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 thei...

Supersymmetry is deeply related to division algebras. For example, nonabelian Yang–Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green–Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this...

Supersymmetry is deeply related to division algebras. For example, nonabelian Yang–Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green–Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this...

LetA be a finite-dimensional commutative algebra over the complex numbers, with identity element e. Thus A is a finite-dimensional complex vector space equipped with an additional binary operation of multiplication which satisfies the usual rules of associativity, commutativity, and distributivity, and e is a nonzero element of A such that e x = x for all x ∈ A. As a basic class of examples, on...

Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers , quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties. The notation and terminology used here have been introduced in the following We use the following convention: u, v, x, y, z, X, Y are sets and r, s ar...

in this paper we consider selberg-type square matrices integrals with focus on kummer-beta types i & ii integrals. for generality of the results for real normed division algebras, the generalized matrix variate kummer-beta types i & ii are defined under the abstract algebra. then selberg-type integrals are calculated under orthogonal transformations.

It is shown that a) it is possible to define the topology of any topological algebra by a collection of F -seminorms, b) every complete locally uniformly absorbent (complete locally A-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of metrizable locally uniformly absorbent algebras (respectively, A-(k-normed) algebras, where k ∈ (0, 1] varies, c) every complete...

We will present a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topo...

In this paper we consider Selberg-type square matrices integrals with focus on Kummer-beta types I & II integrals. For generality of the results for real normed division algebras, the generalized matrix variate Kummer-beta types I & II are defined under the abstract algebra. Then Selberg-type integrals are calculated under orthogonal transformations.

Disclaimer: This dissertation does not contain plagiarised material; except where otherwise stated all theorems are the author's. Acknowledgement: Many thanks to Joel Feinstein for guidance with the literature, useful suggestions and comments on this work.