The main objective of this work is to show how to construct a ternary \({\mathbb {Z}}_3\)-graded Clifford algebra on two generators by using a group algebra of an extra-special 3-group G of order 27. The approach used is an extension of the method implemented to classify \({\mathbb {Z}}_2\)-graded Clifford algebras as images of group algebras of Salingaros 2-groups [2]. We will show how non-equivalent

On various occasions, when working with quaternionic linear spaces, there is a need to restrict them to their complex linear structure, then it becomes essential to understand whether the pre-existing internal product or norm in the quaternionic space will continue to be compatible with the complex structure of the new space obtained. There are other situations in which these types of questions arise

This paper extends approach of our recent paper together with Kähler to building special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media by means of different generalizations of the Cauchy–Riemann system with variable coefficients. A new class of three-dimensional solutions of the static Maxwell system in some special cylindrically layered media is obtained

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. An algebraist’s perspective on the many subgroups and subalgebras of Clifford algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental

Conformal geometric algebra (CGA) is a framework that allows the representation of objects, such as points, planes and spheres, and deformations, such as translations, rotations and dilations as uniform vectors, called multivectors. In this work, we demonstrate the merits of multivector usage with a novel, integrated rigged character simulation framework based on CGA. In such a framework, and for the

Recent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of \(H_3\rightarrow H_4\) in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple

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

Several topological methods have been used successfully in the study of the hypercomplex analysis; for example, in the theory of functions of several complex variables (Grauert et al., in: Encyclopaedia of mathematical science, vol. 74, Springer, 1991, Hirzebruch, in: Topological methods in algebraic geometry, Classics in Mathematics. Reprint of the 1978 Edition. Springer, Berlin, 1995, Krantz, in

Let \((g,~[-,-],~\omega )\) be a finite-dimensional complex \(\omega \)-Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra \({\mathrm{Der}}(g)\) and the automorphism group \({\mathrm{Aut}}(g)\) of \((g,~[-,-],~\omega )\). We study \({\mathrm{Der}}^{\omega }(g)\) and \({\mathrm{Aut}}^{\omega }(g)\), which are superalgebra of \({\mathrm{Der}}(g)\) and subgroup of \({\mathrm{Aut}}(g)\)

In this paper, we complement some recent results of L. Márki, J. Meyer, J. Szigeti and L. van Wyk, by investigating the constant-trace representations of a Clifford algebra \(C(V)\) of an arbitrary quadratic form \(q:V\rightarrow F\) (possibly degenerate) and we present some relevant applications. In particular, the existence of the polynomial identities of \(C(V)\) of particular form when the characteristic

Previously we have shown that the algebra $$\begin{aligned} J_3 [{\mathbb {C}}\otimes {\mathbb {O}}] \otimes C \ell (4,{\mathbb {C}}), \end{aligned}$$ given by the tensor product of the complex exceptional Jordan \(J_3 [{\mathbb {C}}\otimes {\mathbb {O}}]\) and the complex Clifford algebra \(C \ell (4,{\mathbb {C}})\), can describe all of the spinorial degrees of freedom of three generations of fermions

Alternative versions of the energy–momentum complex in general relativity are given compact new formulations with spacetime algebra. A new unitary form for Einstein’s equation greatly simplifies the derivation and analysis of gravitational superpotentials. Interpretation of Einstein’s equations as a gauge field theory on flat spacetime is shown to resolve ambiguities in energy–momentum conservation

This paper deals with the inframonogenic functions which are solutions of the sandwich equation \(DfD=0\), where D is the Dirac operator in \({\mathbb {R}}^n\). The inframonogenic functions plays an important role in Fischer decomposition for homogeneous polynomials and applications in elasticity theory. In this paper we introduce a general structure of the inframonogenic functions in star-like domains

In this paper, we study relationships between generalized derivations, quasiderivations and centroids of ternary Jordan superalgebra. We prove that the generalized derivation algebras of a ternary Jordan superalgebra are the sum of quasiderivation algebras and centroids where centroids are ideals of generalized derivation algebras. We also show that quasiderivations can be embedded into larger ternary

