Lie Groups & Lie Algebras

1. Continuous Symmetries and Lie Groups

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

The dimension of a Lie group is the dimension of the underlying manifold. Key examples include:

• $\mathrm{GL}(n, \mathbb{R})$: the general linear group of invertible $n \times n$ real matrices, dimension $n^2$
• $\mathrm{O}(n)$: orthogonal matrices with $R^T R = I$, dimension $n(n-1)/2$
• $\mathrm{SO}(n)$: special orthogonal matrices (det = +1), same dimension as $\mathrm{O}(n)$
• $\mathrm{U}(n)$: unitary matrices with $U^\dagger U = I$, dimension $n^2$
• $\mathrm{SU}(n)$: special unitary matrices (det = 1), dimension $n^2 - 1$

Consider the rotation group $\mathrm{SO}(2)$. Every element can be parameterized by a single angle $\theta \in [0, 2\pi)$:

This is a one-parameter Lie group. The group operation is $R(\theta_1)R(\theta_2) = R(\theta_1 + \theta_2)$, showing that $\mathrm{SO}(2) \cong U(1)$, topologically a circle $S^1$.

For $\mathrm{SO}(3)$, we need three parameters (e.g., Euler angles). An arbitrary rotation can be written as a rotation by angle $\theta$ about an axis $\hat{n}$:

where $\mathbf{J} = (J_1, J_2, J_3)$ are the generators of rotation. This is our first encounter with the exponential map, which we will develop systematically below.

2. The Lie Algebra as Tangent Space at the Identity

The Lie algebra $\mathfrak{g}$ of a Lie group $G$ is the tangent space at the identity element $e \in G$, equipped with a bracket operation. To construct it, consider a smooth curve $\gamma(t)$ through the identity:

For a matrix Lie group, the elements of the Lie algebra are matrices. Consider $G = \mathrm{SU}(2)$. If $U(t) \in \mathrm{SU}(2)$ with $U(0) = I$, then $U(t)^\dagger U(t) = I$ implies:

So $X$ must be anti-Hermitian: $X^\dagger = -X$. Similarly, $\det U(t) = 1$ implies$\mathrm{tr}(X) = 0$. Thus $\mathfrak{su}(2)$ consists of traceless anti-Hermitian$2 \times 2$ matrices.

In physics, we conventionally use Hermitian generators by writing $X = iT$, where $T$ is Hermitian and traceless. A basis for $\mathfrak{su}(2)$ is then given by the Pauli matrices (up to a factor of $1/2$):

A basis $\{T_a\}$ for $\mathfrak{g}$ defines the structure constants$f_{abc}$ through:

For $\mathfrak{su}(2)$, the structure constants are $f_{abc} = \epsilon_{abc}$ (the Levi-Civita symbol):

This is precisely the angular momentum algebra of quantum mechanics! The structure constants encode the entire local structure of the Lie group.

3. The Exponential Map

The exponential map connects the Lie algebra (linear, easy to work with) to the Lie group (curved, nonlinear). For matrix Lie groups, it is literally the matrix exponential:

To see why this works, define $g(t) = e^{tX}$ for $X \in \mathfrak{g}$. Then:

So $g(t)$ is a one-parameter subgroup of $G$ with tangent vector $X$ at the identity.

Example: $\mathrm{SU}(2)$ rotations. Let $X = -i\theta \hat{n} \cdot \boldsymbol{\sigma}/2$. Using $(\hat{n} \cdot \boldsymbol{\sigma})^2 = I$, the exponential simplifies:

This is the fundamental spinor rotation formula. A $2\pi$ rotation gives $-I$, not $I$ — the hallmark of spin-$1/2$ particles. This demonstrates the double cover: $\mathrm{SU}(2) \to \mathrm{SO}(3)$ is a 2-to-1 homomorphism.

Key properties of the exponential map:

• $\exp(0) = I$ (the identity element)
• $\exp(-X) = [\exp(X)]^{-1}$ (inverses)
• $\exp((s+t)X) = \exp(sX)\exp(tX)$ (one-parameter subgroups)
• If $[X,Y] = 0$, then $\exp(X+Y) = \exp(X)\exp(Y)$
• $\det(\exp(X)) = e^{\mathrm{tr}(X)}$

The last property immediately explains why $\mathfrak{su}(n)$ requires traceless generators:$\det(e^X) = e^{\mathrm{tr}(X)} = 1$ forces $\mathrm{tr}(X) = 0$.

