Worked Classical Examples: $SO(2)$, $SU(2)$, $SO(3)$

The smallest non-trivial Lie groups exercise nearly all of the machinery of this folder — axioms, algebra, exponential map, topology, covers, and representations — without notational overhead. This page works $SO(2)$, $SU(2)$, and $SO(3)$ end-to-end and tabulates the comparison. It ties together lie-groups.md, lie-algebras.md, topology-covers.md, and reps-lie.md, and is the direct background for the Lorentz/Poincaré analysis in QFT/preliminaries.md.

$SO(2)$ — rotations in the plane

$SO(2)$ is the group of $2\times 2$ rotation matrices $$R(\theta )=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right),\theta \in [0,2\pi ).$$

• Group axioms. $R({\theta }_1)R({\theta }_2)=R({\theta }_1+{\theta }_2)$ (closure, associativity); $R(0)=1$; $R(\theta {)}^{-1}=R(-\theta )$. Abelian.
• Topology. Diffeomorphic to the circle $S^1$ — compact and connected but not simply connected; ${\pi }_1(SO(2))=\mathbb{Z}$. Universal cover $\mathbb{R}$ with covering map $\theta \mapsto e^{i\theta }$ and kernel $2\pi \mathbb{Z}$ (see topology-covers.md).
• Lie algebra. $\mathfrak{so}(2)\cong \mathbb{R}$, one-dimensional. Generator $J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ (antisymmetric); the physics-convention Hermitian generator is $T=-iJ$, with $R(\theta )=\exp (i\theta T)$. Automatically abelian, $[T,T]=0$, no structure constants.
• Representations. All irreps are $1$-dimensional (abelian + compact ⇒ irreps are characters), labelled by $n\in \mathbb{Z}$: $${\rho }_n(R(\theta ))=e^{in\theta }.$$ Integrality is forced by $R(2\pi )=1$; non-integer $n$ would give projective reps of $SO(2)$ = ordinary reps of the cover $\mathbb{R}$. These characters are the classical Fourier modes (Peter–Weyl for $S^1$, see compact-groups.md).
• Physics use. Planar rotations; and $U(1)\cong SO(2)$, the gauge group of electromagnetism.

$SU(2)$ — the simplest non-abelian compact group

$SU(2)$ is the group of $2\times 2$ complex unitary matrices of determinant $1$: $$U=\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right),|a{|}^2+|b{|}^2=1.$$

• Group axioms. Matrix multiplication; identity $1_2$; inverse $U^{-1}=U^{†}$. Non-abelian.
• Topology. The constraint $|a{|}^2+|b{|}^2=1$ identifies $SU(2)$ with the unit $3$-sphere $S^3\subset {\mathbb{C}}^2$ — compact, connected, and simply connected, so $SU(2)$ is its own universal cover. It double-covers $SO(3)$: $$SU(2)\stackrel{2:1}{\to }SO(3),\ker =\{\pm 1\}={\mathbb{Z}}_2.$$
• Lie algebra. $\mathfrak{su}(2)=\{X:X^{†}=-X,\mathrm{tr}X=0\}$, real $3$-dimensional. Generators $T^a={\sigma }^a/2$ with the Pauli matrices $${\sigma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$ Hermitian and traceless, so $U=\exp (i{\theta }^a{T}^a)$. Structure constants $[T^a,T^b]=i{ϵ}^{abc}{T}^c$, i.e. $f^{abc}={ϵ}^{abc}$.
• Casimir. Rank $1$ ⇒ one Casimir $T^2=T^a{T}^a=\frac{3}{4}1$ on the defining rep; on the spin-$j$ irrep it is $j(j+1)1$.
• Representations. Irreps labelled by $j\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$, of dimension $2j+1$ (see reps-lie.md). The defining rep ($j=\frac{1}{2}$) is the spinor; $j=1$ is the vector rep that descends to $SO(3)$. Half-integer $j$ are projective reps of $SO(3)$ = ordinary reps of $SU(2)$ — the reason a $2\pi$ rotation flips the sign of a spin-$\frac{1}{2}$ wavefunction.
• Physics use. Spin in QM, isospin, the $SU(2{)}_L$ of the electroweak group, and (complexified) half of the Lorentz algebra.

$SO(3)$ — rotations in space, and the double cover

$SO(3)$ is the group of $3\times 3$ real orthogonal matrices with $\det =1$ — rotations of ${\mathbb{R}}^3$.

• Topology. $SO(3)\cong {\mathbb{RP}}^3$ (the ball of radius $\pi$ with antipodal boundary points identified): compact, connected, not simply connected, ${\pi }_1(SO(3))={\mathbb{Z}}_2$. The two loop classes are the physical content of the belt/plate trick: a $2\pi$ rotation cannot be undone continuously, but a $4\pi$ rotation can.
• Lie algebra. $\mathfrak{so}(3)=\{$antisymmetric $3\times 3\}$, with Hermitian generators $(J^a{)}_{bc}=-i{ϵ}^{abc}$ satisfying the same brackets $[J^a,J^b]=i{ϵ}^{abc}{J}^c$ as $\mathfrak{su}(2)$. The two algebras are isomorphic — $\mathfrak{so}(3)\cong \mathfrak{su}(2)$ — even though the groups differ globally. This is the cleanest illustration that the algebra sees only the identity component, not the global topology (see topology-covers.md).
• The double cover explicitly. Map $SU(2)\to SO(3)$ by $U\mapsto R_U$ where $R_U$ acts on $\vec{x}\cdot \vec{\sigma }$ by $U(\vec{x}\cdot \vec{\sigma })U^{†}=(R_U\vec{x})\cdot \vec{\sigma }$. Both $U$ and $-U$ give the same $R_U$, so the map is $2:1$ with kernel $\{\pm 1\}$; hence $SO(3)\cong SU(2)/{\mathbb{Z}}_2$.
• Representations. Only the integer-$j$ irreps of $SU(2)$ descend to genuine reps of $SO(3)$ (dimension $2j+1=1,3,5,\dots$): these are the scalars, vectors, and rank-$\ell$ spherical harmonics. Half-integer $j$ are only projective — realised faithfully on the cover $SU(2)$.

Side-by-side summary

The middle two columns share an algebra; the outer two share a topology type (non-simply-connected). Together they display every phenomenon the general theory predicts: abelian vs. non-abelian, the role of ${\pi }_1$, and how covers manufacture the half-integer spins of quantum mechanics.

References

• Hall, Lie Groups, Lie Algebras, and Representations, Ch. 1, 4.
• Stillwell, Naive Lie Theory, Ch. 2–4.
• Sakurai & Napolitano, Modern Quantum Mechanics, Ch. 3 (the $SU(2)$–$SO(3)$ connection through spin).