Splitting of quaternions and octonions over purely inseparable extensions in characteristic 2

Matrix Representations of Octonions and Their Applications

As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras over the real number field ${\cal R}$, because ${\cal O}$ is a non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an extension of ${\cal H}$ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper…

Total linkage of quaternion algebras and Pfister forms in characteristic two

We study the subfields of quaternion algebras that are quadratic extensions of their center in characteristic 2. We provide examples of the following: two non-isomorphic quaternion algebras that share all their quadratic subfields, two quaternion algebras that share all their inseparable but not all their separable quadratic subfields and two algebras that share all their separable but not all their inseparable quadratic subfields. We also discuss quaternion algebras over global fields and fields of Laurent series over a perfect field of characteristic 2 and show that the quaternion algebras o…

Box-Kites III: Quizzical Quaternions, Mock Octonions, and Other Zero-Divisor-Suppressing "Sleeper Cell" Structures in the Sedenions and 2^n-ions

Building on two prior studies of zero-divisors (ZD's) generated by the Cayley-Dickson process, algebras we call "lariats" (Line Algebras of Real and Imaginary Axis Transforms), linkable to quantum measurement, are discovered in the Sedenions, complementing the 7 isomorphic "box-kites" (pathway systems spanning octahedral lattices) interconnecting all primitive ZD's. By switching "edge-signs," products among the diagonal line-pairs associated with each of a box-kite's 4 triangular, vertex-joined, "sails" generate not 6-cyclic ZD couplings when circuited, but 28 pairs of structures with Quaterni…

Integral Octonions, Octonion XY-Product, and the Leech Lattice

The integral octonions arise from the octonion XY-product. A parallel is shown to exist with the quaternion Z-product. Connections to the laminated lattices, $Λ_{4}$, $Λ_{8}$, $Λ_{16}$ and $Λ_{24}$ (Leech), are developed.

Finding 3x3 Hermitian Matrices over the Octonions with Imaginary Eigenvalues

We show that any 3-component octonionic vector which is purely imaginary, but not quaternionic, is an eigenvector of a 6-parameter family of Hermitian octonionic matrices, with imaginary eigenvalue equal to the associator of its elements.

一种记忆矩阵规则用于(split)八元数乘法及其对凯莱-迪科森塔的扩展

A Mnemonic Matrix Rule for (Split) Octonionic Multiplication and its Extension to the Cayley--Dickson Tower

本文提出了一种基于记忆矩阵的八元数乘法规则，并将其扩展到凯莱-迪科森塔中的所有代数，为非结合代数提供了高效的计算方法。

Symmetry of the triple octonionic product

The Hermitian decomposition of a linear operator is generalized to the case of two or more operations. An additive expansion of the product of three octonions into three parts is constructed, wherein each part either preserve or change the sign under the action of the Hermitian conjugation and operation of inversion of the multiplicative order of three hypercomplex numbers, as well as under the composition of specified operations. The product of three octonions, in particular quaternions, with conjugate central factor is presented as the sum of mutually orthogonal triple anticommutator, triple…

Non-associativity, supersymmetry and hidden variables

It is shown that the supersymmetric quantum mechanics has an octonionic generalization. The generalization is based on the inclusion of quaternions into octonions. The elements from the coset octonions/quaternions are unobservables bacause they cannot be considered as quantum operators as a consequence of their non-associative properties. The idea that the octonionic generalization of the supersymmetric quantum mechanics describes an observable particle formed with unobservable ``particles'' is presented.

Some remarks about Fibonacci elements in an arbitrary algebra

In this paper, we prove some relations between Fibonacci elements in an arbitrary algebra. Moreover, we define imaginary Fibonacci quaternions and imaginary Fibonacci octonions and we prove that always three arbitrary imaginary Fibonacci quaternions are linear independents and the mixed product of three arbitrary imaginary Fibonacci octonions is zero.

A note on polynomial equations over algebras

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three.

八元数代数上的广义交换子

Generalized Commutator on Octonion Algebra

本文研究八元数代数上广义交换子的像，给出其满射的充要条件，并证明迹零八元数为交换子，同时推广了四元数代数上的相关结果。