Mathematics

Lie Groups & Algebras

Lie groups are continuous symmetry groups that combine group theory with differential geometry. They are fundamental to modern physics: gauge theories, particle physics, General Relativity, and quantum gravity.

1. Lie Groups - Definition and Structure

A Lie group G is a smooth manifold that is also a group, where the group operations (multiplication and inversion) are smooth maps.

Definition

G is a Lie group if:

  • G is a smooth manifold
  • Multiplication μ: G × G → G is smooth
  • Inversion i: G → G, g ↦ g⁻¹ is smooth
  • There exists an identity element e ∈ G

Examples of Lie Groups

GL(n, ℝ): General linear group, n×n invertible real matrices

$$\text{GL}(n, \mathbb{R}) = \{A \in \mathbb{R}^{n \times n} : \det(A) \neq 0\}$$

O(n): Orthogonal group, preserving dot product

$$\text{O}(n) = \{A \in \mathbb{R}^{n \times n} : A^T A = I\}$$

SO(n): Special orthogonal group, rotations (det = +1)

$$\text{SO}(n) = \{A \in \text{O}(n) : \det(A) = 1\}$$

U(n): Unitary group, complex matrices preserving Hermitian product

$$\text{U}(n) = \{A \in \mathbb{C}^{n \times n} : A^\dagger A = I\}$$

SU(n): Special unitary group (det = 1)

$$\text{SU}(n) = \{A \in \text{U}(n) : \det(A) = 1\}$$

2. Lie Algebras

The Lie algebra 𝔤 of a Lie group G is the tangent space at the identity, with a bracket operation encoding infinitesimal group structure.

Definition

The Lie algebra 𝔤 = T_e G (tangent space at identity) with Lie bracket [·,·]:

$$[X, Y] = XY - YX \quad \text{(for matrix groups)}$$

Properties of Lie Bracket

  • Bilinearity: $[aX + bY, Z] = a[X,Z] + b[Y,Z]$
  • Antisymmetry: $[X, Y] = -[Y, X]$
  • Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$

Exponential Map

The exponential map connects the Lie algebra to the Lie group:

$$\exp: \mathfrak{g} \to G, \quad X \mapsto e^X = \sum_{n=0}^\infty \frac{X^n}{n!}$$

For matrix groups, this is the usual matrix exponential. It maps a neighborhood of 0 ∈ 𝔤 to a neighborhood of e ∈ G.

Examples of Lie Algebras

𝔰𝔬(3): Lie algebra of SO(3), 3×3 antisymmetric matrices

$$\mathfrak{so}(3) = \left\{\begin{pmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{pmatrix} : a,b,c \in \mathbb{R}\right\}$$

𝔰𝔲(2): Lie algebra of SU(2), traceless anti-Hermitian 2×2 matrices

$$\mathfrak{su}(2) = \left\{i\begin{pmatrix} a & b+ic \\ -b+ic & -a \end{pmatrix} : a,b,c \in \mathbb{R}\right\}$$

Remarkably, 𝔰𝔬(3) and 𝔰𝔲(2) are isomorphic as Lie algebras (both 3-dimensional with same commutation relations).

3. SO(3) and Rotations

SO(3) is the group of rotations in 3D space, fundamental to classical mechanics and angular momentum.

Generators

The Lie algebra 𝔰𝔬(3) has basis (generators) $\{L_x, L_y, L_z\}$ satisfying:

$$[L_i, L_j] = \epsilon_{ijk} L_k$$

where ε_ijk is the Levi-Civita symbol. These are the angular momentum commutation relations in quantum mechanics.

Rotation Matrices

Rotation by angle θ about axis n̂:

$$R(\theta, \hat{n}) = \exp(\theta \, \hat{n} \cdot \vec{L})$$

For rotation about z-axis by θ:

$$R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Topology

SO(3) is not simply connected. Rotations by 2π and 0 are distinct paths, but 4π and 0 are homotopic. This leads to spinors and the double cover SU(2).

4. SU(2) and Spin

SU(2) is the group of 2×2 unitary matrices with determinant 1, essential for describing spin-½ particles and the double cover of SO(3).

Pauli Matrices

The Lie algebra 𝔰𝔲(2) has basis given by $\{i\sigma_x/2, i\sigma_y/2, i\sigma_z/2\}$ where:

$$\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Commutation Relations

$$[\sigma_i, \sigma_j] = 2i\epsilon_{ijk} \sigma_k$$

SU(2) as Double Cover of SO(3)

There is a 2-to-1 homomorphism SU(2) → SO(3). Each rotation in SO(3) corresponds to two elements ±g in SU(2).

This is why spin-½ particles require a 4π rotation (not 2π) to return to original state—they transform under SU(2) representations, not SO(3).

Spinors

Spinors are objects that transform under SU(2) (or its Lorentz group analogue SL(2,ℂ)). A spin-½ particle state is a 2-component spinor:

$$|\psi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}, \quad |\alpha|^2 + |\beta|^2 = 1$$

Under rotation: $|\psi\rangle \to U(\theta, \hat{n})|\psi\rangle$ where $U = \exp(-i\theta \hat{n} \cdot \vec{\sigma}/2) \in \text{SU}(2)$

5. The Lorentz Group

The Lorentz group O(1,3) preserves the Minkowski metric in special relativity.

