Lie Groups and Their Algebras

Lie Groups: The Geometry of Continuous Symmetries

What is a Lie Group?

The symmetries of physical systems—rotations, translations, gauge transformations—form not just groups, but continuous families of transformations. These objects are called Lie groups—smooth manifolds equipped with a group structure.

The key idea of Lie (1870s): continuous symmetries are described by “infinitesimal” transformations—elements of the Lie algebra. Instead of studying the entire group (a complex nonlinear object), we study its algebra (a linear space!) and “reconstruct” the group via the exponential.

Definition and Examples

A Lie group is a smooth manifold $G$ with operations: multiplication $(a, b) \mapsto ab$ and inversion $a \mapsto a^{-1}$, both smooth.

Main examples:

$\mathbb{R}^n$ (addition)—an Abelian Lie group, $\dim = n$.

$GL(n, \mathbb{R})$—invertible $n\times n$ matrices, an open subset of $M(n, \mathbb{R}) \cong \mathbb{R}^{n^2}$, $\dim = n^2$.

$SL(n, \mathbb{R}) = {\det A = 1}$—a hypersurface in $GL(n)$, $\dim = n^2 - 1$.

$O(n) = {A^\mathsf{T}A = I}$—orthogonal matrices, $\dim = n(n-1)/2$.

$SO(n) = O(n) \cap {\det A = 1}$—the connected component $e \in O(n)$.

$U(n) = {A^\dagger A = I}$—unitary matrices, $\dim = n^2$ (over $\mathbb{R}$).

$SU(n) = U(n) \cap {\det A = 1}$, $\dim = n^2 - 1$.

$SO(3)$: A 3-dimensional compact manifold—the group of rotations in $\mathbb{R}^3$. Parameterization: three Euler angles $(\varphi, \theta, \psi)$ or axis–angle (axis $\hat{n} \in S^2$, angle $\theta \in [0, \pi]$).

$SU(2) \cong S^3$—the 3-sphere! Element: $U = aI + i(b\sigma_1 + c\sigma_2 + d\sigma_3)$, $|a|^2 + |b|^2 + |c|^2 + |d|^2 = 1$. $SU(2) \to SO(3)$—a double covering (twofold covering: $\pm U \to$ one rotation).

Lie Algebra

The Lie algebra $\mathfrak{g}$ of a group $G$ is the tangent space at the identity: $\mathfrak{g} = T_e G$, $\dim \mathfrak{g} = \dim G$.

Lie bracket $[\cdot, \cdot]$: $\mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$—defined via the commutator: $[X, Y] = XY - YX$ for matrix groups. Properties: antisymmetry $[X, Y] = -[Y, X]$; Jacobi identity $[[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0$.

Algebras of classical groups:

$\mathfrak{gl}(n) = M(n, \mathbb{R})$: all matrices, $[X, Y] = XY - YX$.

$\mathfrak{sl}(n) = {\operatorname{tr} X = 0}$: trace zero.

$\mathfrak{so}(n) = {X + X^\mathsf{T} = 0}$: skew-symmetric matrices, $\dim = n(n-1)/2$.

$\mathfrak{su}(n) = {X + X^\dagger = 0,, \operatorname{tr} X = 0}$: anti-Hermitian with zero trace.

$\mathfrak{so}(3)$: Skew-symmetric $3\times 3$ matrices. Basis: $e_1 = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & -1 \ 0 & 1 & 0 \end{bmatrix}$, $e_2 = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ -1 & 0 & 0 \end{bmatrix}$, $e_3 = \begin{bmatrix} 0 & -1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}$. Brackets: $[e_1, e_2] = e_3$, $[e_2, e_3] = e_1$, $[e_3, e_1] = e_2$—the angular momentum algebra!

Exponential Map

$\exp: \mathfrak{g} \to G,\ X \mapsto e^X = I + X + X^2/2! + \ldots$

Properties: A local diffeomorphism near $0 \in \mathfrak{g}$ and $e \in G$. For compact connected groups: $\exp$ is surjective. One-parameter subgroup: $t \mapsto e^{tX}$ (a straight line through $e$ in $G$).

Baker–Campbell–Hausdorff Formula:

$ e^X e^Y = e^{X + Y + [X,Y]/2 + ([X,[X,Y]] - [Y,[X,Y]])/12 + \ldots} $

A series in terms of Lie brackets. It shows: group multiplication $\leftrightarrow$ bracket in the algebra.

Representations of Lie Groups

A representation $\rho: G \to GL(V)$ is a homomorphism of Lie groups ($V$ is a vector space). Differential: $d\rho: \mathfrak{g} \to \mathfrak{gl}(V)$—a representation of the Lie algebra.

Classification for $SO(3)$ / $SU(2)$: The irreducible finite-dimensional representations of $SU(2)$ are parametrized by $j = 0, 1/2, 1, 3/2, 2, \ldots$ (total spin). Dimension: $2j + 1$. Physically: $j = 0$ (scalar), $j = 1/2$ (spinor, electron), $j = 1$ (vector, photon), $j = 2$ (graviton).

For $SO(3)$: only integer $j$ (no spinor representations). Spin $1/2$ is an “illegal” representation for $SO(3)$, but legal for $SU(2)$ $\to$ quantum mechanics requires $SU(2)$, not $SO(3)$!

Real Applications

Quantum mechanics: Angular momentum is described by the algebra $\mathfrak{so}(3)$. Ladder operators $L_{\pm} = L_x \pm iL_y$ and $[L_z, L_{\pm}] = \pm \hbar L_{\pm}$—these are $su(2)$ brackets. The quantization rules for angular momenta are the representation theory of $SU(2)$.

Particle physics: The Standard Model is based on the gauge symmetry $SU(3) \times SU(2) \times U(1)$. $SU(3)$—quantum chromodynamics (quark color, 8 gluons $= \dim, su(3) - 1 = 8$). $SU(2) \times U(1)$—the electroweak interaction.

Robotics: The groups $SO(3)$ and $SE(3) = SO(3) \ltimes \mathbb{R}^3$—the configurations of a rigid body. Control algorithms use the exponential map to compute orbits.

Lie Groups in Physics and Robotics

Lie groups and their algebras are working tools of theoretical physics and modern engineering. In particle physics, the Standard Model is based on the gauge symmetry group $U(1) \times SU(2) \times SU(3)$: each group generates a corresponding fundamental type of interaction (electromagnetic, weak, strong). The representations of the groups $SU(2)$ and $SU(3)$ classify elementary particles: multiplets of quarks, leptons, and bosons are determined by the dimensions of irreducible representations of these groups. In general relativity, the Lorentz group $O(1,3)$—the Lie group of spacetime—determines the tensorial nature of physical quantities. In robotics, the configuration space of a manipulator with $n$ links is the product of $n$ copies of $SO(2)$ or $SO(3)$, and trajectory planning is performed directly on Lie groups using the exponential map. Satellite and quadcopter attitude control is carried out on $SO(3)$, not in angular parameters, avoiding singularities (gimbal lock). In computer vision, the group $SE(3) = SO(3) \ltimes \mathbb{R}^3$ describes rigid body transformations, and camera calibration, SLAM algorithms operate with exponential maps on $SE(3)$.

Assignment: (a) $SO(2) \cong S^1$: describe the Lie algebra $\mathfrak{so}(2)$ and the exponential map $\exp: \mathfrak{so}(2) \to SO(2)$. (b) Compute $[X, Y]$ for $X = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}$ and $Y = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}$ in $sl(2, \mathbb{R})$. (c) Why is spin $1/2$ possible only in quantum mechanics ($SU(2)$), but not in classical mechanics ($SO(3)$)?