Lie Group and Lie Algebra Representations | Highest Weights and Continuous Symmetry

From discrete symmetry to continuous symmetry

A finite group has isolated elements. A Lie group is both a group and a smooth manifold, with multiplication and inversion compatible with the smooth structure.

Examples include:

• the circle group U(1);
• the rotation group SO(3);
• the unitary group U(n);
• the special unitary group SU(n);
• the general linear group GL(n,R) or GL(n,C);
• the special linear group SL(n,R) or SL(n,C).

A representation of a Lie group G is a homomorphism ρ:G→GL(V) satisfying an appropriate continuity or smoothness condition. In finite dimensions, smooth matrix representations are the standard setting.

Why the identity matters

Every Lie group has an identity element e. The tangent space at e, denoted 𝔤=T_eG, carries more than vector-space structure: it has a bilinear bracket [X,Y] that records the first noncommutative correction to composing small motions.

This tangent-space object is the Lie algebra of G.

Lie group: global continuous symmetry. Lie algebra: infinitesimal symmetry near the identity.

Matrix Lie groups

For matrix groups, Lie algebras can often be described concretely as matrices satisfying linear conditions.

• Lie algebra of GL(n,C): all n×n complex matrices.
• Lie algebra of SL(n,C): trace-zero complex matrices.
• Lie algebra of U(n): skew-Hermitian matrices.
• Lie algebra of SU(n): skew-Hermitian trace-zero matrices.
• Lie algebra of SO(n): real skew-symmetric matrices.

The bracket is the matrix commutator

[X,Y]=XY−YX.

The exponential map

The matrix exponential connects infinitesimal generators to finite group elements:

exp(X)=I+X+X²/2!+X³/3!+···.

If X lies in the Lie algebra, t↦exp(tX) is a one-parameter subgroup. Differentiating at t=0 recovers X.

This gives a useful mental model: Lie algebra elements are velocity vectors of continuous symmetry paths through the identity.

Differentiating a representation

Suppose ρ:G→GL(V) is a smooth finite-dimensional representation. Differentiating ρ at the identity gives a linear map

dρ:𝔤→End(V).

This map respects Lie brackets:

dρ([X,Y])=[dρ(X),dρ(Y)].

Therefore dρ is a Lie algebra representation.

In practical terms, a potentially complicated nonlinear group action is converted into a linear family of infinitesimal operators.

One-parameter subgroups and generators

If g(t)=exp(tX), then

ρ(exp(tX)) = exp(t dρ(X))

under the usual finite-dimensional compatibility. The group representation along that one-parameter subgroup is determined by the infinitesimal generator dρ(X).

This is why generators matter so much in physics and differential equations: continuous motion can be reconstructed from infinitesimal operators.

Lie algebra representations

A representation of a Lie algebra 𝔤 on V is a linear map ρ:𝔤→End(V) satisfying

ρ([X,Y])=ρ(X)ρ(Y)−ρ(Y)ρ(X).

The bracket is the structure to preserve. Compare this with group representations, where multiplication must be preserved. The two theories mirror each other at different scales.

The model algebra sl₂

The Lie algebra sl₂(C) consists of 2×2 trace-zero complex matrices. A standard basis is

H = [ 1  0 ]   E = [ 0  1 ]   F = [ 0  0 ]
    [ 0 -1 ]       [ 0  0 ]       [ 1  0 ]

They satisfy

• [H,E]=2E,
• [H,F]=−2F,
• [E,F]=H.

This tiny algebra is a laboratory for highest-weight theory. Many features of general semisimple Lie algebras already appear here in a completely calculable form.

Weight vectors

In an sl₂-representation, a vector v is a weight vector of weight λ if

H·v=λv.

The operator H acts diagonally on weight vectors. The relations tell us what E and F do to weights.

If H·v=λv, then

• E·v has weight λ+2, if nonzero;
• F·v has weight λ−2, if nonzero.

So E raises weight and F lowers weight.

Highest-weight vectors

A highest-weight vector v is a nonzero weight vector annihilated by E:

E·v=0.

Starting from v, repeated application of F generates a descending string of weights. In a finite-dimensional irreducible representation, this process eventually stops.

The remarkable theorem for sl₂ is that finite-dimensional irreducible representations are classified by a nonnegative integer n, the highest weight. The representation has dimension n+1 and weights

n, n−2, n−4, …, −n.

Worked sl₂ example: highest weight 2

Take highest weight n=2. The irreducible representation has dimension 3 and weights 2,0,−2.

Let v₀ be a highest-weight vector. Then

• v₀ has weight 2;
• Fv₀ has weight 0;
• F²v₀ has weight −2;
• F³v₀=0.

The whole irreducible representation is generated from one highest-weight vector by the lowering operator.

highest weight + lowering operations → entire irreducible module.

Symmetric powers and sl₂

Let V=C² be the defining representation of sl₂. Its k-th symmetric power Sym^k(V) has dimension k+1 and is irreducible of highest weight k.

