Remark

We have seen above that $\mathbf{S}\mathbf{U}(2,\mathbf{C})$ is simply-connected and that $\pi_1(\mathbf{S}\mathbf{O}(3,\mathbf{R}))$ is of order 2. We shall see much later that these results generalize to $\mathbf{S}\mathbf{U}(n,\mathbf{C}) (n\geq 1)$ and $\mathbf{S}\mathbf{O}(n,\mathbf{R}) (n\geq 3)$, respectively (cf. also §3, Exerc. 4 and 5).

Recall (Chap. VIII, §1, no. 1) that the canonical basis of $\mathfrak{s}\mathfrak{l}(2,\mathbf{C})$ is the basis $(X_+, X_-, H)$, where

$(01)(00)(1$ 0 $)$

$X_+=$ 0 0 $, X_-=-1$ 0 $, H=$ 0 $-1$.

We thus obtain a basis $(U,V,iH)$ of $\mathfrak{su}(2,\mathbf C)$, also called canonical, by putting

We have

If B denotes the Killing form of $\mathfrak{su}(2,\mathbf C)$ an immediate calculation gives

so that, if we identify $\mathfrak{su}(2,\mathbf C)$ with $\mathbf R^3$ by means of the canonical basis, the adjoint representation of $\mathbf{SU}(2,\mathbf C)$ defines a homomorphism $\mathbf{SU}(2,\mathbf C)\to\mathbf{SO}(3,\mathbf R)$ (cf. above).

Further, note that $\mathbf RiH$ is a Cartan subalgebra of $\mathfrak{su}(2,\mathbf C)$, that the maximal torus T of $\mathbf{SU}(2,\mathbf C)$ that corresponds to it consists of the diagonal matrices

where $a\bar a=1$, and that the exponential map maps $xH$, for $x\in\mathbf Ri$, to the matrix and thus has kernel $\mathbf Z.K$, where K is the element of $\mathfrak{su}(2,\mathbf C)$ defined by

Further, the centre of $\mathbf{SU}(2,\mathbf C)$ consists of the identity and $\exp(K/2)$.

Put

By Chap. VIII, §1, no. 5,

Finally, for $t= \begin{pmatrix} a&0\\ 0&\bar a \end{pmatrix} \in T$, we have