Compact Lie Groups, Roots, Weights, and Weyl Structure

For a compact connected Lie group, the representation problem becomes discrete after one chooses a maximal torus. Its commuting generators split every finite-dimensional complex representation into simultaneous eigenspaces, called weight spaces. The nonzero weights of the adjoint representation are the roots; their root operators move vectors between weight spaces. Coroots define reflections, those reflections generate the Weyl group, and a choice of positive roots selects one dominant chamber.

The resulting classification has an essential global qualification. Every irreducible representation has one dominant highest weight, but that weight must be a character of the maximal torus of the chosen group. For a simply connected compact semisimple group every dominant integral Lie-algebra weight is allowed. A central quotient can remove some of them, even though the Lie algebra and root system do not change. Abelian central factors add characters but no roots.

Thus roots, weights, and Weyl symmetry organize compact-group representations by converting matrices into a finite root system, an integral character lattice, and a dominant label. The method does not classify arbitrary noncompact, disconnected, or infinite-dimensional representations, and a weight diagram alone does not determine physical interactions.

Required background. Lie Groups, Lie Algebras, and Exponential and Adjoint Maps supplies maximal tori, complexification, adjoint actions, and exponentiation; Representations, Intertwiners, Invariants, and Tensor Decomposition supplies weight spaces, complete reducibility, tensor products, and descent.

Compact groups, maximal tori, and global form

Unless stated otherwise, $K$ is a compact connected Lie group, $T\subset K$ is a maximal torus, and $V$ is a finite-dimensional continuous complex representation. Write

$$\mathfrak k=\operatorname{Lie}(K), \qquad \mathfrak t=\operatorname{Lie}(T), \qquad \mathfrak g=\mathfrak k_{\mathbb C}, \qquad \mathfrak h=\mathfrak t_{\mathbb C}.$$

For a matrix group, $\mathfrak k$ and $\mathfrak t$ consist of anti-Hermitian tangent matrices. To use the site’s Hermitian gauge generators, set

$$\mathfrak a=i\mathfrak t, \qquad X=-iH\in\mathfrak t \quad\text{for }H\in\mathfrak a.$$

Then $\mathfrak h=\mathfrak a_{\mathbb C}$ as a complex vector space. This is the local translation of the Lie-group page’s $t^a=-iT^a$ convention. It makes the weights below real eigenvalues of Hermitian Cartan generators. Below, $\mathrm d\rho$ also denotes its complex-linear extension from $\mathfrak k$ to $\mathfrak g$.

The root system lives only in the semisimple directions. When an invariant inner product is used there, its scale could be chosen independently on each simple factor; here it is fixed by assigning the long roots of every factor squared length $2$. The character–cocharacter pairing, rather than a raw plotted length, is the intrinsic datum.

Maximal tori turn the group problem into commuting data

A torus is a compact connected Abelian Lie group, hence isomorphic to $U(1)^r$ for some $r$. A maximal torus is one not contained in a larger torus. For compact connected $K$:

• every element of $K$ lies in some maximal torus;
• all maximal tori are conjugate; and
• $\mathfrak t$ is maximal Abelian in $\mathfrak k$, while $\mathfrak h=\mathfrak t_{\mathbb C}$ is a Cartan subalgebra of the complex reductive algebra $\mathfrak g$.

These statements are the reason that diagonalizing one maximal torus loses no conjugacy-invariant representation data. Precise compact-group versions appear in Hall 2015, Chapter 11, especially Theorems 11.9 and 11.36 and Etingof 2024, §§43.1 and 44.1, PDF.

