Emergence of Non-Abelian Algebraic Structures from Extended Clifford Algebra Representation Spaces

Abstract:This work proves that non-Abelian Lie algebras su(2) and su(3) emerge as algebraic necessities of Clifford algebra in extended representation spaces. The central result is a classification theorem: when the Dirac spinor space is extended to V_4 ⊗ V_N with N=2,3, the commutator constraint with the Clifford algebra generators uniquely yields su(2) and su(3) as the complete non-trivial Lie algebras. The numbers of generators—3 and 8—and the structure constants are determined entirely by the dimension of the extended space via the formula N^2−1. The derivation uses only the anticommutation relations of the Clifford algebra and standard Lie algebra theory.