Narrow Results

In this paper, triality refers to the $S_3$-symmetry of the language of quasigroups, which is related to, but distinct from, the notion of triality as the $S_3$-symmetry of the Dynkin diagram $D_4$. The paper investigates a homogeneous method for rendering the linearization of quasigroups (over a commutative ring) naturally invariant under the action of the triality group, on the basis of an appropriate algebra generated by three invertible, non-commuting coefficient variables that is isomorphic to the group algebra of the free group on two generators. The algebra has a natural quotient given by setting the square of each generating variable to be $-1$. The quotient is an algebra of quaternions over the underlying ring, in a way reminiscent of how symmetric groups appear as quotients of braid groups on declaring the generators to be involutions. The corresponding quasigroups (which are described as quaternionic) are characterized by three equivalent pairs of quasigroup identities, permuted by the triality symmetry. The three pairs of identities are logically independent of each other. Totally symmetric quasigroups (such as Steiner triple systems) are quaternionic.

Let $D$ be an integral domain that is algebraic over $\mathbb{Z}$. It is shown that each directed maximal partial order on $D$ is an Archimedean total order. Let $F$ be a subfield of $ℝ$ and $C=F(i)$ be the complex field over $F$. As a consequence of the above result, if $F$ is algebraic over $\mathbb{Q}$, then $C$ does not have a directed partial order making it a partially ordered ring. In particular, $C$ cannot be a lattice-ordered ring. The result is proved for certain partially ordered algebras of quaternions as well.

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Starting with the complexes, quaternions and the octonions we study five families of groups and homogeneous spaces each building geometrically on the previous families. For each family a particular theorem is studied from the perspective of the structure developed. The evolution of the spaces and the corresponding results form a sort of mathematical memoir of a career spanning more than 50 years.

A Dirac fermion is expressed by a four-component spinor, which is a combination of two quaternions and can be treated as an octonion. The octonion possesses the triality symmetry, which defines symmetry of fermion spinors and bosonic vector fields. The triality symmetry relates three sets of spinors and two sets of vectors, which are transformed among themselves via transformations G₂₃, G₁₂, G₁₃, G₁₂₃ and G₁₃₂. If the electromagnetic (EM) interaction is sensitive to the triality symmetry, i.e. EM probe selects one triality sector, EM signals from the five transformed world would not be detected and be treated as the dark matter. According to an astrophysical measurement, the ratio of the dark to ordinary matter in the universe as a whole is almost exactly 5. We expect quarks are insensitive to the triality, and triality will appear as three times larger flavor degrees of freedom in the lattice simulation.

A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space–time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism, the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to ${𝒩}=1$ supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

In this paper, we propose a quaternion form of equations describing electromagnetic field in a homogeneous isotropic medium without dispersion. It is shown that by renormalizing the values of field inductions and sources, one can transform the asymmetrical Maxwell equations to a highly symmetric form. This provides a possibility to introduce the scalar and vector field potentials and represent the generalized equation for electromagnetic field in the form of a single second-order quaternionic wave equation. We demonstrate that this equation reduces to the system of ordinary hyperbolic wave equations for the field potentials and on the other hand, the same equation is equivalent to the system of Maxwell equations for renormalized field intensities. The symmetry of the renormalized Maxwell equations allows one to obtain the second-order relations for energy and momentum, as well as for Lorentz invariants of the renormalized fields, which formally have the same form as for the fields in a vacuum. In addition, the generalization of renormalized equations to the case of magnetic sources corresponding to the models of Dirac magnetic monopoles and Schwinger dyons is discussed.

The performance of face recognition systems tends to decrease when images are affected by illumination. Feature extraction is one of the main steps of a face recognition process, where it is possible to alleviate the illumination effects on face images. In order to increase the accuracy of recognition tasks, different methods for obtaining illumination invariant features have been developed. The aim of this work is to compare two different ways to represent face image descriptions in terms of their illumination invariant properties for face recognition. The first representation is constructed following the structure of complex numbers and the second one is based on quaternion numbers. Using four different face description approaches both representations are constructed, transformed into frequency domain and expressed in polar coordinates. The most illumination invariant component of each frequency domain representation is determined and used as the representative information of the face image. Verification and identification experiments are then performed in order to compare the discriminative power of the selected components. Representative component of the quaternion representation overcame the complex one.

The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z²+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.

In this paper, we show how generalized quaternions including some 2 × 2 matrices, can be used to find solutions of the equation

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation

In this paper, based on a talk given at the Oberwolfach research centre in May 2008 I will describe how biquandles and their big brother, biracks, can be used to differentiate isotopy classes of virtual (and welded) knots and links.

In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, adds the angles and multiplies the radii of the spherical coordinates. We show that the algebraic structure of pure quaternions with the spherical product is a commutative unital magma. Then, we present several generalizations of the Mandelbrot set. Among them, we present a set that is visually identical to the so-called Mandelbulb. We show that this set is bounded and that it can be generated by an escape time algorithm. We also define another generalization, the bulbic Mandelbrot set. We show that one of its 2D cuts has the same dynamics as the Mandelbrot set and that we can generate this set only with a quaternionic product, without using the spherical product.

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ_0,2, or to the real Clifford algebra ℝ_0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ_0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ_0,0 ≃ ℝ, ℝ_0,1 ≃ ℂ and the trivial case ℝ_1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝ_p,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.

We consider the arithmetic background of integral representations of finite groups over $p$-adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite $p$-groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed.

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every $4$-dimensional (4D) real unital algebra ${𝒜}$ into an 8D real unital algebra denoted by $\mathrm{\text{FD}}({𝒜}).$ We also find the conditions on ${𝒜}$ under which $\mathrm{\text{FD}}({𝒜})$ is a division algebra. This covers the most classes of known $4$D real division algebras. The second process allows us to embed particular classes of $4$D RDAs into $8$D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic $8$D real division algebras whose derivation algebras contain $\mathrm{\text{su}}(2)$.

We study realization fields and integrality of characters of finite subgroups of ${\mathrm{\text{GL}}}_n(C)$ and related lattices with a focus on the integrality of characters of finite groups $G$. 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.

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.