Quaternion Algebra Research Articles

The sum of two quadratic matrices: Exceptional cases

Let p and q be polynomials with degree 2 over an arbitrary field F. A square matrix with entries in F is called a (p,q)-sum when it can be split into A+B for some pair (A,B) of square matrices such that p(A)=0 and q(B)=0.A (p,q)-sum is called regular when none of its eigenvalues is the sum of a root of p and of a root of q. A (p,q)-sum is called exceptional when each one of its eigenvalues is the sum of a root of p and of a root of q. In a previous work [7], we have shown that the study of (p,q)-sums can be entirely reduced to the one of regular (p,q)-sums and to the one of exceptional (p,q)-sums. Moreover, regular (p,q)-sums have been characterized thanks to structural theorems on quaternion algebras, giving the problem a completely unified treatment.The present article completes the study of (p,q)-sums by characterizing the exceptional ones. The new results here deal with the case where at least one of the polynomials p and q is irreducible over F.

On Motohashi’s formula

We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of L L -functions for G L 1 GL_1 with the third moment of L L -functions for G L 2 GL_2 over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on M 2 ( A ) M_2(\mathbb {A}) instead of G L 2 ( A ) GL_2(\mathbb {A}) , on whose spectral side the matrix coefficients are replaced by the standard Godement-Jacquet zeta integrals. This is also a generalization of Bruggeman-Motohashi’s other proof of Motohashi’s formula. We give a variation of our method in the case of division quaternion algebras instead of M 2 M_2 , yielding a new spectral reciprocity, for which we are not sure if it is within the period formalism given by Michel-Venkatesh. We also indicate a further possible generalization, which seems to be beyond what the period method can offer.

Hida theory for special orders

This note is devoted to the study of families of quaternionic modular forms arising from orders defined by Pizer. In this situation, the Hecke eigenspaces are 2-dimensional contrary to the classical case of Eichler orders. The main result is a Control Theorem in the spirit of Hida, interpolating these 2-dimensional Hecke eigenspaces. We restrict our attention to a definite rational quaternion algebra ramified at a single odd prime [Formula: see text].

Diffusion in Cauchy Elastic Solid

It is commonly accepted that a starting point of the science of diffusion is the phenomenological diffusion equation postulated by German physiologist Adolf Fick inspired by experiments on diffusion by Thomas Graham and Robert Brown. Fick’s diffusion equation has been interpreted decades later by Albert Einstein and Marian Smoluchowski. Here we will show that the theory of diffusion has its elegant mathematical foundations formulated three decades before Fick by French mathematician Augustin Cauchy (~1822). The diffusion equation is straightforward consequence of his model of the elastic solid - the classical balance equations for isotropic, elastic crystal. Basing on the Cauchy model and using the quaternion algebra we present a rigorous derivation of the quaternion form of the diffusion equation. The fundamental consequences of all derived equations and relations for physics, chemistry and the future prospects are presented.

On the Second Cohomology of the Norm One Group of a p-Adic Division Algebra

Let F be a p-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F, and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H2(SL1(D),R/Z) is a cyclic p-group whose order is bounded from below by the number of p-power roots of unity in F, unless D is a quaternion algebra over Q2. In this paper we give an explicit upper bound for the order of H2(SL1(D),R/Z) for p≥5 and determine H2(SL1(D),R/Z) precisely when F is cyclotomic, p≥19, and the degree of D is not a power of p.

On the Quaternion Transformation and Field Equations in Curved Space-Time

In this paper, we use four-dimensional quaternionic algebra to describe space-time geodesics in curvature form. The transformation relations of a quaternionic variables are established with the help of basis-transformations of quaternion algebra. We deduce the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and the geodesic equation are also discussed to describe the quaternionic line element in curved space-time. We examine a quaternionic metric tensor equation for the Riemannian Christoffel curvature tensor. We present the quaternionic Einstein’s field-like equation, which indicates that quaternionic matter and geometry are equivalent. Relevance of the work:In recent decades, hypercomplex algebra, viz., quaternion and octonions, has been widely used to explain various branches of physics. In this way, we have investigated quaternionic transformations and field equations in curved space-time. The present novel work will help to explain the characteristics of the curved space-time universe in terms of quaternion algebra. It can also be used to describe quaternionic gravitational waves, the black hole formulation, and so on.

