Quaternion Algebra Research Articles

On quaternion algebras over some extensions of quadratic number fields

Let p and q be two positive primes, let \(\ell\) be an odd positive prime and let F be a quadratic number field. Let K be an extension of F of degree \(\ell\) such that K is a dihedral extension of \({\mathbb {Q}}\), or else let K be an abelian \(\ell\)-extension of F unramified over F whenever \(\ell\) divides the class number of F. In this paper, we provide a complete characterization of division quaternion algebras \(H_{K}(p, q)\) over K.

On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis

Abstract In this paper, we introduce generalizations of Quaternacci sequences (Quaternaccis), called Split Quaternacci sequences, which arose on a base of split quaternion algebras. Explicit and recurrent formulae for Split Quaternacci sequences are given, as well as generating functions. Also, matrices related to Split Quaternaccis sequences are investigated. Moreover, new identities connecting Horadam sequences with other known sequences are generated. Analogous identities for Horadam quaternions and split Horadam quaternions are proved.

Counting and equidistribution in quaternionic Heisenberg groups

AbstractWe develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebraAover${\mathbb{Q}}$in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points overAin quaternionic Heisenberg groups.

Dual-complex quaternion representation of gravitoelectromagnetism

In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.

Rank one sheaves over quaternion algebras on Enriques surfaces

Abstract Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

On the Lie triple derivations

This article aims to interpret some Lie triple derivations of Tensor algebras , using generalized quaternion algebra over a field F and assuming T as an F-linear associative algebra. The study concludes with complete description of all Lie triple derivations of A ⊗ F L. It is pertinent to mention here that A is assumed to be a finite-dimensional commutative algebra and L is a simple Lie algebra over an algebraically closed field.

The octonion linear canonical transform: Definition and properties

The linear canonical transform (LCT) is a kind of integral transforms with wide applications in signal analysis. There have been numerous studies in the literature to generalize the LCT by making use of the quaternion algebra. In this paper, we first define the octonion linear canonical transform (OCLCT). Based on the definition of OCLCT, we extend the relationship between the LCT and the Fourier transform (FT) to the OCLCT and the octonion Fourier transform (OFT). Then explore related properties for the OCLCT such as shift property, inversion formula, isometry and Riemann-Lebesgue lemma. The relation between OCLCT and 3-D LCT is also builded. Moreover, based on these properties, we obtain Heisenberg’s uncertainty principle and Donoho-Stark’s uncertainty principle associated with the OCLCT. Finally, some potential applications are presented to show the effectiveness of the OCLCT.

A quaternionic construction of p-adic singular moduli

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of \({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article, we present a construction of cohomology classes inspired by that of Darmon–Vonk, in which \({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) is replaced by an order in an indefinite quaternion algebra over a totally real number field F. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. We also report on extensive numerical evidence for this algebraicity conjecture.

Spectral halo for Hilbert modular forms

Let F be a totally real field and p be an odd prime which splits completely in F. We prove that the eigenvariety associated to a definite quaternion algebra over F satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the \(U_\mathfrak {p}\)-slopes of points and the p-adic valuations of the \(\mathfrak {p}\)-parameters are bounded by explicit numbers, for all primes \(\mathfrak {p}\) of F over p. Applying Hansen’s p-adic interpolation theorem, we are able to transfer our results to Hilbert modular eigenvarieties. In particular, we prove that on every irreducible component of Hilbert modular eigenvarieties, as a point moves towards the boundary, its \(U_p\) slope goes to zero. In the case of eigencurves, this completes the proof of Coleman–Mazur’s ‘halo’ conjecture.

On quaternion algebras over the composite of quadratic number fields

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

Hyperquaternion Conformal Groups

The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as $$O(p+1,q+1)$$ isomorphic to the nD conformal group with $$n=p+q.$$ The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory.

General theory of the higher-order quaternion linear difference equations via the complex adjoint matrix and the quaternion characteristic polynomial

Since the non-commutativity and particular structure of the quaternion algebra, the quternion difference equations (short for QDCEs) have a large difference from the classical theory of difference equations. In this paper, we establish a general theory of the higher-order linear QDCEs including the criteria of the linear independence (or dependence) of the discrete functions in the quaternion space, Liouville formulas, the structure theorems of the general solutions and the particular solutions with the quaternion power and trigonometric form, etc. By introducing the complex adjoint difference equations and the quaternion characteristic polynomial of the higher-order linear QDCEs, some basic results of the homogeneous and non-homogeneous difference equations are obtained. Through the analysis of the complex adjoint matrix and the quaternion eigenvalue, the general solutions of the higher-order linear QDCEs with variable and with constant coefficients are established. Several methods of obtaining the general solutions for the higher-order linear QDCEs are demonstrated and several examples are provided to illustrate the feasibility of our obtained results.