Two integral lattices record the periodicity of $T$:

$$\begin{aligned} X^*(T)&=\operatorname{Hom}(T,U(1)), &&\text{the character lattice},\\ X_*(T)&=\operatorname{Hom}(U(1),T), &&\text{the cocharacter lattice}. \end{aligned}$$

If $\chi\in X^*(T)$ and $\eta\in X_*(T)$, their composition is a circle homomorphism, so for one unique integer

$$\chi\circ\eta(z)=z^{\langle\chi,\eta\rangle}, \qquad \langle\chi,\eta\rangle\in\mathbb Z.$$

This pairing is independent of a $2\pi$ coordinate, generator scale, or matrix realization. Those choices enter only when characters are written as linear functionals. To translate a cocharacter into the Hermitian convention, define its generator $H_\eta\in\mathfrak a$ by

$$\eta(e^{-i\theta}) = \exp(-i\theta H_\eta).$$

Using the same symbol $\chi$ for the differentiated real functional then gives

$$\langle\chi,\eta\rangle=\chi(H_\eta).$$

After equipping $V$ with the $K$-invariant Hermitian form supplied by compactness on the representation page, restricting $\rho:K\to GL(V)$ to $T$ gives commuting unitary operators. Therefore

$$V=\bigoplus_{\lambda\in X^*(T)}V_\lambda, \qquad V_\lambda = \{v\in V:\rho(t)v=\lambda(t)v \text{ for every }t\in T\}.$$

The nonzero $V_\lambda$ are the weight spaces, $\lambda$ are the weights, and $\dim V_\lambda$ are their multiplicities. Use the same symbol $\lambda$ for the character and its differentiated real functional on $\mathfrak a$. In Hermitian coordinates, if $X=-iH\in\mathfrak t$ and $v\in V_\lambda$, then

$$\mathrm d\rho(H)v=\lambda(H)v, \qquad \rho(\exp X)v=e^{-i\lambda(H)}v.$$

The second equation is the exponentiation check: a proposed infinitesimal weight belongs to $X^*(T)$ only if it is single-valued around every period of the actual torus.

Roots are the nonzero adjoint weights

Apply the same torus decomposition to the adjoint representation on $\mathfrak g$. The zero-weight space is $\mathfrak h$, and the nonzero weights form the root set $\Phi$:

$$\mathfrak g = \mathfrak h \oplus \bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha, \qquad \mathfrak g_\alpha = \{Y\in\mathfrak g: \operatorname{Ad}_tY=\alpha(t)Y \text{ for every }t\in T\}.$$

Equivalently, after differentiating in the Hermitian Cartan direction,

$$[H,Y]=\alpha(H)Y, \qquad H\in\mathfrak a, \quad Y\in\mathfrak g_\alpha.$$

A root is therefore a functional—or, intrinsically, a character—not a matrix and not a state in $V$. For a complex reductive Lie algebra, roots occur in pairs $\pm\alpha$, each root space is one-dimensional, and the roots vanish on the center. Zero is an adjoint weight of multiplicity $\dim\mathfrak h$, but zero is not a root. The root decomposition and its rank-one $\mathfrak{sl}_2$ subalgebras are developed in Etingof 2020, §§19.3–19.4, PDF and Kirillov 2008, §§6.5–6.6, PDF.

Root operators explain the geometry of every other weight diagram. If $Y_\alpha\in\mathfrak g_\alpha$ and $v\in V_\lambda$, then

$$\begin{aligned} \mathrm d\rho(H)\bigl(\mathrm d\rho(Y_\alpha)v\bigr) &= \mathrm d\rho(Y_\alpha)\mathrm d\rho(H)v +\mathrm d\rho([H,Y_\alpha])v\\ &= \bigl(\lambda(H)+\alpha(H)\bigr) \mathrm d\rho(Y_\alpha)v. \end{aligned}$$

Hence

$$\mathrm d\rho(\mathfrak g_\alpha)V_\lambda \subseteq V_{\lambda+\alpha}.$$

The image may be zero, especially at the end of a weight string. When it is nonzero, the root is the exact displacement between the two weights. Representation weights need not themselves be roots.