We introduce three kinds of new weighted quaternion-matrix minimization problems in order to extend some well-known constrained approximation problems. The main result is the claim that these new minimization problems have unique solutions which are expressed in terms of expressions involving weighted core-EP inverse and its dual for adequate quaternion matrices. Also, determinantal expressions of

This paper outlines an approach for studying operators on stratified spaces \(M \in \mathfrak {M}_k\) with regular singularities of higher order k. Smoothness corresponds to \(k=0.\) Manifolds with smooth boundaries belong to the category \(\mathfrak {M}_1.\) The case \(k=1\) generally indicates conical or edge singularities. Boutet de Monvel’s algebra of boundary value problems (BVPs) with the transmission

In this paper we prove Clarkson-type and Nash-type inequalities in the (right-sided) Quaternion Linear Canonical transform (QLCT) for \(L^p\)-functions. Next, we show Heisenberg-type inequalities and Matolcsi–Szücs-type inequality for the QLCT. Finally, we deduce local-type uncertainty inequalities for the Quaternion Linear Canonical transform.

The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators \((A_1,\ldots ,A_n)\). A second way is to consider hyperholomorphic functions of quaternionic

We give a proof of the fact that the third cohomology group \(H^3_\alpha (L,M)\) of the Hom–Lie algebra \((L,\alpha _L)\) with coefficients in the Hom-L-module \((M,\alpha _M)\) is in bijection to the set \({{\,\mathrm{{\mathrm {Ext}}}\,}}^2_\alpha (L,M)\) of the equivalence classes of \(\alpha \)-crossed module extensions of \((L,\alpha _L)\) by \((M,\alpha _M)\).

This article is based on a talk given by the author at the 12th International Conference on Clifford Algebras and their Applications in Mathematical Physics. A generalization, introduced by Cassidy and the author, of a classical Clifford algebra is discussed together with connections between that generalization and a generalization of a graded Clifford algebra. A geometric approach to studying the

The well-known quaternion algebra is a four-dimensional natural extension of the field of complex numbers and plays a significant role in various aspects of signal processing, particularly for representing signals wherein several instincts are to be controlled simultaneously. For efficient analysis of such quaternionic signals, we introduce the notion of linear canonical wavelet transform in quaternion

Carroll’s group is presented as a group of transformations in a 5-dimensional space (\(\mathcal {C}\)) obtained by embedding the Euclidean space into a (4,1)-de Sitter space. Three of the five dimensions of \(\mathcal {C}\) are related to \(\mathcal {R}^3\), and the other two to mass and time. A covariant formulation of Caroll’s group, analogous as introduced by Takahashi to Galilei’s group, is deduced

In this paper, inverses of matrices over zeon algebras are discussed and methods of computation are presented. As motivation the zeon combinatorial Laplacian of a simple finite graph is defined, and its inverse is shown to enumerate paths and cycles in its associated graph.

A robotic manipulator is a networked assemblage of kinematic pairs, mobile platform(s) and/or tool(s) that generate motion and forces at its end-effector. A manipulator’s theoretical dynamic, kinematic and workspace capabilities are governed by the design parameters (mass, geometry, dimensions, etc.) of its kinematic pairs and its architecture (number of limbs, degrees of freedom, actuation ability

This article carries out the investigation to obtain the condition under which the generalized \(\mathcal {Z}\)-recurrent spacetimes (briefly, \(G\mathcal {Z}_{4}\) spacetimes) become a perfect fluid spacetime. It is shown that in a \(G \mathcal {Z}_{4}\) spacetime obeying Codazzi type and cyclic parallel \(\mathcal {Z}\) tensor represents a perfect fluid spacetime, provided the basic vector fields

In this paper we study a generalization of Clifford algebra depending on parameters introduced by Tutschke and Vanegas in 2008. We also introduce some related notions, a Cauchy–Pompeiu integral formula, and two boundary value problems for monogenic functions with values in this Cliford algebra.

