Fibonacci Lie algebras: decomposition of local braiding subalgebras in the Fibonacci anyon chain

We prove that the Lie algebra generated by k consecutive braiding operators in the Fibonacci anyon chain has dimension F(2k+1) − 1 and decomposes asLie_k ≅ su(F(k+1)) ⊕ su(F(k)) ⊕ u(1) where F(n) denotes the n-th Fibonacci number. The proof combines the boundary-charge decomposition of the fusion Hilbert space (which produces blocks of Fibonacci dimensions), the semisimplicity of the Temperley–Lieb algebra at δ = φ (Wenzl 1988), and the Larsen–Wang density theorem for braid group representations (2005). The decomposition is independently confirmed by Killing form analysis, structure constant verification, Cartan–root system identification, and commutant computation. At k = 3 this yields su(3) ⊕ su(2) ⊕ u(1), the gauge algebra of the Standard Model. At k = 4 this yields su(5) ⊕ su(3) ⊕ u(1), reproducing the Georgi–Glashow grand unification structure. A complete breaking theorem (via Vajda's Fibonacci identity) shows that the number of broken generators at the descent k+1 → k equals the dimension of Lie_k if and only if k ∈ {2, 3}. The repository contains the paper (LaTeX + PDF) and five Python verification scripts that independently confirm all numerical claims. Dependencies: Python ≥ 3.8, NumPy, SciPy.