Lie Group Catalog

A reference catalog of the principal Lie groups, their Lie algebras, structure theory, representations, and applications. Primary references: Hall (2015) Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer; Fulton-Harris (1991) Representation Theory, GTM 129; Bröcker-tom Dieck (1985) Representations of Compact Lie Groups; Knapp (2002) Lie Groups Beyond an Introduction; Helgason (1978) Differential Geometry, Lie Groups, and Symmetric Spaces; Varadarajan (1984) Lie Groups, Lie Algebras, and Their Representations; Sepanski (2007) Compact Lie Groups. For robotics and applications: Lynch and Park (2017) Modern Robotics: Mechanics, Planning, and Control, Cambridge; Murray-Li-Sastry (1994) A Mathematical Introduction to Robotic Manipulation, CRC.

1. Foundations

1.1 Lie group definition

A Lie group G is a smooth manifold equipped with group operations:

Multiplication μ: G × G → G, (g, h) ↦ g h, smooth
Inversion ι: G → G, g ↦ g^{-1}, smooth
Identity element e ∈ G

Equivalently: a group object in the category of smooth manifolds. Examples are typically presented as matrix subgroups of GL(n, K).

1.2 Lie algebra

The Lie algebra g of G is the tangent space at the identity, T_e G, equipped with the Lie bracket [·, ·]: g × g → g, a bilinear alternating map satisfying the Jacobi identity: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0

For a matrix group G ⊂ GL(n, R), the bracket is the matrix commutator: [X, Y] = XY - YX.

The Lie algebra encodes the local structure of G near the identity; the connected component of the identity is determined by g.

1.3 Exponential map

exp: g → G via the matrix exponential exp(X) = Σ_{k=0}^∞ X^k / k! for matrix groups; in general, exp(X) = γ_X(1) with γ_X(t) the one-parameter subgroup with γ_X’(0) = X.

Properties:

exp(0) = e
exp(-X) = exp(X)^{-1}
exp((s + t) X) = exp(s X) exp(t X)
exp(X) exp(Y) = exp(X + Y) iff [X, Y] = 0
d/dt|_0 exp(t X) = X

Baker-Campbell-Hausdorff (Campbell 1898, Baker 1905, Hausdorff 1906): log(exp X exp Y) = X + Y + [1/2](X, Y) + (1/12)([X, [X, Y]] + [Y, [Y, X]]) - [1/24](Y, [X, [X, Y)]] + …

For compact connected G, exp is surjective; in general it need not be (e.g. SL(2, R)).

1.4 Lie subgroups and the closed subgroup theorem (Cartan 1930)

Closed subgroups of Lie groups are embedded Lie subgroups; their Lie algebras are subalgebras of g.

1.5 Lie’s third theorem

Every finite-dimensional Lie algebra over R is the Lie algebra of some simply-connected Lie group G̃, unique up to isomorphism. Other connected Lie groups with the same algebra are quotients G̃ / D for D ⊂ Z(G̃) discrete central. (Lie 1880-1893, Cartan 1936.)

1.6 Adjoint representation

Ad: G → GL(g), Ad_g X = (d/dt|_0)(g exp(t X) g^{-1}); for matrix groups Ad_g X = g X g^{-1}. Differential ad = d Ad: g → End(g), ad_X Y = [X, Y].

1.7 Killing form

K(X, Y) = tr(ad_X ∘ ad_Y) is a symmetric bilinear form on g. Cartan’s criterion: g semisimple ⟺ K non-degenerate. K is negative definite ⟺ g is the algebra of a compact group.

1.8 Compact / semisimple / solvable / nilpotent

Abelian: [X, Y] = 0 for all X, Y. Example R^n, T^n = (S^1)^n.
Nilpotent: lower central series g ⊃ [g, g] ⊃ [g, [g, g]] ⊃ … terminates at 0. Example Heisenberg.
Solvable: derived series g ⊃ [g, g] ⊃ [[g, g], [g, g]] ⊃ … terminates. Example upper triangular.
Simple: g non-abelian, no proper non-trivial ideal.
Semisimple: direct sum of simples; equivalently no abelian ideal; K non-degenerate.
Reductive: g = z(g) ⊕ [g, g] with [g, g] semisimple.

1.9 Levi decomposition (Levi 1905, Malcev 1942)

Every finite-dim Lie algebra g over R or C splits as g = r ⋊ s where r is the maximal solvable ideal (the radical) and s is semisimple (unique up to conjugation).