Coroots make integrality and reflection intrinsic

Each root has a corresponding coroot $\alpha^\vee\in X_*(T)$, characterized within its rank-one subgroup so that

$$\langle\alpha,\alpha^\vee\rangle=2.$$

The character–cocharacter definition keeps its mathematical type visible. After choosing a Weyl-invariant positive inner product on the semisimple Cartan dual and using it to identify that dual with the Cartan,

$$H_{\alpha^\vee} = \frac{2\alpha^\sharp}{(\alpha,\alpha)}, \qquad \langle\lambda,\alpha^\vee\rangle = \frac{2(\lambda,\alpha)}{(\alpha,\alpha)}.$$

The second formula is the one to use for a dominance or integrality test. It remains invariant if the inner product on a simple factor is rescaled. Coroots cannot be discarded in a non-simply-laced system: long roots and short roots are interchanged in the dual root system.

The root lattice and the full semisimple weight lattice are

$$Q=\operatorname{span}_{\mathbb Z}\Phi, \qquad P= \left\{ \lambda\in\operatorname{span}_{\mathbb R}\Phi: \langle\lambda,\alpha^\vee\rangle\in\mathbb Z \text{ for every }\alpha\in\Phi \right\}.$$

For a compact connected semisimple group,

$$Q\subseteq X^*(T)\subseteq P.$$

The simply connected global form has $X^*(T)=P$; the adjoint global form has $X^*(T)=Q$. Intermediate central quotients give intermediate lattices. This is the first place where the global group changes the answer while the Lie algebra and roots stay fixed. Root, coroot, and weight lattices are treated in Etingof 2020, §21.6, PDF and Kirillov 2008, §7.5, PDF.

Positive roots select a chamber; Weyl symmetry removes redundancy

Choose a regular linear functional that is nonzero on every root. It splits

$$\Phi=\Phi_+\sqcup\Phi_-, \qquad \Phi_-=-\Phi_+.$$

The simple roots $\Delta=\{\alpha_1,\ldots,\alpha_r\}\subset\Phi_+$ are the positive roots that cannot be written as sums of two positive roots. Every root is an integer combination of the $\alpha_i$ with coefficients either all nonnegative or all nonpositive. The choice of $\Phi_+$ is a choice of orientation, not an invariant of the representation.

For every root, define the reflection

$$s_\alpha(\lambda) = \lambda-\langle\lambda,\alpha^\vee\rangle\alpha.$$

It satisfies

$$s_\alpha^2=1, \qquad s_\alpha(\alpha)=-\alpha.$$

The Weyl group has equivalent compact-group and root-system descriptions:

$$W=\frac{N_K(T)}{T} = \langle s_\alpha:\alpha\in\Phi\rangle.$$

It is finite, preserves the root and character lattices, and is generated by reflections in the simple roots. If $V$ is a representation of $K$, $W$ permutes its weights without changing multiplicities. This follows directly from the action of a representative in $N_K(T)$ on the weight spaces.

The reflecting hyperplanes divide the semisimple Cartan dual into Weyl chambers. Extend a closed semisimple chamber by every central dual direction; $\overline C_+$ below denotes this cylinder in the full real Cartan dual:

$$\overline C_+ = \left\{ \lambda: \langle\lambda,\alpha_i^\vee\rangle\geq0 \text{ for }i=1,\ldots,r \right\}.$$

Every Weyl orbit meets $\overline C_+$ exactly once. Thus choosing positive roots supplies one canonical representative relative to that choice; changing the chamber changes the representative convention, not the representation. See Etingof 2020, §§21.2, 21.4, and 22.1–22.2, PDF and Kirillov 2008, §§7.2, 7.4, and 7.6–7.7, PDF.

Fundamental weights $\omega_i$ are defined in the semisimple weight space by

$$\langle\omega_i,\alpha_j^\vee\rangle=\delta_{ij}.$$

They generate $P$. They need not all lie in $X^*(T)$ for the chosen global group, so a list of nonnegative Dynkin labels is not yet a completed group-representation test.

