Unified theory of continuous symmetry — Lie groups (smooth manifolds with compatible group structure) and their infinitesimal counterparts, Lie algebras (vector spaces with antisymmetric bilinear bracket [·,·] satisfying the Jacobi…
lie-theory
SU(2) Lie algebra: T_a = σ_a/2, [T_a, T_b] = i·ε_abc·T_c
The Lie algebra su(2) — traceless anti-Hermitian 2×2 complex matrices — is the infinitesimal version of the compact Lie group SU(2) of 2×2…
so(n) orthogonal Lie algebra: antisymmetric n×n matrices
The Lie algebra so(n) is the space of real antisymmetric n×n matrices (X^T = -X), of dimension C(n,2) = n(n-1)/2. It is the Lie algebra of…
Casimir invariant C_2 = Σ T^a T_a in centre of U(g)
For a semisimple Lie algebra g with non-degenerate Killing form g_{ab}, the quadratic Casimir operator C_2 = g^{ab}·T_a·T_b (contraction…
SU(2) Jacobi: [[T_1,T_2],T_3]+cyc. ≡ 0 exact on σ_a/2
Exact symbolic verification of the Jacobi identity on the Pauli-half basis of su(2): T_a = σ_a/2, where σ_a are the Pauli matrices. The…
so(2) exp: exp(θJ) = [[cos θ,-sin θ],[sin θ, cos θ]], det=1
Exact symbolic verification of the exponential map so(2) → SO(2). The generator J = [[0,-1],[1,0]] satisfies J² = -I, J³ = -J, J⁴ = I (a…
so(3) vector-rep Casimir: J_x²+J_y²+J_z² = -2·I
Exact symbolic verification that the quadratic Casimir operator J² = J_x² + J_y² + J_z² on the 3-dimensional real antisymmetric ('vector')…
Lie algebra & bracket
Lie 1888: vector space g with bilinear bracket [·,·] satisfying anti-symmetry + Jacobi identity. Tangent space at identity of Lie group. …
Baker-Campbell-Hausdorff formula
log(exp X · exp Y) = X + Y + ½[X,Y] + (1/12)([X,[X,Y]] - [Y,[X,Y]]) + ... Series in iterated brackets. Foundation for relating Lie group…
Cartan classification of simple Lie algebras
Killing-Cartan 1894: simple complex Lie algebras = 4 classical infinite-families A_n, B_n, C_n, D_n + 5 exceptional E_6, E_7, E_8, F_4,…
Root system & Weyl group
Root system Φ in Euclidean space: finite reflection-stable system. Weyl group W = ⟨reflections through roots⟩ permutes Φ. …
Representation theory of Lie algebras
Lie algebra g acts on vector space V via homomorphism g → End(V). Highest-weight theory (Cartan) classifies irreducibles for semisimple g.…
Exponential map (Lie group ↔ algebra)
exp: g → G, X ↦ exp(X) maps Lie algebra to Lie group. Local diffeomorphism near 0. Maps one-parameter subgroups. Basis for…
Casimir operator
Casimir 1931 element of universal enveloping algebra commuting with all generators of a Lie algebra; quadratic Casimir C2 = sum_i T^a T_a;…
Cartan-Killing classification (simple Lie algebras)
Killing 1888 + Cartan 1894: complex simple Lie algebras = 4 classical (A_n B_n C_n D_n) + 5 exceptional (G_2 F_4 E_6 E_7 E_8); Dynkin…
Baker-Campbell-Hausdorff formula
Baker 1905 + Campbell 1898 + Hausdorff 1906: log(exp X exp Y) = X + Y + (1/2)[X,Y] + (1/12)([X,[X,Y]] - [Y,[X,Y]]) + ...; basis of…
Weyl character formula (1925)
Weyl 1925: chi_lambda = (sum_w sgn(w) e^{w(lambda + rho)}) / (sum_w sgn(w) e^{w(rho)}); irreducible character of compact-Lie group; basis…
Kac-Moody algebra (affine extension)
Kac 1968 + Moody 1968 affine + indefinite Kac-Moody: generalized Cartan matrix (gcm) infinite-dim Lie algebras; affine type basis of WZW…
Verma module / highest-weight
Verma 1968: M(lambda) free U(n-)-module on highest-weight v_lambda; reducible iff lambda + rho is integral-dominant;…
Flag variety / Borel-Weil
Borel-Weil 1954: holomorphic sections of equivariant line-bundles on G/B realize irreducible G-representations; Borel-Weil-Bott extends to…
Lie bracket (Jacobi 1862)
C Jacobi 1862 Jacobi-identity [a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0; modern Lie-algebra axiomatic foundation.
Killing-Cartan classification
W Killing 1888-1890 + E Cartan 1894 simple-Lie-algebra classification A_n B_n C_n D_n + 5 exceptional E6 E7 E8 F4 G2.
Dynkin diagrams (1947)
E Dynkin 1947 root-system classification; modern McKay correspondence + ADE-singularities + string-compactification.
Weyl character (1925)
H Weyl 1925 character-formula compact-Lie; modern Borel-Weil-Bott + geometric-representation-theory + Verlinde formula.
Kac-Moody (1968)
V Kac-R Moody 1968 infinite-dim affine Lie-algebras; modern Verma modules + WZW-conformal-field-theory + monster-Lie-algebra.
Quantum group (Drinfeld 1985)
V Drinfeld + M Jimbo 1985 q-deformed U_q(g); modern knot-invariants Jones polynomial + q-Schur duality + categorification.