2. Classical matrix groups — connected and complex

2.1 GL(n, R), GL(n, C), GL(n, H)

General linear group: invertible n × n matrices.

dim_R GL(n, R) = n²; non-compact; 2 connected components (det > 0, det < 0)
dim_C GL(n, C) = n² complex, 2 n² real; connected; non-compact
Lie algebra gl(n) = all n × n matrices with [X, Y] = X Y - Y X

2.2 SL(n, R), SL(n, C)

Special linear: det = 1.

dim_R SL(n, R) = n² - 1
dim_C SL(n, C) = n² - 1 complex
sl(n) = traceless n × n matrices
Non-compact; simple for n ≥ 2
SL(2, R) is the universal cover of PSL(2, R) = SO^+(1, 2), 2:1 cover via SL(2, R) → PSL(2, R)
SL(2, R) and SL(2, C) ubiquitous in hyperbolic geometry, modular forms, Möbius transformations

2.3 O(n) and SO(n)

Orthogonal: O(n) = {R ∈ GL(n, R) : R^T R = I}.

Two components: det = ±1
dim O(n) = dim SO(n) = n(n - 1)/2
Compact

SO(n) = {R ∈ O(n) : det R = 1} = connected component of identity. Proper rotations of R^n.

so(n) = {X : X^T = -X} (skew-symmetric); dim n(n-1)/2
Connected; simply connected for n = 2 (= S^1) and simply connected double cover Spin(n) for n ≥ 3

2.4 SO(2)

Circle group S^1 = R/Z; abelian, 1-dimensional.

so(2) ≅ R; bracket [·, ·] = 0
Elements: R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]]
exp(θ J) = R(θ), J = [[0, -1], [1, 0]]
Universal cover: R → SO(2), t ↦ R(2π t); kernel Z

2.5 SO(3)

3D rotations. dim 3; compact; connected but not simply connected (π_1 = Z/2).

so(3) ≅ R³ with [u, v]_{so(3)} = u × v (cross product)
Basis L_x, L_y, L_z (angular momentum) with [L_i, L_j] = ε_{ijk} L_k
exp(θ [n̂]*×) = I + sin θ [n̂]*× + (1 - cos θ) [n̂]_×² (Rodrigues’ rotation formula 1840)
with [n̂]_× = skew-symmetric matrix representing cross product with unit vector n̂
Universal cover Spin(3) = SU(2) (2:1)
Parameterizations: Euler angles (singular at gimbal lock), axis-angle (θ, n̂), unit quaternions (Hamilton 1843), rotation matrix R ∈ SO(3), Cayley parameters
Logarithm: log(R) = (θ/(2 sin θ)) (R - R^T), θ = arccos((tr R - 1)/2)

2.6 SE(2)

Planar rigid-body motions: g = (R, t) with R ∈ SO(2), t ∈ R².

dim 3; semi-direct product SO(2) ⋉ R²
Matrix form: [[R, t], [0, 1]] ∈ GL(3, R)
se(2) = {([ω]_×, v) : ω ∈ R, v ∈ R²}, [(ω₁, v₁), (ω₂, v₂)] = (0, ω₁ J v₂ - ω₂ J v₁)

2.7 SE(3)

3D rigid-body / pose. The workhorse of robotics.

dim 6
Semi-direct product SO(3) ⋉ R³
Matrix form g = [[R, t], [0, 1]] ∈ GL(4, R) with R ∈ SO(3), t ∈ R³
se(3) = {[[[ω]_×, v], [0, 0]]: ω, v ∈ R³}; bracket [(ω_1, v_1), (ω_2, v_2)] = (ω_1 × ω_2, ω_1 × v_2 - ω_2 × v_1)
Twist ξ = (ω, v); screw = (axis, pitch); Chasles’ theorem (Chasles 1830): every rigid motion is a screw motion
Adjoint Ad_g = [[R, [t]_× R], [0, R]]; coadjoint operates on wrenches (forces + moments)
Exponential: exp((ω, v)) = ([R, t], 1) with R = Rodrigues and t = (I + (1-cos θ)/θ² [ω]*× + (θ - sin θ)/θ³ [ω]*×²) v if θ = ‖ω‖

Reference: Murray, Li, and Sastry (1994); Park, F.C. (1995). “Distance metrics on the rigid-body motions with applications to mechanism design,” J. Mechanical Design 117(1), 48-54.

2.8 O(p, q) and SO(p, q)

Indefinite orthogonal: preserves quadratic form of signature (p, q): R^T η R = η, η = diag(1_p, -1_q). dim p(p-1)/2 + q(q-1)/2 + pq.

Lorentz O(1, 3) and SO(1, 3)

4-dim spacetime symmetry; non-compact. Four connected components based on time and parity orientation; restricted Lorentz SO^+(1, 3) is the identity component.

