4x4 Matrix Representation of Hybrid Numbers, Gram Matrix and Golden Ratio

Abstract

This study investigates the Lie algebra structure and geometric properties of hybrid numbers, under a specific non-associative multiplication table. Within the scope of the study, the structure constants of the system were derived through commutator relations, and 4x4 matrix representation was constructed. By defining the Gram matrix, complex and hybrid conjugates, the connection of the eigenvalues of Gram matrices to the golden ratio has been obtained. Theoretical analysis reveal that this hybrid Lie algebra possesses a semi-simple structure and that the characteristic spectrum of its Killing form is directly related to the Golden Ratio (ϕ). Furthermore, it is proven that the Complex and Hyperbolic conjugation operations are symmetry transformations that leave the Killing form invariant.