Representations of Lie Algebras

The representation theory of a semisimple Lie algebra is governed by weights and a single distinguished highest weight per irreducible. This is the machinery that produces the spin-$j$ multiplets of $SU(2)$ and the quark multiplets $3,8,10$ of $SU(3)$. This page builds on lie-algebras.md and the classification of lie-classification.md, and specialises the general linear-representation theory of reps-basics.md to the continuous case.

Weights

Fix a Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$ — a maximal abelian subalgebra of simultaneously diagonalisable generators (dimension $=$ rank $r$; see lie-classification.md). In a representation $V$, the commuting operators $\{H_i{\}}_{i=1}^r$ representing $\mathfrak{h}$ can be simultaneously diagonalised, splitting $V$ into weight spaces: $$V=\underset{\lambda }{⨁}{V}_{\lambda },H_i|\lambda ⟩={\lambda }_i|\lambda ⟩\text{ for }|\lambda ⟩\in V_{\lambda }.$$ The eigenvalue tuple $\lambda =({\lambda }_1,\dots ,{\lambda }_r)$ is a weight — a vector in the dual ${\mathfrak{h}}^{\ast }$. The weights of the adjoint representation are the roots $\alpha$; the remaining (raising/lowering) generators $E_{\alpha }$ shift weights by roots: $$H_i{E}_{\alpha }=E_{\alpha }(H_i+{\alpha }_i),E_{\alpha }:V_{\lambda }\to V_{\lambda +\alpha }.$$ So the $E_{\alpha }$ move a state from one weight space to another, exactly as the ladder operators $J_{\pm }$ do for angular momentum.

The highest-weight theorem

Order the weights (pick a set of positive roots). A highest-weight vector is annihilated by every raising operator, $E_{\alpha }|\Lambda ⟩=0$ for all positive $\alpha$. Then:

Highest-weight (Cartan–Weyl) theorem. Every finite-dimensional irreducible representation of a semisimple Lie algebra has a unique highest weight $\Lambda$, and is generated from its highest-weight vector by lowering operators. Two irreps are equivalent iff their highest weights agree, and the admissible highest weights are exactly the dominant integral weights — non-negative integer combinations $$\Lambda =\sum _{i=1}^r{n}_i{\omega }_i,n_i\in {\mathbb{Z}}_{\ge 0},$$ of the fundamental weights ${\omega }_i$.

The non-negative integers $(n_1,\dots ,n_r)$ are the Dynkin labels — a finite combinatorial address for every irrep. Building the representation is then the purely mechanical process of applying lowering operators until they annihilate, and the Weyl character/dimension formulas give the multiplicities and $\dim V_{\Lambda }$ in closed form.

$\mathfrak{su}(2)$ — angular momentum

Rank $1$, one Dynkin label $n=2j$. The Cartan generator is $J_3$, the roots are $\pm 1$ with ladder operators $J_{\pm }=J_1\pm i{J}_2$: $$[J_3,J_{\pm }]=\pm J_{\pm },[J_{+},J_{-}]=2J_3.$$ Starting from the highest weight $j$, lowering generates the states $m=j,j-1,\dots ,-j$ — a $(2j+1)$-dimensional irrep. The quadratic Casimir $J^2=J_3^2+\frac{1}{2}(J_{+}{J}_{-}+J_{-}{J}_{+})$ acts as $j(j+1)$ (see lie-algebras.md § Casimir operators). This is the quantum theory of angular momentum and spin — worked concretely in examples-classical.md.

$\mathfrak{su}(3)$ — the quark model

Rank $2$, two Dynkin labels $(p,q)$; weights live in a plane and irreps are hexagonal/triangular weight diagrams. The fundamental representations are the quark triplet $3=(1,0)$ and antiquark $\bar{3}=(0,1)$; the adjoint is the gluon octet $8=(1,1)$. The physically central tensor decompositions are $$3\otimes \bar{3}=1\oplus 8(\text{mesons}),3\otimes 3\otimes 3=1\oplus 8\oplus 8\oplus 10(\text{baryons}),$$ the $8$ and $10$ being the Eightfold-Way meson octet and baryon decuplet. The two Casimirs of rank-$2$ $\mathfrak{su}(3)$ supply the two labels. The full physics is in QFT/theories/standard-model/from-postulates.md.

Tensor products and branching

Two operations dominate applications:

• Clebsch–Gordan decomposition — writing $V_{\Lambda }\otimes V_{{\Lambda }^{\prime }}$ as a direct sum of irreps. For $\mathfrak{su}(2)$ this is the addition of angular momenta, $j_1\otimes j_2=|j_1-j_2|\oplus \cdots \oplus (j_1+j_2)$.
• Branching rules — decomposing an irrep of $\mathfrak{g}$ under a subalgebra ${\mathfrak{g}}^{\prime }\subset \mathfrak{g}$. This is how a Standard-Model multiplet splits when a symmetry is broken $G\to H$ (see physics-bridge.md), and how GUT multiplets decompose into Standard-Model ones.

Compact-group reps are the same data

For a compact group, the finite-dimensional irreps of the group and of its (complexified) Lie algebra are in bijection, and all are unitary (compact-groups.md). So the highest-weight combinatorics above simultaneously classifies the representations of $SU(n)$, $SO(n)$, $\mathrm{Sp}(n)$ — the multiplet structure of every internal symmetry in physics.

References

• Fulton & Harris, Representation Theory: A First Course, Part II–III.
• Humphreys, Introduction to Lie Algebras and Representation Theory, Ch. VI.
• Georgi, Lie Algebras in Particle Physics, Ch. 6–8 ($SU(3)$ and the quark model).