SO^+(1, 3) ≅ PSL(2, C); double cover SL(2, C) (left-handed Weyl spinors)
so(1, 3) split into rotations J_i and boosts K_i with [J_i, J_j] = ε_{ijk} J_k, [K_i, K_j] = -ε_{ijk} J_k, [J_i, K_j] = ε_{ijk} K_k

Poincaré group ISO(1, 3) = SO^+(1, 3) ⋉ R^{1, 3}, dim 10. Foundational symmetry of special relativity and QFT (Wigner 1939 classifies unitary irreps of Poincaré: massive m > 0 with spin s = 0, 1/2, 1, …; massless with helicity ±s).

2.9 U(n)

Unitary: U(n) = {U ∈ GL(n, C) : U* U = I}.

dim_R U(n) = n²
Compact, connected; π_1(U(n)) = Z (det)
u(n) = anti-Hermitian: X* = -X
U(1) = S^1 (electromagnetic gauge); U(2) = (U(1) × SU(2))/Z_2

2.10 SU(n)

Special unitary: det U = 1.

dim_R SU(n) = n² - 1
Compact, simply connected; simple for n ≥ 2

SU(2)

dim 3, double cover of SO(3).

su(2) basis: τ_a = -iσ_a/2 where σ_a are Pauli matrices (Pauli 1927): σ_1 = [[0, 1], [1, 0]], σ_2 = [[0, -i], [i, 0]], σ_3 = [[1, 0], [0, -1]]
[σ_a, σ_b] = 2 i ε_{abc} σ_c; {σ_a, σ_b} = 2 δ_{ab} I
exp(-i θ n̂ · σ/2) = cos(θ/2) I - i sin(θ/2) (n̂ · σ)
Quaternions H = R + R i + R j + R k with i² = j² = k² = i j k = -1; unit quaternions Sp(1) = SU(2)
Foundation of qubits in quantum information

SU(3)

dim 8. Gell-Mann basis λ_a (Gell-Mann 1962). Color SU(3) of QCD (Yang-Mills 1954, Gross-Politzer-Wilczek 1973 asymptotic freedom — Nobel 2004).

SU(5)

dim 24. Georgi-Glashow GUT (Georgi-Glashow 1974) unifying U(1)_Y × SU(2)_L × SU(3)_c. Proton decay predictions ruled out the simplest version by Super-Kamiokande.

2.11 Sp(2n, R), Sp(n) (compact)

Symplectic.

Sp(2n, R) preserves Ω = [[0, I_n], [-I_n, 0]]: g^T Ω g = Ω
dim n(2n + 1); non-compact
sp(2n, R) = {X : X^T Ω + Ω X = 0}
Phase space symmetries in Hamiltonian mechanics; canonical transformations
Maslov index, Weil representation

Sp(n) compact: Sp(n) = U(2n) ∩ Sp(2n, C); dim n(2n + 1); compact, simply connected.

2.12 Spin(n), Pin(n)

Universal double cover of SO(n) (for n ≥ 3): 1 → Z/2 → Spin(n) → SO(n) → 1. dim Spin(n) = n(n-1)/2.

Spin(3) = SU(2)
Spin(4) = SU(2) × SU(2)
Spin(5) = Sp(2)
Spin(6) = SU(4)
Spin(7), Spin(8) — Spin(8) has triality (S_3 outer automorphism)

Pin(n) is the double cover of O(n); includes both parity-orientations.

Reference: Lawson-Michelsohn (1989). Spin Geometry, Princeton.

2.13 Heisenberg group

Upper triangular 3 × 3 with 1’s on diagonal: H_3 = {[[1, a, c], [0, 1, b], [0, 0, 1]] : a, b, c ∈ R}

dim 3, nilpotent of class 2
h_3 basis P, Q, Z with [P, Q] = Z, all other brackets = 0
Canonical commutation relations of QM
Stone-von Neumann theorem (Stone 1930, von Neumann 1931): unique unitary irrep with central character

Generalization H_{2n+1} of dim 2n + 1; symplectic structure on R^{2n} ⊂ h_{2n+1}.

2.14 Affine group Aff(n) and similarity group Sim(n)

Aff(n) = GL(n) ⋉ R^n: maps x ↦ A x + b.

dim n² + n

Sim(n) = (R^+ × SO(n)) ⋉ R^n: similarities (rotation + scale + translation).

dim n + 1 + n(n-1)/2

Used in computer vision (homographies), graphics, robotics.