This gives a concrete realisation of every finite-dimensional irreducible sl₂-representation. Homogeneous polynomials of degree k in two variables provide a useful model.

The earlier tensor-power article therefore feeds directly into highest-weight theory: symmetric powers are not side constructions; they produce the basic irreducible ladder.

From sl₂ to semisimple Lie algebras

For a semisimple Lie algebra 𝔤, one chooses a Cartan subalgebra 𝔥: a maximal abelian subalgebra with good diagonalisation properties. The representation decomposes into weight spaces

V = ⊕_λ V_λ,

where each weight λ is a linear functional on 𝔥 and

H·v=λ(H)v for H∈𝔥 and v∈V_λ.

Weight theory is therefore simultaneous eigenvalue theory for a commuting family of operators.

Roots

The Lie algebra itself carries the adjoint representation on 𝔤, defined by ad_X(Y)=[X,Y]. Relative to a Cartan subalgebra, 𝔤 decomposes into root spaces.

𝔤 = 𝔥 ⊕ ⊕_{α∈Φ} 𝔤_α.

The nonzero weights α of the adjoint representation are the roots. Root vectors shift weights in other representations: if v has weight λ and X_α lies in the root space 𝔤_α, then X_αv has weight λ+α when nonzero.

This generalises the raising/lowering behaviour of E and F in sl₂.

Positive roots and highest weights

Choose a set of positive roots. This divides root directions into raising and lowering directions. A highest-weight vector is killed by all positive-root operators and is an eigenvector for the Cartan subalgebra.

For complex semisimple Lie algebras, finite-dimensional irreducible representations are classified by dominant integral highest weights.

This is one of the great compression results of algebra:

large matrix representation → one highest-weight label + root system rules.

Why integrality appears

Each simple root generates an sl₂-like subsystem. Finite-dimensionality forces the corresponding highest-weight coordinates to be nonnegative integers. Thus the general classification inherits the discrete integrality already visible in sl₂.

The global theory is more elaborate, but the local rank-one control comes repeatedly from sl₂ triples.

The weight lattice

Weights live in a vector space dual to the Cartan subalgebra, but finite-dimensional irreducible representations occupy a discrete lattice. Fundamental weights form a natural coordinate system, and dominant integral combinations label irreducibles.

For a rank-r semisimple Lie algebra, an irreducible highest weight may be recorded by r nonnegative integers. This is why Dynkin labels can encode large representations compactly.

Dynkin diagrams

Root systems can be compressed into Dynkin diagrams. Nodes represent simple roots; edges record angle and length relationships. The connected Dynkin diagrams classify complex simple Lie algebras into the classical families A_n, B_n, C_n, D_n and the exceptional families E₆, E₇, E₈, F₄, G₂.

This classification is not merely a taxonomy. The root system controls highest-weight representation theory, Weyl groups, character formulas and tensor-product structure.

SU(2) and spin

SU(2) is the compact Lie group closely related to sl₂(C). Its irreducible finite-dimensional complex representations have dimensions 1,2,3,4,… and can be realised through symmetric powers of the defining two-dimensional representation.

Physicists commonly label them by spin j=0,1/2,1,3/2,…, with dimension 2j+1. Mathematically, the corresponding highest weight is n=2j.

This explains the familiar sequence:

• spin 0 → dimension 1;
• spin 1/2 → dimension 2;
• spin 1 → dimension 3;
• spin 3/2 → dimension 4.

SU(2) versus SO(3)

SU(2) is a double cover of SO(3). Integer-spin irreducible representations descend to SO(3), while half-integer-spin representations do not. They are genuine representations of SU(2) and projective representations of SO(3).

This is an important lesson about global topology. The Lie algebras of SU(2) and SO(3) are closely related, but the groups have different global structure. A Lie algebra representation may integrate to one group but not descend to another quotient without an additional integrality condition.

Not every Lie algebra representation integrates to every Lie group

For simply connected Lie groups, finite-dimensional Lie algebra representations integrate uniquely to group representations under standard hypotheses. For non-simply-connected groups, some Lie algebra representations fail to descend because loops in the group can act nontrivially in the covering representation.

This is where local and global information separate:

Lie algebra controls infinitesimal structure; group topology controls which infinitesimal representations globalise.

The adjoint representation

A Lie group G acts on its Lie algebra by conjugation, differentiated to the adjoint representation. At the Lie algebra level, ad_X(Y)=[X,Y].

The adjoint representation is fundamental because the Lie algebra acts on itself using its own bracket. Roots are weights of this representation. Structural properties of 𝔤 are therefore encoded in one of its own canonical representations.

Casimir operators

For semisimple Lie algebras, one can build central elements in the universal enveloping algebra, such as the quadratic Casimir. In an irreducible representation, central elements act by scalars by Schur’s lemma.

The Casimir eigenvalue therefore labels irreducible symmetry types and appears naturally in physics, spectral theory and harmonic analysis.

