Related papers: An equivalent definition for multiplicities of a q…
We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another.…
Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and…
An algebra A not encountered in either the usual algebraic varieties or supervarieties is introduced. A is a graded and deformed version of the quaternions, with structure similar to that of a Jordan-Lie superalgebra as defined by Okubo and…
In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities…
Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.
{\small In this paper, we find a class of division quaternion algebras over the field }$\mathbb{Q}\left( i\right) ${\small \ and a class of division symbol algebras over a cyclotomic field.}
Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran. The aim of this paper is to extend their Quaternionic Nullstellensatz to matrix…
We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological…
We ask whether the only multiplicities in the spectrum of the clamped round plate are trivial, i.e., whether all existing multiplicities are due to the isometries of the sphere, or, equivalently, whether any eigenfunction is separated. We…
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 analysis is considered in full details where a new analyticity condition in complete analogy to complex analysis is found. The extension to octonions is also worked out.
We introduce some basic notions and results for quaternionic linear operators analogous to those for complex linear operators. Our main result is to prove the additive and multiplicative Jordan-Chevalley decompositions for quaternionic…
The use of complexified quaternions and $i$-complex geometry in formulating the Dirac equation allows us to give interesting geometric interpretations hidden in the conventional matrix-based approach.
Second order circularity, also called properness, for complex random variables is a well known and studied concept. In the case of quaternion random variables, some extensions have been proposed, leading to applications in quaternion signal…
The functions studied in the paper are quaternion-valued functions of a quaternionic variable. It is show that the left slice regular functions and right slice regular functions are related by a particular involution. The relation between…
We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…
A variety of universal similarity factorization equalities over real Clifford algebras ${\cal R}_{p,q}$ are established. On the basis of these equalities, real, complex and quaternion matrix representations of elements in ${\cal R}_{p,q}$…
Quaternionic formulation of supersymmetric quantum mechanics has been developed consistently in terms of Hamiltonians, superpartner Hamiltonians, and supercharges for free particle and interacting field in one and three dimensions.…
Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided…
The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the…