Highest weights classify irreducibles—with the group test included

Order weights by

$$\mu\leq\lambda \quad\Longleftrightarrow\quad \lambda-\mu\in Q_+ = \sum_i\mathbb Z_{\geq0}\alpha_i.$$

For the chosen positive roots, an irreducible finite-dimensional representation has a highest-weight vector $v_\lambda$ satisfying

$$v_\lambda\in V_\lambda\setminus\{0\}, \qquad \mathrm d\rho(\mathfrak g_\alpha)v_\lambda=0 \quad\text{for every }\alpha\in\Phi_+.$$

Its highest-weight space is one-dimensional, and every other weight lies in $\lambda-Q_+$. The exact compact-group classification is:

Highest-weight theorem. For compact connected $K$, equivalence classes of finite-dimensional irreducible complex representations are in bijection with dominant characters $\lambda\in X^*(T)\cap\overline C_+$. Equivalently, $\langle\lambda,\alpha_i^\vee\rangle\in\mathbb Z_{\geq0}$ for every simple root, and $\lambda$ must exponentiate on the actual torus $T$.

At the complex semisimple Lie-algebra level, the last group condition is absent and all

$$\lambda=\sum_i a_i\omega_i, \qquad a_i\in\mathbb Z_{\geq0},$$

occur. This is the version proved in Etingof 2020, §25.3, Theorem 25.17, PDF and Kirillov 2008, §8.3, Corollary 8.24, PDF. Exponentiation is automatic for the corresponding simply connected group; a quotient requires its kernel to act trivially, exactly as on the groups-and-actions page. Hall’s compact-group statement uses characters of the actual torus (Hall 2015, Chapter 12, Theorem 12.6).

The complete-reducibility theorem on the representation page now finishes the organizational step: an arbitrary finite-dimensional $K$-representation is a direct sum of these irreducibles, recorded by dominant highest weights and their irreducible multiplicities. Those multiplicities are different from the dimensions of individual weight spaces.

For the semisimple part, one useful independent dimension check is Weyl’s formula. With

$$\varrho=\frac12\sum_{\alpha\in\Phi_+}\alpha,$$

one has

$$\dim L(\lambda) = \prod_{\alpha\in\Phi_+} \frac{ \langle\lambda+\varrho,\alpha^\vee\rangle }{ \langle\varrho,\alpha^\vee\rangle }.$$

The formula is Etingof 2020, §26.5, Proposition 26.8, PDF; see also Kirillov 2008, §8.5, Corollary 8.40, PDF. Every factor uses a coroot pairing, so the result is insensitive to an overall root-length rescaling. A nonpositive or nonintegral outcome signals that conventions or dominance data were mixed.

Central tori and finite quotients carry extra information

A compact connected Lie algebra is reductive:

$$\mathfrak k = \mathfrak z(\mathfrak k) \oplus [\mathfrak k,\mathfrak k].$$

Its root system describes the semisimple summand. The connected center is a torus whose representation data are ordinary characters, with no nonzero roots and with trivial Weyl action. At group level, one may write

$$K\cong \frac{Z_0\times K_{\mathrm{ss}}^{\mathrm{sc}}}{\Gamma},$$

where $Z_0$ is a torus, $K_{\mathrm{ss}}^{\mathrm{sc}}$ is compact, semisimple, and simply connected, and $\Gamma$ is finite and central. An irreducible label consists of a central character and a dominant semisimple weight for which $\Gamma$ acts trivially. This structure is stated in Etingof 2024, §43.1, Corollaries 43.5–43.6, PDF.

This formulation explains both possible omissions:

• roots alone cannot recover Abelian charges; and
• Lie-algebra weights alone cannot decide descent through a finite central quotient.

Disconnected compact groups add a third omission. Root data describe the identity component $K^\circ$, but the component group can permute its representations and require additional extension or induction data. The highest-weight theorem above is therefore not a classification theorem for all disconnected compact groups.

