3 research outputs found
The twisted group algebra structure of the Cayley-Dickson algebra
The Cayley-Dickson algebra has long been a challenge due to the lack of an
explicit multiplication table. Despite being constructible through inductive
construction, its explicit structure has remained elusive until now. In this
article, we propose a solution to this long-standing problem by revealing the
Cayley-Dickson algebra as a twisted group algebra with an explicit twist
function . We show that this function satisfies the equation
and provide a formula for the
relationship between the Cayley-Dickson algebra and split Cayley-Dickson
algebra, thereby giving an explicit expression for the twist function of the
split Cayley-Dickson algebra. Our approach not only resolves the lack of
explicit structure for the Cayley-Dickson algebra and split Cayley-Dickson
algebra but also sheds light on the algebraic structure underlying this
fundamental mathematical object
The Octonions
The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction