Example: $SU(2)$ and its Lie algebra

The special unitary group

SU(2)=\left\{\begin{pmatrix}a&b\\-\overline b&\overline a\end{pmatrix} : |a|^2+|b|^2=1\right\}

is a compact, connected, simply connected Lie group (topologically $S^3$ ).

Its Lie algebra is $\mathfrak{su}(2)$ , the traceless skew-Hermitian $2\times 2$ matrices.

A concrete basis and commutators

Using the Pauli matrices $\sigma_1,\sigma_2,\sigma_3$ , set

X_i := \frac{i}{2}\sigma_i \in \mathfrak{su}(2).

A standard multiplication calculation yields

[X_1,X_2] = -X_3,\qquad [X_2,X_3]=-X_1,\qquad [X_3,X_1]=-X_2,

so (up to an overall sign convention) $\mathfrak{su}(2)$ has the same structure constants as $\mathfrak{so}(3)$ (compare the $\mathfrak{so}(3)$ example).

Exponentials (explicit)

For $t\in\mathbb R$ ,

\exp(tX_3)=\exp\!\left(\frac{it}{2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}\right) = \begin{pmatrix}e^{it/2}&0\\0&e^{-it/2}\end{pmatrix}\in SU(2).

This shows directly that one-parameter subgroups arise from the exponential map.

The 2-to-1 covering

SU(2)\to SO(3)

Let $SU(2)$ act on $\mathfrak{su}(2)$ by conjugation $g\cdot X=gXg^{-1}$ . This preserves the (negative) Killing form inner product, giving a homomorphism

\mathrm{Ad}:SU(2)\to SO(\mathfrak{su}(2))\cong SO(3)

(compare adjoint action). One checks:

$\mathrm{Ad}$ is surjective,
$\ker(\mathrm{Ad})=\{\pm I\}$ .

Hence $\mathrm{Ad}$ is a covering Lie group map $SU(2)\to SO(3)$ with discrete central kernel $\{\pm I\}$ .

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Definition

Basic structure and context