to homotopy groups, some facts are:

  • \({\pi_{1}(G)}\) is abelian for any Lie group \({G}\) (in fact for any H-space)
  • \({\pi_{2}(G)=0}\) for any Lie group \({G}\)
  • \({\pi_{1}\left(SO\left(n\right)\right)=\mathbb{Z}_{2}}\) for \({n>2}\); \({\pi_{3}\left(SU\left(n\right)\right)=\mathbb{Z}}\) for \({n>1}\)

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