2.15 Conformal group Conf(p, q)

Maps preserving angles. In R^n for n ≥ 3: O(n + 1, 1) (acting on Möbius compactification S^n), dim (n+2)(n+1)/2.

In R² conformal group is infinite-dimensional (analytic / antiholomorphic); central to 2D CFT.

2.16 Diffeomorphism group Diff(M)

Smooth diffeomorphisms of manifold M; infinite-dimensional Fréchet Lie group. Lie algebra = vector fields with Lie bracket.

Subgroups: Symp(M, ω) (symplectic), Cont(M, α) (contact), Diff_vol(M) (volume-preserving, gauges of incompressible fluids, Arnol’d 1966).

2.17 Loop groups LG = Map(S^1, G)

Smooth maps S^1 → G with pointwise multiplication. Infinite-dimensional but well-studied (Pressley-Segal 1986). Affine Kac-Moody algebras as central extensions. Appear in string theory, 2D Yang-Mills, integrable systems.

3. Exceptional simple Lie groups

Classification by Killing-Cartan (Killing 1888-1890, Cartan 1894): the simple complex Lie algebras are A_n (sl(n+1)), B_n (so(2n+1)), C_n (sp(2n)), D_n (so(2n)), plus five exceptional: G_2, F_4, E_6, E_7, E_8.

3.1 G_2

dim 14. Automorphism group of octonions O (Cayley 1845, Cartan 1914 realization). Rank 2. Holonomy of certain 7-manifolds (Bryant 1987); G_2 manifolds in M-theory compactifications.

3.2 F_4

dim 52, rank 4. Automorphism group of exceptional Jordan algebra J_3(O) of 3×3 Hermitian octonion matrices (Albert 1934, Freudenthal 1951).

3.3 E_6

dim 78, rank 6. Symmetry group of 27 lines on cubic surface (Cayley 1849); GUT candidate (Gürsey-Ramond-Sikivie 1976).

3.4 E_7

dim 133, rank 7. Largest containing E_6 × U(1).

3.5 E_8

dim 248, rank 8. Largest simple exceptional. E_8 × E_8 heterotic string theory (Gross-Harvey-Martinec-Rohm 1985). E_8 lattice has densest sphere packing in 8 dim (Viazovska 2017, Fields Medal 2022). The 7-dim flag manifold E_8/(E_7 × SU(2)/Z_2) appears in supergravity.

3.6 Magic square (Freudenthal-Tits 1957, 1966)

Construction relating R, C, H, O (real, complex, quaternion, octonion) algebras to exceptional and classical groups:

	R	C	H	O
R	SO(3)	SU(3)	Sp(3)	F_4
C	SU(3)	SU(3)²	SU(6)	E_6
H	Sp(3)	SU(6)	SO(12)	E_7
O	F_4	E_6	E_7	E_8

4. Lie algebras: structure and classification

4.1 Cartan subalgebra

A maximal abelian subalgebra h ⊂ g consisting of semisimple (ad-diagonalizable) elements. Dimension = rank of g.

4.2 Root system

Under ad action of h on g, g decomposes: g = h ⊕ ⊕_{α ∈ Φ} g_α, g_α = {X ∈ g : [H, X] = α(H) X ∀H ∈ h}

The nonzero α ∈ h* form the root system Φ. Each g_α has dim 1.

4.3 Cartan-Killing classification (Cartan 1894)

Irreducible reduced root systems are:

Type	Rank	Group	Dim	Coxeter #
A_n	n	SU(n+1)	n(n+2)	n+1
B_n	n	SO(2n+1)	n(2n+1)	2n
C_n	n	Sp(2n)	n(2n+1)	2n
D_n	n	SO(2n)	n(2n-1)	2n-2
G_2	2	G_2	14	6
F_4	4	F_4	52	12
E_6	6	E_6	78	12
E_7	7	E_7	133	18
E_8	8	E_8	248	30

4.4 Dynkin diagrams

Graphs encoding root system: nodes = simple roots; edges = inner products (1, 2, or 3 edges or single edge with arrow). Classification: A_n, B_n, C_n, D_n connected linear chains plus exceptional G_2, F_4, E_6, E_7, E_8. (Dynkin 1947, Coxeter 1934.)

4.5 Weyl group

Generated by reflections s_α through hyperplanes ⊥ α for α ∈ Φ; permutes Φ.

|W(A_n)| = (n+1)! (symmetric group) |W(B_n)| = |W(C_n)| = 2^n n! (hyperoctahedral) |W(D_n)| = 2^{n-1} n! |W(E_6)| = 51840, |W(E_7)| = 2903040, |W(E_8)| = 696729600

4.6 Iwasawa decomposition (Iwasawa 1949)

For semisimple G: G = K A N (compact × abelian × nilpotent), unique factorization. Example SL(n, R) = SO(n) × diag^+ × upper unipotent (QR decomposition).

4.7 Cartan decomposition

g = k ⊕ p with [k, k] ⊂ k, [k, p] ⊂ p, [p, p] ⊂ k. K compact subgroup; symmetric space G/K.

4.8 Bruhat decomposition

G = ⊔_{w ∈ W} B w B, B Borel (maximal solvable); gives cell decomposition of flag variety G/B.

5. Representation theory

5.1 Definitions

A representation ρ: G → GL(V) (homomorphism into invertibles of V). Irreducible (irrep) = no proper invariant subspace. Completely reducible: any rep = ⊕ irreps. True for compact G (Maschke / Weyl unitary trick).

5.2 Schur’s lemma (Schur 1905)

For irrep V: End_G(V) = C (over C). Intertwiners between non-isomorphic irreps = 0.

5.3 Adjoint and fundamental representations

Adjoint: g acting on itself by [·, ·]; ad: g → End(g)
Fundamental: defining rep of matrix group on K
Dual: ρ*(g) = ρ(g^{-1})
Tensor product, symmetric, antisymmetric, exterior powers

5.4 Highest weight (Cartan-Weyl theory)

Choose Borel B ⊂ G; positive roots; dominant integral weights P^+ ⊂ h*. Each λ ∈ P^+ determines a unique finite-dim irrep V_λ.

For SU(2): weights are non-negative half-integers j ∈ {0, 1/2, 1, 3/2, …}; dim V_j = 2j + 1; called “spin j” reps.

Clebsch-Gordan decomposition for SU(2): V_j ⊗ V_k = ⊕_{l = |j-k|}^{j+k} V_l, in integer steps. Coefficients given by 3j symbols.

5.5 Weyl character formula (Weyl 1925-1926)

For irrep V_λ: χ_λ(e^H) = (Σ_{w ∈ W} (-1)^{l(w)} e^{w(λ + ρ)(H)}) / (Σ_{w ∈ W} (-1)^{l(w)} e^{w(ρ)(H)})

with ρ = half-sum of positive roots; l(w) = length in W.

Dimension: dim V_λ = Π_{α > 0} ⟨λ + ρ, α⟩ / ⟨ρ, α⟩ (Weyl dimension formula).

5.6 Casimir invariants

C_2 = Σ_a g^{ab} X_a X_b ∈ U(g) (center of universal enveloping algebra); acts as scalar c_λ on V_λ.

SU(2): C_2 = J² = J_x² + J_y² + J_z²; c_j = j(j+1). SU(3): C_2 + cubic C_3.

5.7 Peter-Weyl theorem (Peter-Weyl 1927)

For compact G: L²(G) = ⊕̂_{[V] ∈ Ĝ} V ⊗ V* (decomposition into matrix coefficients of irreps); orthonormal basis = {sqrt(dim V) ρ^V_{ij}}.

5.8 Borel-Weil-Bott theorem (Borel-Weil 1953, Bott 1957)

Irreps of compact G realized as holomorphic sections of line bundles over flag variety G/B; cohomology of non-dominant bundles gives shifted irreps. Foundation of geometric representation theory.

5.9 Wigner 3j, 6j, 9j coefficients (Wigner 1931, 1959; Racah 1942)

Coupling of angular momenta for SU(2):

3j symbol relates Clebsch-Gordan to symmetric coupling
6j Wigner (Racah W) recouples three angular momenta
9j recouples four angular momenta

Connects to spectroscopy, atomic and nuclear physics.

5.10 Spinor representations

Real Clifford algebra Cl(p, q): generated by e_i with e_i² = ±1, e_i e_j + e_j e_i = 0. Pin(p, q), Spin(p, q) acting on spinor module via Clifford. For SO(n) the spin rep has dim 2^{⌊n/2⌋}.

Dirac spinors (Dirac 1928) for Lorentz SO(1, 3) live in 4-dim complex Clifford rep; split into left/right Weyl spinors under SL(2, C).

5.11 Infinite-dimensional unitary representations

Bargmann (1947) for SL(2, R): principal series, complementary series, discrete series, limits. Harish-Chandra (1950s-1960s): Plancherel decomposition for semisimple Lie groups; admissible representations; (g, K)-modules.

5.12 Howe duality (reductive dual pairs, Howe 1989)

Pairs (G, G’) ⊂ Sp(2N) act on metaplectic representation; representations of G and G’ related. Examples: (O(p, q), Sp(2n)); (GL(m), GL(n)).

