An Octonion Algebra Originating in Combinatorics
نویسنده
چکیده
C.H. Yang discovered a polynomial version of the classical Lagrange identity expressing the product of two sums of four squares as another sum of four squares. He used it to give short proofs of some important theorems on composition of δ-codes (now known as T -sequences). We investigate the possible new versions of his polynomial Lagrange identity. Our main result shows that all such identities are equivalent to each other.
منابع مشابه
Some Properties of Octonion and Quaternion Algebras
In 1988, J.R. Faulkner has given a procedure to construct an octonion algebra on a finite dimensional unitary alternative algebra of degree three over a field K. Here we use a similar procedure to get a quaternion algebra. Then we obtain some conditions for these octonion and quaternion algebras to be split or division algebras. Then we consider the implications of the found conditions to the u...
متن کاملGauging the Octonion Algebra
By considering representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realized. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of ...
متن کاملOctonion Song
The Octonions and their multiplication table, the Fano Plane, bear an isomorphic relationship to the Pythagorean music system, which this paper shows is not a coincidence. In fact, the Octonions vibrate at one stage during the formation of new particles, in what might be termed the Octonion Song. This explains why Octonion structure matches musical structure. This paper explains how this happen...
متن کاملAn Octonionic Gauge Theory
The nonassociativity of the octonion algebra necessitates a bimodule representation, in which each element is represented by a left and a right multiplier. This representation can then be used to generate gauge transformations for the purpose of constructing a field theory symmetric under a gauged octonion algebra, the nonassociativity of which appears as a failure of the representation to clos...
متن کاملNew Values for the Level and Sublevel of Composition Algebras
Constructions of quaternion and octonion algebras, suggested to have new level and sublevel values, are proposed and justified. In particular, octonion algebras of level and sublevel 6 and 7 are constructed. In addition, Hoffmann’s proof of the existence of infinitely many new values for the level of a quaternion algebra is generalised and adapted.
متن کامل