Computing roots and weights

Given a compact representation problem, use the following procedure.

1. Fix the group, not just the algebra. Record $K$, its connectedness, global quotient, maximal torus $T$, and the kernel of $\exp:\mathfrak t\to T$.
2. Translate generators. Convert anti-Hermitian tangent matrices $X=-iH$ to the site’s Hermitian $H$, and declare trace and root-length normalizations separately.
3. Find the adjoint eigencharacters. Diagonalize the adjoint action of $T$; remove the zero weight to obtain $\Phi$ and retain the zero space $\mathfrak h$.
4. Choose a chamber. Select $\Phi_+$ and its simple roots, construct coroots, and compute the simple reflections.
5. Decompose the representation. Restrict to $T$, list weights with multiplicities, and verify that root operators shift them by roots.
6. Reduce to dominant data. Move a candidate highest weight into $\overline C_+$ with Weyl reflections and evaluate every simple-coroot pairing.
7. Apply the global-form test. Check that the candidate lies in $X^*(T)$, or equivalently that the cover kernel acts trivially.
8. Validate. Check Weyl-invariant multiplicities, total dimension, tensor-product weight addition, and at least one center or exponentiation condition.

The output is a root datum, a weight multiset, a dominant highest-weight label, and a global descent verdict. Work grows with the rank, the number of positive roots, and the number of weights; large multiplicities and tensor products usually call for character formulas or dedicated representation software. Stop this method when the problem asks for Clebsch–Gordan basis coefficients, noncompact unitary duals, disconnected-group extension data, or physical interaction and anomaly constraints: those require additional structure.

Worked example: SU(2), weights, and global descent

Use the site normalization

$$T_a=\frac{\sigma_a}{2}, \qquad [T_a,T_b]=i\epsilon_{ab}{}^cT_c, \qquad \operatorname{tr}_{\mathbf2}(T_aT_b) =\frac12\delta_{ab}.$$

Choose the Hermitian Cartan generator $T_3$ and

$$T_\pm=T_1\pm iT_2.$$

Direct calculation gives

$$[T_3,T_\pm]=\pm T_\pm.$$

If $\alpha(T_3)=1$, the two roots are $\pm\alpha$. The coroot and its Hermitian generator are

$$\alpha^\vee(z) = \operatorname{diag}(z,z^{-1}), \qquad H_{\alpha^\vee}=2T_3.$$

Therefore

$$\langle\alpha,\alpha^\vee\rangle=2.$$

For the maximal torus

$$T= \left\{ \operatorname{diag}(z,z^{-1}):z\in U(1) \right\},$$

let $\omega(\operatorname{diag}(z,z^{-1}))=z$. The adjoint action on $T_+$ has character $z^2$, hence

$$\alpha=2\omega, \qquad \langle\omega,\alpha^\vee\rangle=1.$$

The irreducible representation $L(n\omega)$ with highest weight $\lambda=n\omega$, $n\in\mathbb Z_{\geq0}$, has

$$j=\frac n2, \qquad m=j,j-1,\ldots,-j, \qquad \dim L(n\omega)=n+1.$$

Here $m$ is the eigenvalue of $T_3$. The operators $T_\pm$ shift $m\mapsto m\pm1$, and the Weyl group $W\cong\mathbb Z_2$ acts by $m\mapsto-m$.

Representation	Highest label $n$	$T_3$ weights	Descends to $SO(3)$?
Singlet $\mathbf1$	$0$	$0$	Yes
Doublet $\mathbf2$	$1$	$+\tfrac12,-\tfrac12$	No
Triplet $\mathbf3$	$2$	$+1,0,-1$	Yes

The global-form distinction is visible in the torus lattices:

$$X^*(T_{SU(2)})=\mathbb Z\omega, \qquad X^*(T_{SO(3)})=\mathbb Z\alpha =2\mathbb Z\omega.$$

Equivalently, the central element

$$-\mathbf1 = \exp(-i2\pi T_3)$$

acts on $L(n\omega)$ as $(-1)^n$. Therefore the representation descends through $SU(2)\to SO(3)=SU(2)/\{\pm\mathbf1\}$ exactly when $n$ is even, or $j$ is an integer. This recovers the representation page’s tensor check:

$$\mathbf2\otimes\mathbf2 = \mathbf3\oplus\mathbf1,$$

because both summands have even highest label and the center acts trivially on the tensor square.