Definition

The Lorentz group consists of all linear transformations Λ that preserve the spacetime interval:

$$\Lambda^T \eta \Lambda = \eta, \quad \eta = \text{diag}(-1, 1, 1, 1)$$

Generators

The Lie algebra 𝔰𝔬(1,3) has 6 generators:

  • 3 rotations: J_i (angular momentum)
  • 3 boosts: K_i (velocity transformations along x, y, z)

Commutation Relations

$$[J_i, J_j] = \epsilon_{ijk} J_k, \quad [J_i, K_j] = \epsilon_{ijk} K_k, \quad [K_i, K_j] = -\epsilon_{ijk} J_k$$

Boost Transformation

A boost along x-direction with velocity v (rapidity η where tanh η = v/c):

$$\Lambda = \begin{pmatrix} \cosh\eta & -\sinh\eta & 0 & 0 \\ -\sinh\eta & \cosh\eta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Spinor Representation

The proper orthochronous Lorentz group is locally isomorphic to SL(2,ℂ). Dirac spinors (4-component) transform under this representation, essential for relativistic quantum mechanics.

6. Representations

A representation of a Lie group G is a homomorphism from G to GL(V), the group of invertible linear operators on a vector space V.

Definition

$$\rho: G \to \text{GL}(V), \quad \rho(g_1 g_2) = \rho(g_1)\rho(g_2)$$

This induces a Lie algebra representation:

$$\rho_*: \mathfrak{g} \to \mathfrak{gl}(V), \quad \rho_*([X,Y]) = [\rho_*(X), \rho_*(Y)]$$

Irreducible Representations

A representation is irreducible if V has no proper invariant subspaces.

Examples:

  • SO(3): Irreducible representations labeled by spin j = 0, ½, 1, 3/2, ... (dimension 2j+1)
  • SU(2): Same as SO(3) but includes half-integer spins
  • SU(3): Quark color symmetry in QCD (fundamental rep is 3-dimensional)

Adjoint Representation

The adjoint representation acts on the Lie algebra itself:

$$\text{Ad}_g: \mathfrak{g} \to \mathfrak{g}, \quad \text{Ad}_g(X) = gXg^{-1}$$

The infinitesimal version:

$$\text{ad}_X(Y) = [X, Y]$$

Gauge fields in physics transform under the adjoint representation (e.g., gluons for SU(3)).

7. Gauge Transformations and Gauge Theory

Gauge theory describes interactions in particle physics using local symmetries (Lie groups at each spacetime point).

Global vs Local Symmetry

Global symmetry: Transformation g ∈ G acts uniformly at all spacetime points

$$\psi(x) \to g\psi(x), \quad g = \text{constant}$$

Local (gauge) symmetry: Transformation depends on spacetime point

$$\psi(x) \to g(x)\psi(x), \quad g: M \to G$$

Gauge Field (Connection)

To make the theory locally symmetric, introduce a gauge field A_μ (connection) that transforms as:

$$A_\mu \to g A_\mu g^{-1} + g \partial_\mu g^{-1}$$

Replace ordinary derivative with covariant derivative:

$$D_\mu = \partial_\mu + igA_\mu$$

Field Strength Tensor

The field strength (curvature) measures non-commutativity of covariant derivatives:

$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + ig[A_\mu, A_\nu]$$

It transforms covariantly: $F_{\mu\nu} \to g F_{\mu\nu} g^{-1}$

Examples in Physics

  • U(1): Electromagnetism, A_μ is photon field, F_μν is electromagnetic field strength
  • SU(2): Weak force, W± and Z bosons
  • SU(3): Strong force (QCD), 8 gluons
  • SU(2) × U(1): Electroweak unification

8. Applications to General Relativity and Quantum Gravity

Diffeomorphism Group

General Relativity has diffeomorphism invariance—coordinate transformations are gauge symmetries. The gauge group is Diff(M), the group of diffeomorphisms of spacetime.

Lorentz Group in GR

At each point p ∈ M, the tangent space has Lorentz symmetry SO(1,3). The frame bundle is a principal SO(1,3)-bundle over spacetime.

Spinors in curved spacetime require the spin connection ω_μ^{ab}, an SO(1,3) gauge field.

Loop Quantum Gravity

LQG reformulates GR as an SU(2) gauge theory using Ashtekar variables:

$$A_a^i = \Gamma_a^i + \gamma K_a^i$$

where A is an SU(2) connection, Γ is the spin connection, K is extrinsic curvature, and γ is the Immirzi parameter.

Quantum states are spin networks—graphs with SU(2) representations on edges. Area and volume become discrete operators with eigenvalues involving SU(2) representation labels (spins j).

String Theory

String theory has gauge symmetries described by Lie groups:

  • E₈ × E₈ or SO(32): Heterotic string theory
  • Conformal group: 2D conformal field theory on worldsheet
  • U-duality groups: E_n series in M-theory
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