Matrix Representation of Octonions and Generalizations

Abstract

We define a special matrix multiplication among a special subset of $2N\x 2N$ matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative non-associative and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac's equation for a free particle. A few other results and remarks arise as byproducts.

18 printed pages

References in corpus (5)

• Quaternions and Special Relativity
• The Quaternionic Dirac Lagrangian
• Quaternions for Guts
• Quaternionic Electroweak Theory
• Translations between Quaternion and Complex Quantum Mechanics

Cited by in corpus (9)

• Black Holes, Qubits and Octonions
• Octonionic Version of Dirac Equations
• Global symmetries of Yang-Mills squared in various dimensions
• From the Geometry of Pure Spinors with their Division Algebras to Fermion's Physics
• The octonionic eigenvalue problem
• Generalized non-associative structures on the 7-sphere
• On the groundstate of octonionic matrix models in a ball
• Lie $3-$algebra and super-affinization of split-octonions
• Generalization of Lagrangian of electro-weak interaction to the octonionic algebra