Rank-two checkpoint: SU(3)

The rank-one example does not show how several simple roots interact. For $SU(3)$ choose

$$T= \left\{ \operatorname{diag}(z_1,z_2,z_3): z_1z_2z_3=1 \right\}.$$

In additive character notation, let $\varepsilon_i$ select $z_i$, subject to $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$. For $H=\operatorname{diag}(h_1,h_2,h_3)$ with $\sum_i h_i=0$,

$$[H,E_{ij}]=(h_i-h_j)E_{ij}.$$

Thus the six roots are

$$\alpha_{ij}=\varepsilon_i-\varepsilon_j, \qquad i\neq j.$$

A simple system is

$$\alpha_1=\varepsilon_1-\varepsilon_2, \qquad \alpha_2=\varepsilon_2-\varepsilon_3,$$

with positive roots $\alpha_1,\alpha_2,\alpha_1+\alpha_2$. The Weyl group is $S_3$, acting by permutations of the $\varepsilon_i$.

This compact table encodes three useful weight diagrams without choosing plot coordinates:

Representation	Highest weight	Weights
$\mathbf3$	$\omega_1$	$\varepsilon_1,\varepsilon_2,\varepsilon_3$
$\overline{\mathbf3}$	$\omega_2$	$-\varepsilon_1,-\varepsilon_2,-\varepsilon_3$
Adjoint $\mathbf8$	$\omega_1+\omega_2$	Six roots, plus $0$ with multiplicity $2$

For the dominant integral weight $\lambda=p\omega_1+q\omega_2$ with $p,q\in\mathbb Z_{\geq0}$, Weyl’s dimension formula becomes

$$\dim(p,q) = \frac{(p+1)(q+1)(p+q+2)}2.$$

It yields $3$, $3$, and $8$ for $(p,q)=(1,0),(0,1),(1,1)$, respectively. The nonzero adjoint weights are roots, while the defining weights are not; the zero weight’s multiplicity is rank two. These facts catch three common diagram-reading errors at once.

QFT-facing example: one electroweak doublet

Consider only the local weight data of the left-handed lepton doublet

$$L_L= \begin{pmatrix} \nu_L\\ e_L \end{pmatrix}.$$

It is in the defining representation of $SU(2)_L$, so with $T_3=\sigma_3/2$ its two weights are

$$m=+\frac12, \qquad m=-\frac12.$$

The root operators $T_\pm$ connect the two weight spaces and change $m$ by $\pm1$. The commuting $U(1)_Y$ generator is central in $\mathfrak{su}(2)\oplus\mathfrak u(1)$, so its eigenvalue is extra weight data rather than an $SU(2)$ root. With the central Lie-algebra weight and charge-generator convention

$$\begin{aligned} Y L_L&=-\frac12L_L,\\ Q&=T_3+Y, \end{aligned}$$

the two electric-charge eigenvalues are

$$\begin{array}{c|cc} &\nu_L&e_L\\ \hline T_3&+\tfrac12&-\tfrac12\\ Y&-\tfrac12&-\tfrac12\\ Q&0&-1 \end{array}$$

This is a controlled weight calculation, supported in the same normalization by Tong 2017, §§5.1 and 5.2.1, PDF. It shows how a non-Abelian multiplet and a central weight combine. It does not choose the global electroweak quotient, derive the hypercharge, specify chirality dynamics, test Yukawa terms or anomalies, or construct the gauge interactions. Those tasks belong to Electroweak Gauge and Matter Structure, which also requires the Yang–Mills action page.

