The Alpha Group 4D Geometry: Symmetric Structures and Topological Transitions

Starting from the representation of De Moivre’s matrix formula combined with the Kronecker product, we derive a 4 × 4 matrix Mθ that encodes algebraic properties with topological implications in four dimensions. This matrix describes trans- formations associated with symmetry and antisymmetry, encompassing distortion, dilation/contraction, and combined shear in a four-dimensional framework. We de- compose Mθ into its symmetric (S) and antisymmetric (Aanti ) components, a result that characterizes key features of the Alpha Group, including the formation of a tan- gent plane at infinity and its corresponding topological and geometric consequences. To investigate the system’s dynamics, we perform Monte Carlo simulations for varying values of the parameter θ, revealing attractor behavior and geometric transitions between Euclidean and hyperbolic regimes in projected three-dimensional trajectories.