Λ-Ring Algebra:Structure, Geometry, and Dynamics Λ-环元代数:结构、几何与动力学

Abstract

Abstract This paper introduces and systematically studies the Λ-ring algebra — a non-associative algebra defined on three-dimensional real space, whose core structural decomposition is u·v = u×v − u⊙v (cross product minus Hadamard product). The commutator Lie algebra of this algebra is so(3), it has no two-sided identity or annihilator, and the squaring operation is strictly diagonalized: u·u = −(a², b², c²). There exists a continuous deformation family connecting this algebra to its opposite algebra, with the midpoint t = 0.5 degenerating to the commutative associative algebra ℝ⊕ℝ⊕ℝ. The standard SU(2) bi-invariant metric induced by the commutator makes this algebra an Einstein manifold (K = 1, R = 6). Spectral statistics of random Hamiltonians exhibit a V-shaped structure in deformation space, with ⟨r⟩ = 0.410 at the integrable point t = 0.5. Lyapunov analysis reveals intermittent chaos: 42% of initial conditions are chaotic, 58% are stable. This paper further generalizes the framework from the three-dimensional special case to a general theory in arbitrary n dimensions, defining the multiplication bundle and non-associative connection γᵢⱼᵏ(p), and proving that square diagonalization is dimension-independent under the antisymmetry condition γᵢⱼᵏ = −γⱼᵢᵏ. Finally, we propose the deformation emerg geometry program: treating the deformation parameter as a vector-valued field t: M → ℝᴺ, whose gradient induces a pullback metric g_μν = δ_ab ∂_μ tᵃ ∂_ν tᵇ on the base manifold, thereby allowing spacetime geometry to emerge from algebraic deformation while circumventing the flatness theorem constraints of traditional non-associative algebra bundles. Keywords: non-associative algebra, so(3), deformation theory, Einstein manifold, quantum chaos, multiplication bundle, non-associative connection, emergent geometry