Mathematics.

lie algebra representations

Representations of sl(2)

Representation Theory75 minDifficulty8 out of 10

You should know: lie algebras, irreducible representations

Overview

The Lie algebra sl(2, ℂ) = {2×2 complex traceless matrices} is the simplest non-abelian semisimple Lie algebra and serves as the prototype for the entire theory of semisimple Lie algebra representations. Its finite-dimensional irreducible representations are completely classified: for each non-negative integer n there is a unique (up to isomorphism) irreducible representation of dimension n+1, with highest weight n. The sl(2) theory is the engine behind the general theory of highest weights.

Intuition

sl(2) has three basis elements e (raising), f (lowering), h (Cartan). In any irreducible representation, h acts diagonally with integer eigenvalues (weights) and e, f shift the weight up and down by 2. The representation is like a 'ladder' of weight spaces, and it must have a top (highest weight vector killed by e) and a bottom (killed by f).

Formal Definition

Definition

The Lie algebra sl(2,ℂ) has standard basis e, f, h satisfying [h,e]=2e, [h,f]=−2f, [e,f]=h. A representation is a Lie algebra homomorphism ρ: sl(2) → End(V).

sl(2,C)=span{e,f,h},e=(0100),  f=(0010),  h=(100−1)\mathfrak{sl}(2,\mathbb{C}) = \mathrm{span}\{e, f, h\}, \quad e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\; f=\begin{pmatrix}0&0\\1&0\end{pmatrix},\; h=\begin{pmatrix}1&0\\0&-1\end{pmatrix}
Standard basis of sl(2,ℂ)
[h,e]=2e,[h,f]=−2f,[e,f]=h[h, e] = 2e, \quad [h, f] = -2f, \quad [e, f] = h
Lie bracket relations
V=⨁λVλ,Vλ={v∈V:h⋅v=λv}(weight space decomposition)V = \bigoplus_{\lambda} V_\lambda, \quad V_\lambda = \{v \in V : h \cdot v = \lambda v\} \quad (\text{weight space decomposition})
Weight space decomposition
Vn:dim⁡Vn=n+1,weights={n,n−2,…,−(n−2),−n}V_n: \quad \dim V_n = n+1, \quad \text{weights} = \{n, n-2, \ldots, -(n-2), -n\}
Unique irrep of dimension n+1 and highest weight n

Notation

NotationMeaning
sl(2,C)\mathfrak{sl}(2,\mathbb{C})Lie algebra of 2×2 traceless complex matrices
VnV_nIrreducible sl(2)-representation of highest weight n (dimension n+1)
VλV_\lambdaWeight space with eigenvalue λ for h
v+v_+Highest weight vector: e·v_+ = 0

Properties

Action formulas

InVnwithbasisv0,…,vn(wherev0=highestweightvector):h⋅vk=(n−2k)vk,e⋅vk=k(n−k+1)vk−1,f⋅vk=vk+1(withvn+1=0).In V_n with basis v₀, …, vₙ (where v₀ = highest weight vector): h·vₖ = (n−2k)vₖ, e·vₖ = k(n−k+1)v_{k−1}, f·vₖ = v_{k+1} (with v_{n+1}=0).

Casimir element

TheCasimirC=h2/4+(ef+fe)/2actsas(n/2)(n/2+1)⋅IdonVn.The Casimir C = h²/4 + (ef+fe)/2 acts as (n/2)(n/2+1)·Id on V_n.

Theorems

Theorem 1: Classification of Finite-Dimensional Irreducibles
Foreachn∈Z≥0,thereisaunique(uptoisomorphism)finite−dimensionalirreduciblerepresentationVnofsl(2,C)withhighestweightnanddimensionn+1.Everyfinite−dimensionalrepresentationdecomposesas⊕Vni.For each n ∈ ℤ_{≥0}, there is a unique (up to isomorphism) finite-dimensional irreducible representation V_n of sl(2,ℂ) with highest weight n and dimension n+1. Every finite-dimensional representation decomposes as ⊕ V_{nᵢ}.
Theorem 2: Complete Reducibility
Everyfinite−dimensionalrepresentationofsl(2,C)iscompletelyreducible(adirectsumofirreduciblesVn).Every finite-dimensional representation of sl(2,ℂ) is completely reducible (a direct sum of irreducibles V_n).
Theorem 3: Weight Symmetry
InVntheweightsaren,n−2,…,−n,eachoccurringwithmultiplicity1.Theweightsetissymmetricabout0.In V_n the weights are n, n−2, …, −n, each occurring with multiplicity 1. The weight set is symmetric about 0.

Worked Examples

  1. 1

    V₁ has dimension 2, highest weight 1. Basis: v₀ (weight 1), v₁ (weight −1).

    V1=span{v0,v1},h⋅v0=v0,h⋅v1=−v1V_1 = \mathrm{span}\{v_0, v_1\}, \quad h \cdot v_0 = v_0, \quad h \cdot v_1 = -v_1
  2. 2

    Action: e·v₀ = 0 (highest weight), f·v₀ = v₁, e·v₁ = v₀, f·v₁ = 0.

    e⋅v0=0,f⋅v0=v1,e⋅v1=1⋅(1−1+1)v0=v0,f⋅v1=0e \cdot v_0 = 0, \quad f \cdot v_0 = v_1, \quad e \cdot v_1 = 1 \cdot (1-1+1)v_0 = v_0, \quad f \cdot v_1 = 0
  3. 3

    In matrix form: e = [[0,1],[0,0]], f = [[0,0],[1,0]], h = [[1,0],[0,−1]], which is just the standard representation.

    ρ(e)=(0100),ρ(f)=(0010),ρ(h)=(100−1)\rho(e) = \begin{pmatrix}0&1\\0&0\end{pmatrix}, \quad \rho(f) = \begin{pmatrix}0&0\\1&0\end{pmatrix}, \quad \rho(h) = \begin{pmatrix}1&0\\0&-1\end{pmatrix}

✓ Answer

V₁ is the standard 2-dimensional representation with weights {1,−1}.

Practice Problems

Mediumfree response

Decompose V₁ ⊗ V₁ into irreducible sl(2)-representations.

Hardproof writing

Show that if v ∈ V is a highest weight vector of weight n (i.e., h·v = nv and e·v = 0), then the vectors fᵏ·v are weight vectors of weight n−2k.

Quiz

The commutator [h, e] in sl(2,ℂ) equals:
The unique irreducible representation of sl(2,ℂ) with highest weight n has dimension:
In the irreducible representation V_n, the action of e on a weight vector v_λ of weight λ:

Summary

  • sl(2,ℂ) has basis e, f, h with [h,e]=2e, [h,f]=−2f, [e,f]=h.
  • In any finite-dimensional representation, h acts semisimply with integer weights.
  • e raises weights by 2, f lowers weights by 2.
  • For each n ≥ 0, there is a unique irreducible V_n of dimension n+1 with weights {n, n−2, …, −n}.
  • Every finite-dimensional sl(2,ℂ)-representation decomposes as a direct sum of V_n's.

References

  1. BookHumphreys, J.E. — Introduction to Lie Algebras and Representation Theory (1972), Chapter 7
  2. BookFulton, W. & Harris, J. — Representation Theory: A First Course (1991), Lecture 11