Part 2: Lie Algebras and Infinitesimal Transformations

From Global to Local: Understanding Infinitesimal Symmetries

When we look closely at any continuous symmetry, we find that large transformations can be built up from infinitesimally small ones. Just as calculus studies functions through their derivatives, Lie algebras study Lie groups through their infinitesimal transformations. This local perspective often simplifies our understanding and calculations while revealing deep structural properties.

Let’s organize our exploration using the mnemonic “ALGEBRA”:

Adjoint representation Lie bracket structure Generators and relations Exponential mapping Basis and coordinates Root decomposition Applications to physics

The Definition of a Lie Algebra

A Lie algebra g is a vector space equipped with a special binary operation called the Lie bracket

that captures the essence of infinitesimal transformations:

1. Axioms For X, Y, Z ∈ g and scalars a, b:

   • Bilinearity:

= a

• b

- Antisymmetry: 

= -

- Jacobi identity: 

] +

] +

] = 0

2. Physical Interpretation

   • Bracket measures non-commutativity

   • Represents infinitesimal version of group multiplication

   • Encodes structure of symmetry transformations

3. Examples

   • so(3): 3×3 antisymmetric matrices

   • su(2): 2×2 traceless anti-Hermitian matrices

   • Vector fields:

= XY - YX

The Adjoint Representation

The adjoint representation provides our first view of how a Lie algebra acts on itself:

1. Definition For X ∈ g:

X,Y

2. Properties

   • Linear map: ad_X ∈ End(g)

   • Represents g on itself

   • Connects to group adjoint action:

3. Physical Applications

   • Angular momentum commutators

   • Gauge transformations

   • Conservation laws

Structure Constants and Classification

The structure of a Lie algebra is encoded in its structure constants:

1. Definition Choose basis {Tᵃ}: T_a,T_b

The fₐbᶜ are structure constants.

2. Properties

   • Antisymmetry: fₐbᶜ = -fbₐᶜ

   • Jacobi identity:

3. Examples For su(2): T_i,T_j

where εᵢⱼₖ is the Levi-Civita symbol.

The Exponential Map in Detail

The exponential map connects Lie algebras to Lie groups:

1. Definition

2. Properties

   • exp(0) = e (identity)

   • exp((s+t)X) = exp(sX)exp(tX)

   • d/dt|ₜ₌₀exp(tX) = X

3. Baker-Campbell-Hausdorff Formula

X,Y

X,[X,Y

Y,[X,Y

Crucial for understanding composition of transformations.

Root Systems and Classification

Simple Lie algebras are classified by their root systems:

1. Cartan Subalgebra Maximal abelian subalgebra h ⊂ g:

   • Simultaneously diagonalizable elements

   • Dimension is the rank of g

   • Determines weight structure

2. Root Decomposition

where Φ is the root system and gα are root spaces.

3. Classification Simple Lie algebras are classified as:

   • Aₙ (SU(n+1))

   • Bₙ (SO(2n+1))

   • Cₙ (Sp(2n))

   • Dₙ (SO(2n))

   • Exceptional: E₆, E₇, E₈, F₄, G₂

Physical Applications

1. Quantum Mechanics Angular momentum algebra: J_i,J_j

Leading to:

• Quantization of angular momentum

• Selection rules

• Clebsch-Gordan coefficients

2. Gauge Theory Yang-Mills theory: T_a,T_b

Determines:

• Field strength

• Interaction vertices

• Conservation laws

3. Particle Physics Standard Model symmetries:

   • SU(3) color

   • SU(2) weak isospin

   • U(1) hypercharge

Advanced Concepts

1. Killing Form Bilinear form on g:

Properties:

• Invariant under adjoint action

• Non-degenerate for semisimple algebras

• Determines compact form

2. Universal Enveloping Algebra U(g) contains all polynomials in generators:

   • Provides representation theory

   • Contains Casimir operators

   • Important for quantization

3. Real Forms Different real versions of complex Lie algebra:

   • Compact forms

   • Split forms

   • Determines physical applications

Practical Calculations

1. Computing Commutators For su(2):

= 2iεᵢⱼₖσₖ

where σᵢ are Pauli matrices

Verify Jacobi identity

2. Structure Constants For so(3):

Choose basis Lᵢ

Compute

Extract structure constants

Common Issues and Their Resolution

1. Abstract vs. Concrete Problem: Getting lost in abstraction Solution: Work with matrix representations Example: su(2) via Pauli matrices

2. Global vs. Local Problem: Confusing group and algebra elements Solution: Keep track of exponential map Example: Rotation angles vs. generators

3. Basis Dependence Problem: Over-relying on specific basis Solution: Focus on invariant properties Example: Casimir operators

Exercises for Understanding

1. Basic Computation Calculate structure constants for su(2):

   • Write out Pauli matrices

   • Compute all commutators

   • Verify antisymmetry

2. Physical Application For angular momentum:

   • Find uncertainty relations

   • Calculate raising/lowering operators

   • Determine spectrum

3. Advanced Problem Prove that the Killing form is:

   • Symmetric

   • Ad-invariant

   • Non-degenerate for su(2)

Looking Ahead

In our next section, we’ll explore representation theory, showing how:

• Lie algebras act on vector spaces

• Weights and roots organize representations

• Physical states form representations

Remember: Lie algebras aren’t just mathematical structures — they encode the fundamental symmetries of physics in their simplest form. Understanding them deeply means understanding how symmetry constrains physical laws.

Next: Part 3 - Representation Theory and Physical Applications