Abstract Algebra
Lie Algebras: Linearizing Symmetry
How do one study a complex nonlinear group (like SO(3) - all rotations in ℝ³)? The answer: linearize it near the identity! The tangent space - the Lie algebra - is linear and finite-dimensional. The bracket [X,Y] = XY−YX encodes non-commutativity. And the beautiful formula exp(tX) carries results back to the group.
- Quantum mechanics: angular momentum operators [Lx,Ly]=iℏLz - Lie brackets of su(2); spin ½ and spin 1 - representations of su(2) of different dimensions
- Gauge field theory: electromagnetism (U(1)), weak interactions (SU(2)), strong (SU(3)) - Lie algebras determine the types of interaction in the Standard Model
Предварительные знания
Lie Algebra: Bracket and Axioms
Unreal Engine 5 (2022) physics engine uses Lie algebras SO(3) for rigid-body rotation interpolation: 60 fps, 0.001 rad precision. A **Lie algebra** g over a field k is a vector space g with an operation [·,·]: g × g → g (Lie bracket) satisfying: 1. **Bilinearity:** [aX+bY, Z] = a[X,Z] + b[Y,Z] 2. **Antisymmetry:** [X,Y] = −[Y,X] (consequence: [X,X] = 0) 3. **Jacobi identity:** [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 **Connection to Lie groups:** If G is a Lie group (smooth manifold + group structure), then g = T_e(G) - the tangent space at the identity - carries a Lie algebra structure via the Lie bracket.
**Physical interpretation:** In quantum mechanics, the angular momentum operators L_x, L_y, L_z satisfy [L_x, L_y] = iℏL_z and cyclically. This is exactly the Lie bracket of so(3) (up to i). Quantum-mechanical failures of commutativity are Lie brackets! Heisenberg's uncertainty principle [x̂, p̂] = iℏ is a Lie bracket in the Weyl algebra.
For the matrix Lie algebra g with bracket [A,B] = AB−BA, what does the Jacobi identity express in matrix terms?
Structure Constants and Classification
Let {X₁, ..., Xₙ} be a basis of the Lie algebra g. Since [Xᵢ, Xⱼ] ∈ g, we can write: [Xᵢ, Xⱼ] = Σₖ fᵢⱼᵏ Xₖ The numbers fᵢⱼᵏ ∈ k are called the **structure constants** of the Lie algebra. They completely determine the bracket. The Jacobi identity imposes algebraic constraints on fᵢⱼᵏ. Classification of finite-dimensional semisimple Lie algebras over ℂ (Wilhelm Killing, Élie Cartan, 1888-1894): - **Series:** Aₙ = sl(n+1), Bₙ = so(2n+1), Cₙ = sp(2n), Dₙ = so(2n) - **Exceptional:** G₂, F₄, E₆, E₇, E₈
**The Lie algebra E₈** has dimension 248 and is connected to the most symmetric lattice in ℝ⁸ (the E₈ lattice, densest sphere packing in 8 dimensions). It appears in heterotic string theory and in the hypothetical 'theory of everything' (Garrett Lisi, 2007). Dynkin diagrams are a beautiful way to encode the structure of a Lie algebra as a combinatorial graph.
The Lie algebra sl(2,ℂ) = {A ∈ Mat(2,ℂ) | tr(A) = 0} has basis: H = [[1,0],[0,-1]], E = [[0,1],[0,0]], F = [[0,0],[1,0]]. What is the bracket [H, E]?
The Exponential Map: Algebra → Group
The **exponential map** exp: g → G is defined for matrix groups as the matrix exponential: exp(A) = I + A + A²/2! + A³/3! + ... It carries the Lie algebra into the Lie group: - exp(0) = I (identity of the group) - exp(tX) - a one-parameter subgroup of G - exp(X)exp(Y) = exp(X + Y + ½[X,Y] + ...) (Baker-Campbell-Hausdorff formula) exp 'linearizes' the group near the identity: the group structure in a neighborhood of e is completely determined by the Lie algebra.
**SU(2) and physics:** The Lie algebra su(2) ≅ so(3) - one algebra, but different groups: SU(2) is a double cover of SO(3). This explains the electron's spinor: a rotation by 360° in SO(3) returns to the starting point, but in SU(2) it gives the opposite point! An electron is not a classical object: its wave function acquires a factor of −1 under a full rotation. The Lie brackets [Lx,Ly] = iLz determine the quantization of angular momenta.
The Lie algebra g and the Lie group G are the same object, just written differently
Correct understanding.
Detailed explanation.
Baker-Campbell-Hausdorff formula: log(exp(X)exp(Y)) = X + Y + ½[X,Y] + ... What happens if [X,Y] = 0?
Key Ideas
- Lie algebra g: vector space + bracket [X,Y] with antisymmetry and Jacobi identity
- Main example: gl(n) with [A,B] = AB − BA
- Structure constants fᵢⱼᵏ: [Xᵢ,Xⱼ] = Σₖ fᵢⱼᵏ Xₖ
- Classification: series Aₙ, Bₙ, Cₙ, Dₙ + exceptional G₂, F₄, E₆, E₇, E₈
- exp: g → G - exponential map, bridge algebra ↔ group
- BCH: log(exp(X)exp(Y)) = X + Y + ½[X,Y] + ...
Further Directions
Representations of Lie algebras (especially sl(2)) are the foundation for classifying quantum numbers in physics. Cartan-Weyl theory describes all irreducible representations via highest weights. Algebraic groups generalize the theory to arbitrary fields.
- Representation Theory — Representations of the Lie algebra g correspond to representations of the Lie group G (via exp)
- Simple Groups — Simple Lie algebras (Killing-Cartan classification) correspond to connected simple Lie groups
Вопросы для размышления
- Prove the Jacobi identity for the commutator bracket [A,B] = AB−BA of matrices, using associativity of matrix multiplication.
- In the algebra sl(2) with basis H,E,F: [H,E]=2E, [H,F]=−2F, [E,F]=H. Find the one-parameter subgroup exp(tE) in SL(2,ℂ).
- SU(2) and SO(3) have isomorphic Lie algebras. Why are the groups themselves not isomorphic? What is the geometric meaning of the double covering SU(2) → SO(3)?