Some properties of the norm in a division quaternion algebra

In this paper we provide some applications of the norm form in some quaternion division algebras over rational field, and we give some properties of Fibonacci sequence and Fibonacci sequence in connection to quaternion elements. We provide some properties of the norm of a rational quaternion algebra, in connection to the famous Lagrange's four‐square theorem and its generalizations given by Ramanujan. Moreover, we prove some results regarding the arithmetic of integer quaternions defined on some division quaternion algebras, and we define and give properties of some special quaternions by using Fibonacci sequences.

A Generalised Quaternion and Clifford Algebra Based Mathematical Methodology to Effect Multi-stage Reassembling Transformations in Parallel Robots

A Generalised Quaternion and Clifford Algebra Based Mathematical Methodology to Effect Multi-stage Reassembling Transformations in Parallel Robots

Why Hamilton Couldn’t Multiply Triples

The history of the discovery of the 4-dimensional algebra of quaternions by William Rowan Hamilton is very well known, but one aspect of the story has received far less attention. This is the question of why he was unable to create a coherent system of complex numbers in 3-space. In fact, even after Hamilton discovered quaternions in 1843, it still took over half a century before anyone was able to prove that no such triple system actually existed. But what is the mathematical reason that Hamilton was unable to find a consistent three-dimensional linear algebra? The purpose of this article is to find out.

Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $\mathbb{Q}(\sqrt{p})$ in all positive characteristic $p\not\equiv 1\pmod{24}$.

Quaternion fractional-order color orthogonal moment-based image representation and recognition

Inspired by quaternion algebra and the idea of fractional-order transformation, we propose a new set of quaternion fractional-order generalized Laguerre orthogonal moments (QFr-GLMs) based on fractional-order generalized Laguerre polynomials. Firstly, the proposed QFr-GLMs are directly constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and fractional-order transformations, which extract only the global features from color images, our proposed QFr-GLMs can extract both the global and local color features. This paper also derives a new set of invariant color-image descriptors by QFr-GLMs, enabling geometric-invariant pattern recognition in color images. Finally, the performances of our proposed QFr-GLMs and moment invariants were evaluated in simulation experiments of correlated color images. Both theoretical analysis and experimental results demonstrate the value of the proposed QFr-GLMs and their geometric invariants in the representation and recognition of color images.

Zeros of slice functions and polynomials over dual quaternions

This work studies the zeros of slice functions over the algebra of dual quaternions and it comprises applications to the problem of factorizing motion polynomials. The class of slice functions over an alternative $*$-algebra $A$ was defined by Ghiloni and Perotti in 2011, extending the class of slice regular functions introduced by Gentili and Struppa in 2006. Both classes strictly include the polynomials over $A$. We focus on the case when $A$ is the algebra of dual quaternions $\mathbb{DH}$. The specific properties of this algebra allow a full characterization of the zero sets, which is not available over general alternative $*$-algebras. This characterization sheds some light on the study of motion polynomials over $\mathbb{DH}$, introduced by Heged\"us, Schicho, and Schr\"ocker in 2013 for their relevance in mechanism science.

Cd Hermite interpolations with spatial Pythagorean hodograph B-splines

Following the recent results of Albrecht et al. (2017, 2020), the problem of Hermite interpolation by clamped spatial Pythagorean hodograph (PH) B-spline curves is thoroughly investigated in this paper. The constructed interpolants are obtained by using beneficial properties of B-spline basis functions and via solving special quadratic and linear equations in quaternion algebra. All the designed procedures are purely symbolic and the main contribution lies in the unifying approach to the formulated problem. The results are confirmed by several computed examples.

Linkage of sets of quaternion algebras in characteristic 2

This note contains two new observations on the linkage properties of quaternion algebras over fields of characteristic 2: first, that a 3-linked field need not be 4-linked (a case which was left open in previous papers) and that three inseparably linked quaternion algebras are also cyclically linked when the base-field is odd-closed.