Diagnostics, limitations, and stop rules

Before trusting a root–weight calculation, run these checks:

• Type check: roots and weights are characters or functionals; coroots are cocharacters, while $H_{\alpha^\vee}$ are their Hermitian Cartan generators.
• Convention round trip: $X=-iH$ must reproduce the same finite torus character, commutator shift, and dimension in anti-Hermitian and Hermitian notation.
• Integrality check: every $\langle\lambda,\alpha_i^\vee\rangle$ is integral, and a highest weight’s values are nonnegative.
• Global check: the exponential kernel or cover kernel acts trivially.
• Weyl check: weights occur with Weyl-invariant multiplicities.
• Dimension check: the sum of weight multiplicities equals $\dim V$; tensor-product dimensions multiply while tensor-product weights add.
• Center check: all weights in one irreducible representation induce the same action of a central element, because their differences lie in the root lattice.

The method stops being complete for a noncompact group, an infinite-dimensional representation, or a disconnected group unless additional theory is supplied. It also stops at representation structure: Clebsch–Gordan coefficients need basis choices, and QFT couplings require statistics, locality, Hermiticity, dynamics, gauge invariance, and anomaly checks beyond a weight diagram.

Common pitfalls

Calling every weight a root. Roots are precisely the nonzero adjoint weights. A matter representation usually has weights that are not roots, and zero can be a weight without being a root.

Using the full weight lattice for every global form. $P$ classifies integrable modules of the simply connected semisimple form. A quotient admits only the sublattice $X^*(T)$ on which its kernel acts trivially.

Forgetting the center. A compact reductive group can have $U(1)$ directions with nontrivial characters and no roots. Root data alone cannot recover their charges.

Treating a chamber as physical. Positive roots, simple roots, and the orientation of a diagram are choices. Weyl-related choices encode the same invariant representation data.

Extending the theorem outside its hypotheses. Root data for $K^\circ$ do not classify a disconnected group, and compact finite-dimensional highest-weight theory does not classify arbitrary noncompact unitary representations.

Promoting a multiplet label to an interaction. Weights identify transformation data. They do not by themselves establish a local, gauge-invariant, anomaly-free, or dynamically realized QFT coupling.

Exercises

1. Prove the root-shift rule

Let $Y\in\mathfrak g_\alpha$ and $v\in V_\lambda$. Starting from the adjoint and representation actions of $t\in T$, show that $\mathrm d\rho(Y)v$ has weight $\lambda+\alpha$ when it is nonzero.

Solution

For $X\in\mathfrak k$, differentiate the group identity

$$\rho(t)\rho(\exp(sX))\rho(t)^{-1} = \rho\!\left(\exp(s\operatorname{Ad}_tX)\right)$$

at $s=0$, then extend the result complex-linearly from $\mathfrak k$ to $\mathfrak g$. For $Y\in\mathfrak g_\alpha$ this gives

$$\rho(t)\mathrm d\rho(Y)\rho(t)^{-1} = \mathrm d\rho(\operatorname{Ad}_tY) = \alpha(t)\mathrm d\rho(Y).$$

Because $\rho(t)v=\lambda(t)v$,

$$\rho(t)\mathrm d\rho(Y)v = \alpha(t)\lambda(t)\mathrm d\rho(Y)v.$$

Characters multiply, which is addition in weight notation, so the image lies in $V_{\lambda+\alpha}$.

2. Test an SU(2) highest weight on both global forms

For $\lambda=5\omega$, compute its Dynkin label, spin, dimension, weights, and Weyl partner of the highest weight. Does it define a representation of $SO(3)$?