6. Differential geometry of Lie groups

6.1 Left-invariant vector fields

For X ∈ g, define X^L(g) = (L_g)_* X. Bracket of left-invariant fields equals Lie bracket in g.

6.2 Maurer-Cartan form

θ: T G → g, θ_g(v) = (L_{g^{-1}})_* v; satisfies dθ + [1/2](θ, θ) = 0 (Maurer-Cartan equation).

6.3 Biinvariant metrics

For compact G: −Killing form is a biinvariant Riemannian metric; geodesics through e are one-parameter subgroups exp(t X); curvature ⟨R(X, Y) Z, W⟩ = (1/4)⟨[X, Y], [Z, W]⟩.

6.4 Symmetric spaces (Cartan 1926)

G/K with involution σ on G fixing K such that [k, p] ⊂ p, [p, p] ⊂ k. Cartan’s classification: spheres S^n, hyperbolic H^n, complex projective CP^n, Grassmannians, Stiefel manifolds, flag varieties.

6.5 Coadjoint orbits and Kirillov method (Kirillov 1962)

Coadjoint orbits in g* carry canonical symplectic structure (Kirillov-Kostant-Souriau symplectic form). Geometric quantization → unitary irreps (orbit method).

7. Applications

7.1 Robotics: SE(3) and the manipulator

SE(3) for pose; SO(3) for orientation; product manifold (SE(3))^N for multi-body.

Twist ξ = (ω, v) ∈ se(3); body twist V^b vs spatial twist V
Wrench F = (m, f) ∈ se(3)*; force-moment screw
Product of exponentials formula (Brockett 1984): forward kinematics T(θ) = exp(ξ̂_1 θ_1) … exp(ξ̂_n θ_n) M
Jacobian columns are twists J_i = Ad_{g_{i-1}} ξ_i
Modern Robotics framework (Lynch-Park 2017)

Reference: Lynch, K.M. and Park, F.C. (2017). Modern Robotics: Mechanics, Planning, and Control, Cambridge University Press.

Screw theory (Ball 1900): rigid-body motion as screw (twist of pitch h about axis). Reciprocal screws (statics/constraint).

Singular value decomposition of Jacobian → kinematic conditioning. Manipulability ellipsoid (Yoshikawa 1985).

Lie-group optimization: tangent space + retraction (exp) + parallel transport. Tools: Pinocchio (Carpentier 2019) for rigid-body dynamics; Theseus (Pineda 2022) for differentiable nonlinear least squares on Lie groups; Sophus C++ template library (Strasdat 2013).

7.2 Hamiltonian mechanics: Sp(2n)

Phase space (T*Q, ω); canonical 2-form ω = dp ∧ dq. Hamilton flow X_H = ω^{-1} dH preserves ω; symplectomorphism. Liouville theorem: phase-space volume preserved.

Symplectic integrators (Verlet 1967, Yoshida 1990, Forest-Ruth 1990): structure-preserving ODE solvers for Hamiltonian systems; leap-frog symplectic to order 2; explicit higher-order via composition.

Marsden-Weinstein symplectic reduction (Marsden-Weinstein 1974): G-invariant Hamiltonian system on TM reduces to symplectic quotient T(M/G).

7.3 Quantum mechanics

SU(2) rotation: |ψ⟩ ↦ e^{-i θ n̂ · J} |ψ⟩. Pauli matrices σ_i = 2 J_i for spin-1/2. Spin-J multiplets dim 2J + 1.

SU(3) flavor (Gell-Mann 1961, Ne’eman 1961): u, d, s quarks → octet/decuplet baryons; Ω^- prediction (Gell-Mann 1962, Nobel 1969).

QED gauge group U(1); QCD gauge group SU(3) (8 gluons in adjoint, dim 8).

Electroweak SU(2)_L × U(1)_Y broken to U(1)_em by Higgs (Higgs-Englert-Brout 1964, Nobel 2013).

7.4 General relativity

Lorentz group SO^+(1, 3) local symmetry of tangent bundle; Poincaré global symmetry of Minkowski. Diffeomorphism group Diff(M) is the gauge group of GR.

ADM 3+1 decomposition (Arnowitt-Deser-Misner 1962): foliate spacetime → Hamiltonian formulation with constraints.

Tetrad / Cartan formalism: SO(1, 3)-valued connection; spinor representations require SL(2, C) double cover.

7.5 Gauge theories — Standard Model

Gauge group G_SM = U(1)_Y × SU(2)_L × SU(3)_c, dim 1 + 3 + 8 = 12. Quark and lepton families transform in specific reps.

