The Alpha Group: Holonomic Structure and Dynamic Coupling of Dual Hopf Topologies in a Nontrivial Topological Space

We define the Alpha Group as a four-dimensional real associative algebra A = spanR{1, i, μ, iμ}, with relations i² = −1, μ² = μ (idempotent invariant), and iμ = −μi (noncommutative). Under the regular representation, elements of A act as 4×4 real matrices. Division is defined projectively via right multiplication by invertible elements of A.

The angular deformation operator M(θ) is the original 4×4 matrix introduced in the Alpha Group framework, whose entries depend analytically on θ and encode the coupling between the imaginary and μ-components of the algebra. This operator governs the deformation of basis directions and induces a θ-dependent metric structure on the associated orbit space.

To study the induced topology, we construct ε-graphs over point clouds generated by iterated application of M(θ). Vertices are connected whenever the induced metric satisfies d(x,y) ≤ ε. The resulting filtration defines a Vietoris–Rips simplicial complex.

Persistent homology groups Hk for k = 0, 1, 2, 3 are computed along the filtration. For 0.4 ≤ ε ≤ 0.8, the second homology group H² exhibits sustained growth, indicating stable 2-cycles generated by the θ-dependent deformation. The third homology group H³ remains constant across the filtration, acting as a structural invariant.

A structural transition occurs near θ ≈ π/2, where the antisymmetric coupling encoded in M(θ) maximizes the generation of higher-order cycles. These results demonstrate that the algebraic structure of the Alpha Group induces a dynamically coupled topology with persistent higher-dimensional invariants, structurally distinct from classical Riemannian models.