Solution

Since $\langle\omega,\alpha^\vee\rangle=1$, the Dynkin label is $n=5$. Thus

$$j=\frac52, \qquad \dim L(5\omega)=6,$$

and the $T_3$ weights are $5/2,3/2,1/2,-1/2,-3/2,-5/2$. The Weyl reflection sends the highest weight $5/2$ to $-5/2$. The label is dominant integral and is allowed for $SU(2)$, but $5$ is odd, so $-\mathbf1$ acts as $-1$ and the representation does not descend to $SO(3)$.

3. Recover the SU(3) roots and a dimension

For trace-zero diagonal $H$, compute $[H,E_{ij}]$. Use the result to list the positive roots for the simple system above, then evaluate $\dim(1,1)$.

Solution

Matrix multiplication gives

$$[H,E_{ij}]=(h_i-h_j)E_{ij},$$

so the roots are $\varepsilon_i-\varepsilon_j$ for $i\neq j$. The positive ones are

$$\alpha_1,\qquad \alpha_2,\qquad \alpha_1+\alpha_2.$$

The dimension formula gives

$$\dim(1,1) = \frac{(2)(2)(4)}2 =8,$$

as required for the adjoint representation. Its six nonzero weights are the roots, and zero has multiplicity two.

4. Find the connectedness failure in O(2)

The identity component of $O(2)$ is $SO(2)\cong U(1)$ and has no roots. Why does that empty root system fail to classify representations of $O(2)$?

Solution

$SO(2)$ has one-dimensional characters labeled by integers $m$. The other component of $O(2)$ contains a reflection $r$ satisfying

$$rR(\theta)r^{-1}=R(-\theta).$$

It therefore exchanges the $SO(2)$ weights $m$ and $-m$. To extend a representation from the identity component, one must specify how the reflection acts and satisfy the component-group relations. The empty root system records neither operation.

5. Transfer one doublet to QFT notation

Let an $SU(2)$ doublet have a common Abelian weight $y$. Compute the two eigenvalues of $Q=T_3+Y$. What remains undecided after this calculation?

Solution

The two $T_3$ weights are $\pm1/2$, so

$$q_+=y+\frac12, \qquad q_-=y-\frac12.$$

This fixes only Lie-algebra weight arithmetic. The period and normalization of the $U(1)$ generator, any finite central quotient, chirality and field content, dynamics, invariant interactions, and anomaly cancellation remain undecided. Those are group-level and QFT-level questions.

Where to continue

• For the first physical application to non-Abelian multiplets and weight diagrams, continue to Electroweak Gauge and Matter Structure. That page also requires The Yang–Mills Action and Gauge Self-Interaction, so this mathematical page alone does not satisfy its preparation.
• Roots and weights are optional in this chapter. They are not a prerequisite for the Lorentz, Clifford, or spinor leaves that follow.

References

• Pavel Etingof (2020, 2024), Lie Groups and Lie Algebras I, PDF and Lie Groups and Lie Algebras II, PDF, MIT OpenCourseWare 18.745/18.755, §§19.3–26.5, 43.1, and 44.1. These open notes develop Cartan subalgebras, roots and coroots, Weyl chambers, weight lattices, highest-weight theory, connected compact global forms, and maximal tori.
• Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second edition, corrected second printing, Graduate Texts in Mathematics 222, Springer, 2015, Chapters 8, 11, and 12. These chapters establish root systems, the compact-group Weyl group, maximal tori, and the group-level highest-weight theorem.
• Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras, Cambridge University Press, 2008, §§6.5–8.5 and Appendix A; an author-posted preliminary version is available as an Open PDF and supplies exact theorem and example locators. This companion authority supports root and weight lattices, Weyl invariance, dominant-integral classification, and the $A_1$ and $A_2$ checks.
• David Tong (2017), Lectures on the Standard Model, §§5.1 and 5.2.1, PDF, Cambridge Part III lecture notes. These sections derive the bounded $SU(2)_L$ lepton-doublet weights, hypercharge convention, and $Q=T_3+Y$ calculation; developed electroweak physics remains at the physical continuation.