Lie groups, global forms, and topology

A Lie algebra determines infinitesimal transformations. To decide which multiplets exist, how they restrict, or what topological sectors are available, also specify the real group, its global form, the relevant map, and the space being studied. This path builds that chain of decisions using $SU(2)$, $SO(3)$, and their common sphere orbit.

Start with linear algebra, tensor products and the definition of a Lie algebra. The representation calculations here are finite-dimensional and complex, in characteristic zero. The main groups are connected and compact; the real-form step explains why that qualification matters. Work through the steps in order, retaining the indicated result from each.

1. Separate the algebra from the real and global groups

Read Lie algebras and global groups, then Real forms. Distinguish the real algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}_2(\mathbb R)$, even though both complexify to $\mathfrak{sl}_2(\mathbb C)$. Their compactness and unitary representation questions differ. Within the compact choice, $SU(2)$ and $SO(3)$ instead have isomorphic real Lie algebras and different topology.

Carry forward: a label such as $A_1$ names a complex simple type; it does not choose between these real or global objects. Compare the SU(2), SO(3) and real-form atlas records. Use Irreducible representations if you need to settle the representation category before proceeding.

2. Turn highest weights into explicit checks

Study Roots and weights, then use Highest weights in practice. For the compact rank-one example, write $V_n=\operatorname{Sym}^n\mathbb C^2$, $n=2j\geq0$. Its weights are $n,n-2,\ldots,-n$ and its dimension is $n+1$. The compact Lie formula sheet fixes the normalization of its quadratic Casimir, $C_2=n(n+2)/4=j(j+1)$.

Carry forward: the doublet has $(n,\dim,C_2)=(1,2,3/4)$ and the triplet has $(2,3,2)$. Calculate both with the highest-weight tool. A dimension and Casimir do not yet decide the global group on which a representation is defined.

3. Test descent to the chosen quotient

Read Covering groups and global forms. The spin covering has kernel $\{I,-I\}$, and $-I$ acts on $V_n$ by $(-1)^n$. Thus only even $n$ descend to $SO(3)$. Reproduce the doublet obstruction and the triplet descent.

Carry forward: $SU(2)$ is simply connected, while $\pi_1(SO(3))\cong\mathbb Z_2$. The relation between a simply connected cover’s discrete kernel and the quotient’s fundamental group follows from Conrad, Fundamental groups of Lie groups, pp. 1–2. For a covering group that is not simply connected, retain the full exact sequence; its kernel alone does not determine the target’s fundamental group.

4. Specify a map before restricting or breaking symmetry

Continue to Embeddings and branching. For the regular $SU(2)\to SU(3)$ map the defining triplet restricts to $V_1\oplus V_0$. For the principal map it restricts to $V_2$, and the image is $SO(3)$ because the kernel is $\{I,-I\}$. The lesson derives the six- and eight-dimensional restrictions and then includes the circle charges of an $SU(5)$ block map.

Carry forward: record domain, codomain, kernel, image and torus map from the map atlas. A fixed vector only establishes that the chosen image lies in its stabilizer. Compute the full stabilizer before using it as the $H$ in a vacuum orbit $G/H$.

5. Compare two sphere orbits without identifying their groups

Read Groups, cosets, and classifying spaces and its topology formula sheet. With the diagonal circle in $SU(2)$ and axial rotations in $SO(3)$,

$$SU(2)/U(1)\cong S^2, \qquad SO(3)/SO(2)\cong S^2.$$

These are quotients by closed subgroups, so their projections are fiber bundles. Compare their exact sequences in the topology workbench. For compatible circle generators, choose the sphere generator so the first boundary is positive. The two boundary maps are then

$$\begin{aligned} \partial_{SU(2)}:\pi_2(S^2)\longrightarrow\pi_1(U(1)), &\qquad k\longmapsto k,\\ \partial_{SO(3)}:\pi_2(S^2)\longrightarrow\pi_1(SO(2)), &\qquad k\longmapsto2k. \end{aligned}$$

The first is the Hopf-bundle isomorphism; the second follows because the spin covering induces $z\mapsto z^2$ on the circle fibers and the identity on the identified sphere orbits. See Hatcher, Theorem 4.41 and the preceding Hopf example, pp. 375–376 for the exact-sequence method. The lesson supplies the maps and sign convention explicitly.

Carry forward: both orbits have $\pi_2=\mathbb Z$ and $\pi_1=0$. Their group fundamental groups differ, and their boundary maps remember that difference. A unit sphere degree gives winding one in the first circle and winding two in the second. Comparing physical magnetic charges requires the electric-charge and generator normalization as well.

6. Ask a separate question about gauge bundles

Return to the classifying-space part of the same lesson. For a Lie group $G$ and a CW complex $X$, isomorphism classes of principal $G$-bundles are classified by $[X,BG]$. For connected $G$, $BG$ is simply connected and $\pi_2(BG)\cong\pi_1(G)$. These statements are Mitchell, Notes on principal bundles and classifying spaces, Theorem 7.4, p. 15, and Corollary 11.2, p. 23.

Carry forward: there is one principal $SU(2)$-bundle class over $S^2$ and two principal $SO(3)$-bundle classes. Neither count is the $\mathbb Z$ of sphere maps into the vacuum orbit. Nor does a bundle class specify a connection or establish a stable classical field solution.

Complete the linked exercise

Work through Global forms and their consequences. For the running example, your finished calculation should distinguish three outputs: the allowed highest-weight parity, the circle winding induced by a unit sphere in the orbit, and the principal-bundle class on the surrounding sphere. Check that an odd $SO(2)$ winding induces the nontrivial $SO(3)$-bundle, so it cannot extend as a smooth bundle over a ball. The exercise also tests an intermediate central quotient and a product kernel, where remembering only a group’s name is insufficient.

These steps cover finite-dimensional compact representation and elementary bundle/homotopy questions. They do not classify noncompact unitary duals, all higher homotopy groups, or the dynamical stability of defects. Continue to global gauge groups and line operators for the additional physical choices that a gauge algebra and matter list leave open.