Exercise 5, § 3

For $n$ an integer $\geq 1$, denote by Spin$(n,\mathbf{R})$ the reduced Clifford group associated to the usual quadratic form on $\mathbf{R}^n($Algebra, Chap. IX, §9, no. 5).

a) Show that Spin$(n,\mathbf{R})$ is a compact Lie group and that the surjective homomorphism $\varphi :$ Spin$(n,\mathbf{R})\rightarrow \mathbf{S}\mathbf{O}(n,\mathbf{R}) ($loc. cit.) is analytic, with kernel $\{+1,-1\}$.

b) Show that, for $n\geq 2$, Spin$(n,\mathbf{R})$ is connected and simply-connected (use Exerc. 2). The group $\pi_1(\mathbf{S}\mathbf{O}(n,\mathbf{R}))$ is cyclic of order 2.

c) Let $Z_n$ be the centre of Spin$(n,\mathbf{R})$. Show that $Z_n=\{+1,-1\}$ if $n$ is odd, and $Z_n=\{+1,-1, \varepsilon ,-\varepsilon \}$ if $n$ is even, where $\varepsilon =e_1. . . e_n$ is the product of the elements of the canonical basis of $\mathbf{R}^n$; the group $Z_{2r}$ is isomorphic to $(\mathbf{Z}/2\mathbf{Z})^2$ (resp. to $\mathbf{Z}/4\mathbf{Z}$) if $r$ is even (resp. odd).

d) Prove that every almost simple connected compact Lie group of type $B_n$ $(n\geq 2)$ is isomorphic to Spin$(2n+ 1,\mathbf{R})$ or $\mathbf{S}\mathbf{O}(2n+ 1,\mathbf{R})$.

e) If $r$ is odd (resp. even) and $\geq 2$, every almost simple connected compact Lie group of type $D_r$ is isomorphic to Spin$(2r,\mathbf{R}),\mathbf{S}\mathbf{O}(2r,\mathbf{R})$ or $\mathbf{S}\mathbf{O}(2r,\mathbf{R})/\{\pm I_{2r}\}$ (resp. to one of the preceding groups or Spin$(2r,\mathbf{R})/\{1, \varepsilon \}$).