GUTs: SU(5) Georgi-Glashow; SO(10) (Fritzsch-Minkowski 1975; Georgi 1975); E_6 (Gürsey-Ramond-Sikivie 1976). Anomaly cancellation, proton decay, fermion-mass hierarchy constraints.

String theory: heterotic E_8 × E_8 and Spin(32)/Z_2; compactifications on Calabi-Yau threefolds; F-theory on elliptically fibered Calabi-Yau fourfolds.

7.6 Machine learning — equivariant neural networks

Group equivariance: f(g · x) = g · f(x). Cohen-Welling (2016) Group equivariant CNNs for translation + rotation + reflection (Z² ⋊ D_4 wallpaper); generalized to homogeneous spaces (Cohen-Geiger-Köhler-Welling 2018).

Spherical CNNs (Cohen-Geiger-Köhler-Welling 2018): convolution on S² and SO(3) via spherical Fourier transform (Wigner-D coefficients).

Gauge-equivariant CNNs (Cohen-Weiler-Kicanaoglu-Welling 2019): on general manifolds with structure group reducing to subgroup of frame bundle.

SE(3)-Transformer (Fuchs-Worrall-Fischer-Welling 2020): self-attention equivariant to 3D rotations + translations; molecular property prediction.

E(n)-equivariant GNN — EGNN (Satorras-Hoogeboom-Welling 2021): minimal architecture equivariant to E(n) for protein folding, molecular dynamics.

Tensor field networks (Thomas et al. 2018): outputs are SO(3) tensors of definite type.

Equivariant message passing (Schütt et al. PaiNN 2021; NequIP Batzner et al. 2022; Allegro Musaelian 2023; MACE Batatia 2022).

Lie group convolution (Bekkers 2020); LieConv (Finzi-Welling-Wilson 2020); Clifford networks (Brandstetter 2022).

7.7 Crystallography

Point groups (32 in 3D): finite subgroups of O(3). Space groups (230 in 3D, Fedorov-Schoenflies 1891-1892): combinations of point groups with Bravais lattice translations. Magnetic groups (1651 Shubnikov groups). International Tables for Crystallography (Hahn 2005).

Bravais lattices (14 in 3D): triclinic, monoclinic (2), orthorhombic (4), tetragonal (2), trigonal/rhombohedral, hexagonal, cubic (3).

7.8 Signal processing — wavelets and scattering

Wavelets (Daubechies 1988): scaling + translation group; Heisenberg-style time-frequency uncertainty.

Scattering networks (Mallat 2012): invariants to translations, deformations; Lipschitz-stable. Generalized to SE(3), SO(3) for medical imaging.

Spherical scattering (Sifre-Mallat 2013).

7.9 Optimization on manifolds

Manopt / Pymanopt (Boumal-Mishra-Absil-Sepulchre 2014): Riemannian gradient descent, conjugate gradient, trust regions on Stiefel, Grassmann, SE(3), Lie groups, fixed-rank matrices.

Geomstats (Miolane et al. 2020): broader differential-geometric tools including Lie groups.

LieTorch / pypose (Wang et al. 2023): PyTorch + Lie groups for differentiable robotics.

Reference: Absil, P.-A., Mahony, R., and Sepulchre, R. (2008). Optimization Algorithms on Matrix Manifolds, Princeton.

7.10 Geometric kernels

Constancy on group orbits; Matérn kernels on compact Lie groups via spectral expansion (Borovitskiy et al. 2020 GeometricKernels). Gaussian processes on manifolds (Lindgren-Rue-Lindström 2011 SPDE).

8. Computational libraries

Pinocchio (Carpentier et al. 2019): rigid-body dynamics on SE(3); C++ and Python; CRBA, RNEA, ABA in O(n) per joint.
Theseus (Pineda et al. 2022, Meta AI): differentiable nonlinear optimization on Lie groups for SLAM, ICP, robotics.
Sophus (Strasdat 2013): C++ templates for SO(2), SO(3), SE(2), SE(3), Sim(3), RxSO(3).
gtsam (Dellaert 2012): factor graphs for SLAM with native Lie-group support.
Manopt / Pymanopt / Manopt.jl: Riemannian optimization.
GeometricKernels (Borovitskiy 2020): kernels on Lie groups + manifolds.
LieGroups.jl / Manifolds.jl (Schichtholz, Axen) Julia.
e3nn (Geiger-Smidt 2022): SO(3)-equivariant networks in PyTorch.
TensorFieldNetworks (Smidt 2018).
Lie.jl, GeometricMachineLearning.jl: Julia.

