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')…