4. Baker-Campbell-Hausdorff Formula

When $[X,Y] \neq 0$, the product of exponentials is not simply $e^{X+Y}$. The Baker-Campbell-Hausdorff (BCH) formula tells us what the correction is:

The remarkable fact is that $Z$ is expressed entirely in terms of $X$, $Y$, and nested commutators. This means the Lie algebra (with its bracket) completely determines the local group multiplication law.

Derivation sketch. Define $Z(t)$ by $e^{Z(t)} = e^X e^{tY}$ with $Z(0) = X$. Differentiating:

where $\mathrm{ad}_X(Y) = [X, Y]$ is the adjoint action and $B_n$ are Bernoulli numbers. Integrating from 0 to 1 and expanding order-by-order in nested commutators gives the BCH series.

Special case: Zassenhaus formula. For decomposing a single exponential:

Physical application: In quantum mechanics, the BCH formula is essential for understanding the relationship between the position and momentum operators. Since$[\hat{x}, \hat{p}] = i\hbar$, which is a c-number, the BCH series truncates:

This is the Weyl relation, fundamental to quantum mechanics and quantum optics.

5. SU(2) and SO(3): The Rotation Groups

$\mathrm{SO}(3)$ is the group of rotations in three dimensions. Its Lie algebra$\mathfrak{so}(3)$ consists of $3 \times 3$ antisymmetric matrices, with basis:

$\mathrm{SU}(2)$ has the same Lie algebra $\mathfrak{su}(2) \cong \mathfrak{so}(3)$(both have $[T_a, T_b] = i\epsilon_{abc}T_c$), but different global topology:

• $\mathrm{SU}(2) \cong S^3$ (the 3-sphere), simply connected
• $\mathrm{SO}(3) \cong \mathbb{RP}^3$ (real projective 3-space), $\pi_1 = \mathbb{Z}_2$
• The covering map $\mathrm{SU}(2) \to \mathrm{SO}(3)$ is 2-to-1: both $U$ and $-U$ map to the same rotation

The Casimir operator $\mathbf{J}^2 = J_1^2 + J_2^2 + J_3^2$ commutes with all generators. Its eigenvalues label irreducible representations:

where $j = 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$ and $m = -j, -j+1, \ldots, j$. Integer $j$ gives representations of $\mathrm{SO}(3)$; half-integer $j$ requires$\mathrm{SU}(2)$. This is why fermions (half-integer spin) pick up a minus sign under $2\pi$ rotation.

6. SU(3) and the Eightfold Way

$\mathrm{SU}(3)$ is the gauge group of quantum chromodynamics (QCD), the theory of the strong interaction. Its Lie algebra $\mathfrak{su}(3)$ has dimension $3^2 - 1 = 8$, with generators given by the Gell-Mann matrices $\lambda_a$ ($a = 1, \ldots, 8$):

The Gell-Mann matrices generalize the Pauli matrices to $3 \times 3$. The first three ($\lambda_1, \lambda_2, \lambda_3$) form an $\mathfrak{su}(2)$ subalgebra (isospin). The diagonal generators $T_3$ and $T_8$ form the Cartan subalgebra:

The Cartan subalgebra is the maximal set of mutually commuting generators. Its rank (2 for SU(3)) determines the number of simultaneously diagonalizable quantum numbers. For quarks, these are isospin $I_3$ and hypercharge $Y$:

The remaining generators are raising and lowering operators (root vectors) that change the eigenvalues of $I_3$ and $Y$. This root structure determines the weight diagrams that classify particles — Gell-Mann's Eightfold Way.

$\mathrm{SU}(3)$ has two independent Casimir operators, corresponding to the two invariants that label representations. The fundamental representation $\mathbf{3}$ describes quarks, the conjugate $\bar{\mathbf{3}}$ describes antiquarks, and the adjoint $\mathbf{8}$describes gluons.

7. Applications to Quantum Mechanics and Particle Physics

In quantum mechanics, symmetries are represented by unitary operators on Hilbert space. If a Lie group $G$ is a symmetry of the Hamiltonian, then by Wigner's theorem there exists a unitary representation $U: G \to \mathcal{U}(\mathcal{H})$:

This has profound consequences:

• Rotational invariance ($\mathrm{SO}(3)$ or $\mathrm{SU}(2)$): Conservation of angular momentum. Energy eigenstates organize into multiplets of dimension $2j+1$.
• Translational invariance (the Lie group $(\mathbb{R}^3, +)$): Conservation of linear momentum.
• Gauge symmetry ($\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$): The Standard Model of particle physics. Each factor determines a force: electromagnetism, weak interaction, and strong interaction respectively.

The representation theory of $\mathrm{SU}(3)_{\mathrm{flavor}}$ classifies hadrons into multiplets. The meson octet and baryon decuplet were predicted by this group theory before being observed experimentally, leading to the discovery of the $\Omega^-$ baryon in 1964.

