Algebra
Representation Theory of Lie Algebras
Цели урока
- Classify irreducible sl(2,C)-modules V_n by highest weight
- Understand Weyl's complete reducibility theorem and its proof
- Master the weight decomposition and the Weyl character formula
- Apply the Clebsch-Gordan rule for tensor products
Предварительные знания
- Lie algebras and sl(2)
- Universal enveloping algebras
- Quantum groups
Physicists predicted the existence of quarks in 1964 using the representation theory of sl(3,C). Not a single quark had been detected experimentally - but the mathematics said: they must exist. Representation theory of sl(2) describes electron spin. sl(3) describes quarks. Same tools.
- Particle physics: the Standard Model is built on representations of su(3) x su(2) x u(1), predicting particles from their quantum numbers
- Quantum mechanics: the operator L^2 is the Casimir of su(2), eigenvalues l(l+1) are characters of V_l
- Computational algorithms: Clebsch-Gordan coefficients are tabulated and used in nuclear physics simulations
- Algebraic geometry: flag varieties G/B realize representations geometrically via cohomology
From Weyl to particle physics
Hermann Weyl in 1925-1926 built the complete theory of finite-dimensional representations of semisimple Lie algebras. The unitary trick - using the compact real form - was his key contribution. In 1961 Murray Gell-Mann applied representation theory of sl(3,C) to classify hadrons. The scheme was called the 'Eightfold Way'. In 1964 Gell-Mann predicted the omega-minus baryon using the dimension-10 irreducible representation - and the particle was found. Nobel Prize 1969.
Irreducible Representations of sl(2,C)
1888. Wilhelm Killing classifies all simple Lie algebras over C. No computers, no standard methods - only logic. Five exceptions: G2, F4, E6, E7, E8. The representation theory of sl(2) is the key to understanding all of it: from quantum spin to string theory.
Uniqueness theorem: V_n is the unique irreducible finite-dimensional sl(2,C)-representation of dimension n+1. By Weyl's theorem, any finite-dimensional representation is a direct sum of V_n.
The irreducible sl(2,C) representation with highest weight 4. Dimension and weights?
V_n: dim = n+1, weights = {n, n-2, ..., -n+2, -n} - a set of n+1 numbers in steps of 2.
Weyl's Complete Reducibility Theorem
Hermann Weyl in 1925 proves: any finite-dimensional representation of a semisimple Lie algebra is a direct sum of irreducibles. This is not automatic: for non-semisimple algebras it can fail. One class of algebras - and an abyss between reducibility and non-reducibility.
Tensor product V_1 tensor V_1
Clebsch-Gordan rule
V_1 tensor V_1 = V_2 + V_0 (dimensions: 2*2 = 3+1). This is the Clebsch-Gordan rule. In physics: combining two spin-1/2 gives spin-1 (triplet) and spin-0 (singlet). Complete reducibility guarantees this decomposition exists and is unique.
Clebsch-Gordan rule: V_m tensor V_n = V_{m+n} + V_{m+n-2} + ... + V_{|m-n|}. Physicists use this every time angular momenta are combined. The total number of summands is min(m,n)+1.
Weyl's theorem: any finite-dimensional representation of a semisimple Lie algebra is a direct sum of irreducibles. The key step in the proof:
Proof via the invariant complement: for W inside V, construct a g-invariant projector pi: V -> W (via the Killing form or compact group). Then V = W + ker(pi).
Highest Weight Theory for Semisimple Algebras
sl(2,C) is the prototype. For a general semisimple g the same idea applies: Cartan subalgebra h, roots alpha, raising and lowering operators e_alpha and f_alpha. The highest weight determines the irreducible representation. Only now the weight lattice is multi-dimensional.
| Algebra | Rank | Irreducibles | Example |
|---|---|---|---|
| sl(2,C) = A_1 | 1 | V_n, n >= 0 | Spin-n/2 in quantum mechanics |
| sl(3,C) = A_2 | 2 | V_{a,b}, a,b >= 0 | Quarks: (1,0)=3, (0,1)=3*, (1,1)=8 |
| sp(4,C) = C_2 | 2 | V_{a,b}, a,b >= 0 | Symplectic mechanics |
| so(5,C) = B_2 | 2 | V_{a,b}, a,b >= 0 | Spinor representations in 5D |
Cartan's highest weight theorem: for semisimple g over C, there is a bijection between dominant integral weights P^+ and isomorphism classes of finite-dimensional irreducible g-modules. A complete classification.
Cartan's highest weight theorem: irreducible finite-dimensional g-modules are classified by:
The bijection P^+ <-> {irreducible modules} is the fundamental theorem of Lie representation theory. lambda in P^+ is dominant if <lambda, alpha^v> >= 0 for all simple roots.
Characters and Tensor Product Decompositions
In particle physics everything reduces to decomposing tensor products. SU(3) x SU(3) -> SU(3) in chromodynamics: how do color states decompose? Mathematics gives the answer via the Littlewood-Richardson rule. A direct connection to symmetric functions.
Quark model SU(3)
Physics via representation theory
In Gell-Mann's quark model, baryons are elements of the triple tensor product: 3 tensor 3 tensor 3 = 10 + 8 + 8 + 1. Here 3 = V_{(1,0)} - the fundamental representation of sl(3,C). The decuplet 10 is the baryon multiplet (including the Delta), the octet 8 is the nucleon-lambda hyperon sector, the singlet 1 is fully antisymmetric.
The Clebsch-Gordan rule for sl(2): V_m tensor V_n = ?
Clebsch-Gordan: V_m tensor V_n = sum V_{m+n-2k} for k=0,...,min(m,n). Total: min(m,n)+1 summands.
Connections to other topics
Lie algebra representation theory is central to mathematical physics.
- alg-28-sym-func — extends
Итоги
- V_n is the unique irreducible sl(2,C)-representation of dimension n+1, weights n, n-2,...,-n
- Weight decomposition: e: V_lambda -> V_{lambda+2}, f: V_lambda -> V_{lambda-2}, h diagonal
- Weyl's theorem: semisimple g implies complete reducibility, invariant complement via compact form
- Clebsch-Gordan: V_m tensor V_n = V_{m+n} + V_{m+n-2} + ... + V_{|m-n|}
- Cartan highest weight theorem: P^+ <-> irreducible modules - complete classification
Вопросы для размышления
- Why does complete reducibility hold for sl(2,C) but not for all Lie algebras over fields of characteristic p?
- How does the character chi_{V_n}(t) = t^n + ... + t^{-n} encode the weight structure?
- What is the physical meaning of the Clebsch-Gordan rule when combining angular momenta of two particles?
Связанные уроки
- alg-24-lie-algebras — A representation is a homomorphism from a Lie algebra
- alg-25-universal-enveloping — U(g)-modules coincide with g-representations
- alg-28-sym-func — Schur functions are characters of GL(n)-representations