A new perspective on quantum field theory revealing possible existence of another kind of fermions forming dark matter

Quantum fields are considered as generators of infinite-dimensional Clifford algebra [Formula: see text], which can be either orthogonal (in the case of fermions) or symplectic (in the case of bosons). A generic quantum state can be expressed as a superposition of the basis elements of [Formula: see text], with the superposition coefficients being multiparticle complex-valued wavefunctions. The basis elements, that are products of the generators of [Formula: see text] in the Witt basis, act as creation and annihilation operators. They create positive and negative energy states that include the bare and the Dirac vacuum as special cases. It is shown that the nonvanishing electric charge arises from an extra dimension or from doubling the number of creation and annihilation operators, which brings an extra imaginary unit [Formula: see text] into the description. A further extension is to consider [Formula: see text] as one of the quaternionic imaginary units and consider a generic state as having values in the quaternionic algebra or, equivalently, in the complexified two-dimensional Clifford algebra, [Formula: see text]. It contains two distinct fundamental representations of [Formula: see text], one associated with the weak isospin doublet [Formula: see text] and the other one with the doublet of new leptons, denoted by [Formula: see text], that together with the new quarks [Formula: see text] can be identified with dark matter.

The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

Let H be the real quaternion algebra and Hm×n denote the set of all m×n matrices over H. For A∈Hm×n, we denote by Aϕ the n×m matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a non-standard involution of H. A∈Hn×n is said to be ϕ-skew-Hermicity if A=−Aϕ. In this paper, we provide some necessary and sufficient conditions for the existence of a ϕ-skew-Hermitian solution to the system of quaternion matrix equations with four unknowns AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4.

Multiple color image encryption based on cascaded quaternion gyrator transforms

In this paper, we propose a novel encryption algorithm using cascaded quaternion gyrator transforms that allows protecting multiple color images with efficiency at once. The originality is reflected in the integration of quaternion algebra with a cascaded encryption structure. First, image RGB channels are constituted into three imaginary parts of a quaternion matrix. A cascaded encryption structure is constructed to cooperate with the vortex phase mask, which is easy to transmit and saves storage. Then, a quaternion gyrator transform is conducted, after which the phase is extracted to serve as the decryption key while the amplitude is used as the real part of the following quaternion matrix. Finally, the amplitude of the last gyrator transform is scrambled to obtain the ciphertext. All phase matrices are further modulated by a secondary image phase mask for additional security. Numerical simulations have been conducted to verify the validity of the proposed algorithm. Experiments on several data test set demonstrate the robustness of the proposed method to various attacks.

One underlying mechanism for two piezoelectric effects in the octonion spaces

The paper aims to apply the algebra of octonions to explore the contributions of external derivative of electric moments and so forth on the induced electric currents, revealing a few major influencing factors relevant to the direct and inverse piezoelectric effects. J. C. Maxwell was the first to adopt the algebra of quaternions to describe the physical quantities of electromagnetic fields. The contemporary scholars utilize the quaternions and octonions to research the physical properties of electromagnetic and gravitational fields. The application of octonions is able to study the physical quantities of electromagnetic and gravitational fields, including the octonion field strength, field source, linear moment, angular moment, torque and force. When the octonion force is equal to zero, it is capable of achieving eight independent equations, including the force equilibrium equation, fluid continuity equation, current continuity equation, and second-precession equilibrium equation and others. One of inferences derived from the second-precession equilibrium equation is that the electric current and derivative of electric moments are able to excite each other. The external derivative of electric moments can induce the electric currents. Meanwhile the external electric currents are capable of inducing the derivative of electric moments. The research states that this inference can be considered as the underlying mechanism for the direct and inverse piezoelectric effects. Further the second-precession equilibrium equation is able to predict several new influencing factors of direct and inverse piezoelectric effects.

Constructing all genus 2 curves with supersingular Jacobian

L. Moret-Bailly constructed families {mathfrak {C}}rightarrow {mathbb {P}}^1 of genus 2 curves with supersingular Jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family.

Biquaternion division algebras and fourth power-central elements

Let A be a biquaternion division algebra over a field F ( char F ≠ 2 ). We prove that an element e ∈ F ⁎ such that A F ( e 4 ) = 0 exists if and only if there is a decomposition A ≃ ( a , b ) ⊗ ( c , d ) with 《 − a , b , c , d 》 = 0 . This statement strengthens the result of Rost, Serre and Tignol ( [7] ), where the additional condition − 1 ∈ F ⁎ holds. Another result is a construction of a field F with − 1 ∈ F ⁎ , a quartic cyclic field extension L / F , and quaternion algebras ( a , b ) and ( c , d ) over F such that ( a , b ) L = ( c , d ) L = 0 , but the quadratic form 〈 a , b , a b 〉 ⊥ 《 c , d 》 is anisotropic. As a consequence, for any algebraically closed field k 0 , we give an example of an anisotropic 7-dimensional quadratic form Ψ over k 0 ( z , x , y ) with ind ( C 0 ( Ψ ) ) = 4 such that for any discrete k 0 ( z ) -valuation v on k 0 ( z , x , y ) the form Ψ becomes isotropic over the completion k 0 ( z , x , y ) v . In other words, the form Ψ provides a counterexample to the strong Hasse principle for 7-dimensional quadratic forms with respect to the rational extension K ( x , y ) / K , where K = k 0 ( z ) .

