混合数、Gram矩阵和黄金比例的4x4矩阵表示
4x4 Matrix Representation of Hybrid Numbers, Gram Matrix and Golden Ratio
浏览论文内容
中文总结 AI 辅助
研究特定非结合乘法表下混合数的李代数结构与几何性质，通过换位关系等推导结构常数并构建4x4矩阵表示，定义相关矩阵和共轭，揭示其与黄金比例联系及具有半单结构等特性。
中文摘要 AI 辅助
本研究在特定非结合乘法表下，研究混合数的李代数结构和几何性质。通过换位关系推导系统的结构常数，并构建4x4矩阵表示。定义Gram矩阵、复共轭和混合共轭后，得到Gram矩阵特征值与黄金比例的联系。理论分析表明该混合李代数具有半单结构，其Killing型特征谱与黄金比例直接相关。此外，证明复共轭和双曲共轭运算是使Killing型不变的对称变换。
英文摘要
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.
