GOD Theory II: The Skinner-Torrado Algebra

We develop the complete mathematical theory of the Skinner-Torrado algebras(K, dH ), a fractal generalisation of Lie algebras that arises as the symmetry algebraof an enrollment subspace K of the GOD manifold M (introduced in [1]).The generators are fractional differential operators DdHa = i DdHa of order dH ∈ R>0 acting on K. The Skinner-Torrado bracket is [Ta, Tb]s = f cab(dH ) Tc+C(dH ) Rcab Tc,where the Skinner-Torrado curvature coefficient C(dH ) = Γ(1 + dH )2/Γ(1 + 2dH ) isexact on any self-similar fractal with Hausdorff measure (Theorem 2.7).Seven theorems are proved: (T1) exactness of C(dH ) via the Moran-Falconertheorem; (T2) the generalised Jacobi identity; (T3) the classical limit s(Ki, n) → gi;(T4) S(M) is a separable Hilbert space with P C(n) = π/(3√3); (T5) convergenceof Cauchy-Torrado sequences; (T6) the Galois-Torrado group GalT = {id}; (T7) theKlein-Torrado classification by the invariant C(dH ).Four open problems are precisely stated.