Classification of involutive automorphisms and anti-automorphisms of the Lie algebra of quaternions

In this article, we treat the space of real quaternions as a Lie algebra equipped with its commutator product. We show that all involutions of this Lie algebra that are automorphisms (respectively, anti-automorphisms) and restrict to the identity on the centre (sometimes called automorphisms of the first kind) are actually algebra automorphisms (resp. anti-automorphisms) of the division algebra of quaternions, which we characterized in an earlier paper. If we compose with scalar multiplication by , we obtain all involutive automorphisms and anti-automorphisms of the second kind, i.e. those for which the centre is contained in the -eigenspace. Together, we have a complete determination of all involutive (anti-)automorphisms on the quaternionic Lie algebra . From this determination of all the involutive (anti-)automorphisms of , one can identify via a standard bijective correspondence all the involutive (anti-)automorphisms for the corresponding simply connected multiplicative quaternion Lie group . We carry out this determination explicitly.

Quadratic Equation in Split Quaternions

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x2+bx+c=0 in split quaternion algebra.

Chaos over order: mapping 3D rotation of triaxial asteroids and minor planets

ABSTRACT Celestial bodies approximated with rigid triaxial ellipsoids in a two-body system can rotate chaotically due to the time-varying gravitational torque from the central mass. At small orbital eccentricity values, rotation is short-term orderly and predictable within the commensurate spin–orbit resonances, while at eccentricity approaching unity, chaos completely takes over. Here, we present the full three-dimensional rotational equations of motion around all three principal axes for triaxial minor planets and two independent methods of numerical solution based on Euler rotations and quaternion algebra. The domains of chaotic rotation are numerically investigated over the entire range of eccentricity with a combination of trial integrations of Euler’s equations of motion and the GALI(k) (Generalized Alignment Index) method. We quantify the dependence of the order–chaos boundaries on shape by changing a prolateness parameter, and find that the main 1:1 spin–orbit resonance disappears for specific moderately prolate shapes already at eccentricities as low as 0.3. The island of short-term stability around the main 1:1 resonance shrinks with increasing eccentricity at a fixed low degree of prolateness and completely vanishes at approximately 0.8. This island is also encroached by chaos on longer time-scales, indicating longer Lyapunov exponents. Trajectories in the close vicinity of the 3:2 spin–orbit resonance become chaotic at smaller eccentricities, but separated enclaves of orderly rotation emerge at eccentricities as high as 0.8. Initial perturbations of rotational velocity in latitude away from the exact equilibrium result in a spectrum of free libration, nutation, and polar wander, which is not well matched by the linearized analysis omitting the inertial terms.

Iterative inverse kinematics for robot manipulators using quaternion algebra and conformal geometric algebra

Iterative inverse kinematics for robot manipulators using quaternion algebra and conformal geometric algebra

Remarks Regarding Computational Aspects in Algebras Obtained by Cayley–Dickson Process and Some of Their Applications

Due to the computational aspects which appear in the study of algebras obtained by the Cayley–Dickson process, it is difficult to obtain nice properties for these algebras. For this reason, finding some identities in such algebras plays an important role in obtaining new properties of these algebras and facilitates computations. In this regard, in the first part of this paper, we present some new identities and properties in algebras obtained by the Cayley–Dickson process. As another remark regarding the computational aspects in these algebras, in the last part of this paper, we solve some quadratic equations in the real division quaternion algebra when their coefficients are some special elements. These special coefficients allowed us to solve interesting quadratic equations, providing solutions directly, without using specialized softs.

On integral representations of finite groups and rings generated by character values

We study realization fields and integrality of characters of finite subgroups of [Formula: see text] and related lattices with a focus on the integrality of characters of finite groups [Formula: see text]. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, order generated by the character values, the number of minimal realization splitting fields, and in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras and some orders generated by character values over the rings of rational and algebraic integers.

MULTIPLICITY ONE FOR THE PAIR (GLn(D);GLn(E))

Let F be a local field of characteristic zero. Let D be a quaternion algebra over F. Let E be a quadratic field extension of F. Let {\mu} be a character of GL(1,E). We study the distinction problem for the pair (GL(n,D), GL(n,E)) and we prove that any bi-(GL(n,E), {\mu})-equivariant tempered generalized function on GL(n,D) is invariant with respect to an anti-involution. Then it implies that dimHom({\pi},{\mu}) is at most 1 by the generalized Gelfand-Kazhdan criterion. Thus we give a new proof to the fact that (GL(2n,F),GL(n,E)) is a Gelfand pair when {\mu} is trivial and D splits.

Low-lying zeros of L-functions for Quaternion Algebras

The density conjecture of Katz and Sarnak predicts that, for natural families of L-functions, the distribution of zeros lying near the real axis is governed by a group of symmetry. In the case of the universal family of automorphic forms on a totally definite quaternion algebra, we determine the associated distribution for a restricted class of test functions in the analytic conductor aspect. In particular it leads to non-trivial results on densities of non-vanishing at the central point.