9. Notation conventions

Lower-case fraktur for Lie algebra of upper-case group: G ↔ g, SO(3) ↔ so(3)
Skew-symmetric matrix bracket: for ω ∈ R^3, [ω]_× ∈ so(3) the cross-product matrix
Hat / vee maps between R^n and Lie algebra: ω̂ ∈ so(3), and similarly for se(3)
exp = expm matrix exponential (computed via Padé approximation + scaling-squaring, Higham 2005)
Adjoint Ad_g X = g X g^{-1} on g; coadjoint Ad_g^on g
BCH formula computed via Magnus series for time-dependent ODEs on Lie groups (Magnus 1954)

10. Reference list of identities and theorems

Cartan-Killing classification (1894)
Lie’s three fundamental theorems
Closed subgroup theorem (Cartan 1930)
Weyl unitary trick + complete reducibility for compact G
Peter-Weyl decomposition
Borel-Weil-Bott theorem
Weyl character + dimension formulas
Schur orthogonality of matrix coefficients
Cartan-Iwasawa-Mostow decomposition for semisimple
Iwasawa decomposition KAN
Bruhat decomposition + cells in G/B
Helgason-Cartan classification of symmetric spaces
Kirillov orbit method
BCH formula
Magnus expansion for time-ordered exponentials
Rodrigues rotation formula
Chasles’ theorem (every motion is a screw motion)
Mostow rigidity (hyperbolic manifolds of dim ≥ 3)
Wigner classification of unitary irreps of Poincaré

Compendium

Explorer

Lie Group Catalog

Lie Group Catalog

1. Foundations

1.1 Lie group definition

1.2 Lie algebra

1.3 Exponential map

1.4 Lie subgroups and the closed subgroup theorem (Cartan 1930)

1.5 Lie’s third theorem

1.6 Adjoint representation

1.7 Killing form

1.8 Compact / semisimple / solvable / nilpotent

1.9 Levi decomposition (Levi 1905, Malcev 1942)

2. Classical matrix groups — connected and complex

2.1 GL(n, R), GL(n, C), GL(n, H)

2.2 SL(n, R), SL(n, C)

2.3 O(n) and SO(n)

2.4 SO(2)

2.5 SO(3)

2.6 SE(2)

2.7 SE(3)

2.8 O(p, q) and SO(p, q)

Lorentz O(1, 3) and SO(1, 3)

2.9 U(n)

2.10 SU(n)

SU(2)

SU(3)

SU(5)

2.11 Sp(2n, R), Sp(n) (compact)

2.12 Spin(n), Pin(n)

2.13 Heisenberg group

2.14 Affine group Aff(n) and similarity group Sim(n)

2.15 Conformal group Conf(p, q)

2.16 Diffeomorphism group Diff(M)

2.17 Loop groups LG = Map(S^1, G)

3. Exceptional simple Lie groups

3.1 G_2

3.2 F_4

3.3 E_6

3.4 E_7

3.5 E_8

3.6 Magic square (Freudenthal-Tits 1957, 1966)

4. Lie algebras: structure and classification

4.1 Cartan subalgebra

4.2 Root system

4.3 Cartan-Killing classification (Cartan 1894)

4.4 Dynkin diagrams

4.5 Weyl group

4.6 Iwasawa decomposition (Iwasawa 1949)

4.7 Cartan decomposition

4.8 Bruhat decomposition

5. Representation theory

5.1 Definitions

5.2 Schur’s lemma (Schur 1905)

5.3 Adjoint and fundamental representations

5.4 Highest weight (Cartan-Weyl theory)

5.5 Weyl character formula (Weyl 1925-1926)

5.6 Casimir invariants

5.7 Peter-Weyl theorem (Peter-Weyl 1927)

5.8 Borel-Weil-Bott theorem (Borel-Weil 1953, Bott 1957)

5.9 Wigner 3j, 6j, 9j coefficients (Wigner 1931, 1959; Racah 1942)

5.10 Spinor representations

5.11 Infinite-dimensional unitary representations

5.12 Howe duality (reductive dual pairs, Howe 1989)

6. Differential geometry of Lie groups

6.1 Left-invariant vector fields

6.2 Maurer-Cartan form

6.3 Biinvariant metrics

6.4 Symmetric spaces (Cartan 1926)

6.5 Coadjoint orbits and Kirillov method (Kirillov 1962)

7. Applications

7.1 Robotics: SE(3) and the manipulator

7.2 Hamiltonian mechanics: Sp(2n)

7.3 Quantum mechanics

7.4 General relativity

7.5 Gauge theories — Standard Model

7.6 Machine learning — equivariant neural networks

7.7 Crystallography

7.8 Signal processing — wavelets and scattering