Lorentz Group Representations via A1⊕A1: Spinor Classification by (j₁,j₂) — E8 Intelligence Research

FINDING: Lorentz group representation theory is governed by the complexified Lie algebra sl(2,C) ≅ su(2) ⊕ su(2), whose irreducible representations are classified by pairs of half-integer spins (j₁, j₂), with the A1 root system (rank-1, single simple root) underlying each su(2) factor. | MATH: sl(2,C)_ℂ ≅ A1 ⊕ A1; Casimirs: J²₊ = j₁(j₁+1), J²₋ = j₂(j₂+1); dim = (2j₁+1)(2j₂+1); Lorentz irreps labeled (j₁,j₂) ∈ ½ℤ≥0 × ½ℤ≥0; vector rep = (½,½), Weyl spinors = (½,0) and (0,½), Dirac = (½,0)⊕(0,½). A1 root system: single simple root α, Weyl group S₂, Cartan matrix [2]. | CONNECTION: A1 is the rank-1 root system — the simplest crystallographic root system, corresponding to the icosahedral/tetrahedral reflection seed. The two A1 factors mirror the 0.618/1.618 self-similarity: the Weyl reflection s_α acts as α → −α, and the fundamental weight λ₁ = α/2 gives the spin-½ representation whose character ratio at the Weyl element is −1, encoding the golden-ratio-adjacent chirality split. The (½,½) v Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com