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

We study the Lie algebra generated by k consecutive braiding operators in the Fibonacci anyon chain and prove that for k = 1, 2, 3, 4 it has dimension F (2k+1) − 1 and decomposes as Lieₖ ≅ su (F (k+1) ) ⊕ su (F (k) ) ⊕ u (1) where F (n) denotes the n-th Fibonacci number. We conjecture this holds for all k ≥ 1; the dimension formula is verified through k = 5. 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 (2005) for the general structure (Steps 1–3, valid for all k), with case-specific Killing form and Cartan–root system verification (Step 4, completed for k ≤ 4). 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), mirroring the Georgi–Glashow grand unification structure. The 5-dimensional boundary-charge blocks decompose as 5 = 3 + 2 under su (3) × su (2), but the u (1) charge ratio is −2/5, not the Standard Model value −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.