The universal enveloping algebra

Lie algebra elements do not multiply internally like associative matrices unless we place them inside a larger associative algebra. The universal enveloping algebra U(𝔤) provides that environment while enforcing the relation

XY−YX=[X,Y].

Representations of 𝔤 correspond to modules over U(𝔤). This brings Lie algebra representation theory into the broader language of associative algebra and modules.

Weyl groups

The root system has a finite reflection group, the Weyl group. It acts on weights and reflects the internal symmetry of the root geometry.

Weights of irreducible representations exhibit strong Weyl-group structure, and the Weyl character formula uses this symmetry to compute characters from the highest weight.

The Weyl character formula

For a dominant integral highest weight λ, the Weyl character formula expresses the irreducible character through an alternating sum over the Weyl group divided by a universal denominator. In schematic form,

ch V_λ = (Σ_{w∈W} sign(w)e^{w(λ+ρ)}) / (Σ_{w∈W} sign(w)e^{wρ}).

The formula is advanced, but its architecture is clear: the highest weight plus root-system symmetry determines the entire character.

Weight multiplicities

A representation may have several independent vectors with the same weight. The dimension of a weight space is its weight multiplicity.

In sl₂ irreducibles, every weight multiplicity is 1. In higher rank, multiplicities can exceed 1 and become a substantial combinatorial problem. Kostant’s multiplicity formula and crystal methods are among the tools used to compute them.

Tensor products and highest weights

If V_λ and V_μ are irreducible highest-weight representations, their tensor product contains a highest-weight component V_{λ+μ}, but usually also many lower components.

For SU(2), the Clebsch–Gordan rule is especially simple. For higher groups, tensor-product multiplicities lead to Littlewood–Richardson coefficients and other combinatorial structures.

This connects directly to the preceding Tensor Products, Duality, Symmetric and Exterior Powers guide.

Continuous symmetry in differential equations

If a differential equation is invariant under a Lie group, infinitesimal generators can be used to study conserved quantities, invariant solution families and symmetry reductions.

Representation theory enters when spaces of functions or solutions decompose into irreducible symmetry types. Operators commuting with the group action preserve those components.

Continuous symmetry in quantum theory

Quantum state spaces are linear, so continuous physical symmetries naturally lead to unitary representations. Infinitesimal generators become observables or conserved quantities under appropriate physical conditions.

Rotational symmetry produces angular-momentum operators satisfying Lie algebra commutation relations. State spaces decompose into SU(2) irreducibles. Tensor products organise composite systems.

The representation theory supplies the symmetry structure; the physical interpretation requires the broader quantum framework.

A learning sequence for Lie representations

• Be fluent with linear algebra, eigenvectors, tensor products and duals.
• Know basic groups, representations, characters and irreducibility.
• Learn smooth manifolds at least at the tangent-space level.
• Study matrix Lie groups and their Lie algebras.
• Understand the exponential map and one-parameter subgroups.
• Master sl₂ representations and raising/lowering operators.
• Learn Cartan subalgebras, roots and weights.
• Move to highest-weight classification and Weyl groups.
• Then study characters, tensor products and applications for specific Lie groups.

Common mistakes

• Confusing the Lie group with its Lie algebra: one is global and nonlinear; the other is infinitesimal and linear.
• Forgetting topology: Lie algebra representations do not automatically descend to every global quotient group.
• Treating weights as ordinary scalar eigenvalues: in higher rank, weights are linear functionals on a Cartan subalgebra.
• Assuming all weight spaces are one-dimensional: higher-rank representations can have multiplicities.
• Assuming all Lie algebras are semisimple: highest-weight theory in this form applies to an important but specific class.
• Skipping sl₂: rank-one structure is the local engine behind much of the general theory.

Practice set

1. Lie bracket

For matrix Lie algebras, what is [X,Y]?

Answer: XY−YX.

2. Highest-weight dimension

What is the dimension of the irreducible sl₂-representation of highest weight 5?

Answer: 6.

3. Weight ladder

List the weights of the sl₂ irreducible of highest weight 4.

Answer: 4,2,0,−2,−4.

4. Spin dimension

What is the dimension of spin j=3/2?

Answer: 2j+1=4.

5. Globalisation

Why can two Lie groups with closely related Lie algebras have different allowed finite-dimensional representations?

Answer: global topology and quotient structure can determine whether a Lie algebra representation descends to the group.

The deeper representation principle

Highest-weight theory is a striking example of mathematical compression. A large family of matrices depending continuously on group parameters can often be understood from a discrete label sitting in a weight lattice.

The path is layered:

continuous group → tangent Lie algebra → Cartan directions → roots → weights → highest weight → irreducible representation.

Each step removes surface complexity while preserving the structure needed for classification.

Continue Representation Mathematics — Batch 02

• Induced Representations, Restriction and Frobenius Reciprocity
• Representation Theory: Tensor Products, Duality, Symmetric and Exterior Powers
• Modular Representation Theory | Semisimplicity, Indecomposables and Extensions
• Return to the BTT Mathematics Learning Hub