In quantum field theory, the gauge principle elevates a global Lie group symmetry to a local one by introducing gauge fields $A_\mu^a$ valued in the Lie algebra:

The covariant derivative $D_\mu$ ensures local gauge invariance. The field strength tensor inherits the Lie algebra structure:

The non-abelian structure constants $f_{abc}$ cause the gauge bosons to interact with each other — gluons carry color charge, unlike photons which are neutral. This is the origin of asymptotic freedom and confinement in QCD.

8. The Killing Form and Classification

The Killing form is a bilinear form on the Lie algebra defined using the adjoint representation:

In terms of structure constants: $\kappa_{ab} = f_{acd} f_{bdc}$. The Killing form plays the role of a metric on the Lie algebra. A Lie algebra is semisimple if and only if the Killing form is non-degenerate (Cartan's criterion).

For $\mathfrak{su}(2)$: $\kappa_{ab} = -2\delta_{ab}$. The negative definiteness reflects the compactness of $\mathrm{SU}(2)$. For compact semisimple Lie algebras,$\kappa$ is negative definite.

Cartan classification. All simple Lie algebras over $\mathbb{C}$ are classified into four infinite families and five exceptional cases:

• $A_n$: $\mathfrak{su}(n+1)$, dimension $n(n+2)$
• $B_n$: $\mathfrak{so}(2n+1)$, dimension $n(2n+1)$
• $C_n$: $\mathfrak{sp}(2n)$, dimension $n(2n+1)$
• $D_n$: $\mathfrak{so}(2n)$, dimension $n(2n-1)$
• Exceptional: $G_2, F_4, E_6, E_7, E_8$

The classification uses root systems and Dynkin diagrams. Each simple root corresponds to a node in the Dynkin diagram, with edges encoding the angles between roots. This elegant classification constrains which Lie groups can appear as gauge groups in physics.

The quadratic Casimir operator can be expressed using the inverse Killing form:

For $\mathfrak{su}(2)$, this gives $C_2 = \mathbf{J}^2$. By Schur's lemma,$C_2$ is proportional to the identity on each irreducible representation, providing quantum numbers that label the representation.

9. The Lorentz and Poincare Groups

The Lorentz group $\mathrm{SO}(3,1)$ is the symmetry group of special relativity. Its Lie algebra has six generators: three rotations $J_i$ and three boosts $K_i$:

The crucial minus sign in $[K_i, K_j]$ reflects the non-compactness of the Lorentz group (boosts do not form a closed subgroup). Defining $N_i^\pm = \frac{1}{2}(J_i \pm iK_i)$:

So $\mathfrak{so}(3,1)_\mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$! Representations are labeled by $(j^+, j^-)$:

• $(0, 0)$: scalar
• $(\frac{1}{2}, 0)$: left-handed Weyl spinor
• $(0, \frac{1}{2})$: right-handed Weyl spinor
• $(\frac{1}{2}, \frac{1}{2})$: four-vector
• $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$: Dirac spinor

The Poincare group extends the Lorentz group by spacetime translations, adding four generators$P_\mu$. The Casimir operators are $P_\mu P^\mu = m^2$ (mass) and the Pauli-Lubanski pseudovector $W^2$ (spin). These label the elementary particles: massive particles by $(m, s)$ and massless particles by helicity.

10. Lie Groups Beyond Particle Physics

Lie groups appear far beyond the Standard Model. In condensed matter physics, the symmetry group of a crystal lattice determines band structure and allowed transitions. The continuous symmetries relevant include:

• Spin-orbit coupling: The total symmetry group is the double group, involving$\mathrm{SU}(2)$ spin rotations combined with spatial point group operations.
• Topological insulators: Protected by time-reversal symmetry ($T^2 = -1$for spin-1/2), classified using K-theory and the homotopy groups of Lie groups.
• Conformal field theory: The conformal group in $d$ dimensions is$\mathrm{SO}(d+1, 1)$ (Euclidean) or $\mathrm{SO}(d, 2)$ (Lorentzian), with infinite-dimensional enhancement in $d = 2$ (the Virasoro algebra).

In general relativity, the diffeomorphism group (an infinite-dimensional Lie group) is the gauge group of gravity. The tangent space at each point carries a Lorentz group action, making gravity a gauge theory of $\mathrm{SO}(3,1)$ in the vierbein formalism.

11. Numerical Exploration: Lie Groups with Python

The following simulation verifies key properties of Lie groups and their algebras numerically. We compute the exponential map, verify the BCH formula, check structure constants, and explore the SU(2)-to-SO(3) covering map.