A Correction to this paper has been published: https://doi.org/10.1007/s00006-020-01074-8

A Correction to this paper has been published: https://doi.org/10.1007/s10711-015-0119-z

In this paper, based on a proposed notion of generalized conjugate harmonic pairs in the framework of complex Clifford analysis, necessary and sufficient conditions for the solvability of inhomogeneous perturbed generalized Moisil–Teodorescu systems in higher dimensional Euclidean spaces are proved. As an application, we derive corresponding solvability conditions for the inhomogeneous Maxwell’s equations

The k-Cauchy–Fueter complex in quaternionic analysis was generalized to the Heisenberg group in a previous paper (Ren et al., in Adv Appl Clifford Algebras 30(2): Paper No. 20, 2020). But it is not known whether this differential complex is exact or not. In this paper, we apply the Penrose transform (the twistor method) to a double fibration of homogeneous spaces of \(\mathrm{SO}(2N,{\mathbb {C}})\)

The aim of this paper is to generalise the construction of n-ary Hom-Lie bracket by means of an \((n-2)\)-cochain of given Hom-Lie algebra to super case inducing n-Hom-Lie superalgebras. We study the notion of generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu and n-Hom-Lie superalgebras and their relation with generalized derivations and Rota-Baxter operators of Hom-Lie superalgebras

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras—subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and

Beltrami fields are complex vector fields \({\mathbf {F}}\) which satisfy the equation \({\text {curl}} {\mathbf {F}} + \lambda {\mathbf {F}}=0.\) Such fields appear in astrophysics, electromagnetics and plasma physics. We construct a complete system of solutions to the differential equation \((D+\lambda (x_3)+M^{\gamma (x_3){\mathbf {e}}_3})u=0\) for a complex quaternionic valued function u in a symmetric

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split

The theory of polynomial bases in one complex variable, as given by Whittaker and Cannon, was extended partially to the scope of Clifford analysis, where many results have been handled from different aspects. The emphasis of this paper is on introducing a certain type of bases of axially monogenic polynomials generated by the maximum modulus base. Determining the convergence properties of such bases

In para-Grassmann algebra a noncommutative, associative star product \(*\) (the Moyal product), which is a direct generalization of the star product in the algebra of Grassmann numbers is introduced. Isomorphism between the algebra of para-Grassmann numbers of order two equipped with the star product and the algebra of creation and annihilation operators \(a_{k}^{\pm }\) obeying the para-Fermi statistics

The CAM (Composition-Algebra based Methodology) (Wolk in Int J Mod Phys A 35:2050037–2050075, 2020; Pap Phys 9:090002–09007, 2017; Phys Scr 94:025301–025307, 2019; Adv Appl Clifford Algebras 27(4):3225–3234, 2017; J Appl Math Phys 6:1537–1538, 2018; Phys Scr 94:105301–105306, 2019; Adv Appl Clifford Algebras 30:4–14, 2020) [46,47,48,49,50,51,52] gauge model’s fiber bundle framework—previously shown

In this paper, we use the equivalence canonical form of four quaternion matrices to consider two systems of quaternion matrix equations. We derive some practical necessary and sufficient conditions for the existence of a solution to these two systems in terms of ranks of the quaternion matrices involved. We also give the general solutions to the systems when the solvability conditions are satisfied

We revisit the topic of two-state quantum systems using the Clifford Algebra in three dimensions \(Cl_3\). In this description, both the quantum states and Hermitian operators are written as elements of \(Cl_3\). By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system

In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real (“motion polynomials”), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition

In this paper, the study of linear differential equations involving one conventional and two nilpotent variables is started. This is a natural extension of the case of one involved nilpotent (para-Grassmann) variable studied earlier. In the case considered here, the two nilpotent variables are assumed to commute, hence they are generators of a (generalized) zeon algebra. Using the natural para-supercovariant

The aim of this paper is to generalise the construction of 3-BiHom-Lie superalgebras. We provide some properties that can be lifted to their \(T^{*}\)-extensions such as nilpotency, solvability and decomposition. We study representations, \(T_{\theta }\)-extensions and \(T^{*}_{\theta }\)-extensions of 3-BiHom-Lie superalgebras and prove the necessary and sufficient conditions for a 2n-dimensional

We show that if Projective Geometric Algebra (PGA), i.e. the geometric algebra with degenerate signature (n, 0, 1), is understood as a subalgebra of Conformal Geometric Algebra (CGA) in a mathematically correct sense, then flat primitives share the same representation in PGA and CGA. Particularly, we treat duality in PGA in the framework of CGA. This leads to unification of PGA and CGA primitives which

Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces

Because of the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\), it is possible to complexify the spacetime Clifford algebra \(C \ell _{1,3}({\mathbb {R}})\) by adding one additional timelike dimension to the Minkowski spacetime. In a recent work we showed how this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the new temporal

Conformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer graphics. In this paper we implement a simple ray tracer in CGA with a Blinn–Phong lighting model, before putting it to use to examine ray intersections with surfaces generated from the direct interpolation of

A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators

For functions \(f\in L^1\left( \mathbb {R}^2,\mathcal {H}\right) \) with the quaternion Fourier transform (QFT) \(\widehat{f}\) we give necessary and sufficient conditions in terms of \(\widehat{f}\) to ensure that f belongs either to one of the generalized Lipschitz classes \(H_{\alpha _1,\alpha _2}^m\) and \(h_{\alpha _1,\alpha _2}^m\) for \(0<\alpha _1,\alpha _2

We consider the technique of lifting frames to higher dimensions with the ridge idea that originally was introduced by Grafakos and Sansing. We pursue a novel approach with regard to a non-commutative setting, concretely the skew-field of quaternions. Moreover, we allow for splitting dimensions and for lifting with regard to multi-ridges. To this end, we introduce quaternionic Sobolev spaces and prove

Ternary Clifford algebras are an essential ingredient in a cubic factorization of the Laplacian and using a ternary Clifford analysis build on such spaces one obtains a Dirac-type operator D such that \(D^3=\Delta \). This paper is a continuation of the work of the authors in describing properties of generalized ternary Clifford algebras. Here we explore a blade decomposition and symmetries of these

In this article, we provide a complete classification of left octonionic modules (finite or infinite dimensions) in terms of new notions such as associative elements and conjugate associative elements. We give a simple approach to determine the irreducible left \(\mathbb {O}\)-modules by utilizing the Clifford algebra \(C\ell _{7}\). We find that every left \(\mathbb {O}\)-module has a basis in some

The Clifford Fourier transform (CFT) has been shown to be a crucial tool in the Clifford analysis. The purpose of this paper is to derive an analog of Titchmarsh’s theorems for the CFT for functions satisfying the Lipschitz and Dini–Lipschitz conditions in the space \(L^p(\mathbb {R}^{p,q},C\ell (p,q)), 1

In this present work, we generalize Titchmarsh’s theorem for the complex- or hypercomplex-valued functions. Firstly, we examine the order of magnitude of the windowed linear canonical transform (WLCT) of complex-valued functions that achieved certain Lipschitz conditions on \({\mathbb {R}}\). Secondly, we studied the order of magnitude of the 2-D continuous quaternion wavelet transform (CQWT) of certain

This paper deals with generalized axially symmetric potentials (GASP) which are solutions of a Weinstein-type equation in \(\mathbb {R}^3\) \(\dfrac{\partial ^2\Phi }{\partial x^2}+\dfrac{\partial ^2\Phi }{\partial y^2}+\dfrac{\partial ^2\Phi }{\partial z^2}+\dfrac{2(m+1)}{z}\dfrac{\partial \Phi }{\partial z}=0,\ m\in \mathbb {N}.\) GASP have been investigated by several generations of mathematicians

In this paper, the concept of the quaternionic adjoint matrix \(\chi _A\) of an elliptic biquaternion matrix A is introduced, which enable one to discuss the elliptic biquaternion problems through the quaternion ones. By this new concept, some fundamental problems, such as the right eigenvalues and eigenvectors, the singular value decomposition and the inverse can be investigated. Moreover, the least-squares

Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), González Cervantes

This paper explains that the definition of Moore–Penrose inverses in a given algebra A does not at all require any matrix representation of A. The pseudo-inverses y of a given element x of A (such that \(xyx=x\) and \(yxy=y\)) involve the right ideals complementary to xA and the left ideals complementary to Ax; the Moore–Penrose inverse corresponds to the complementary right and left ideals selected

Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: \(\min \Vert \mathbf {A}\mathbf {X}\mathbf {B}-\mathbf {C}\Vert _F\) subject to the constraints imposed to the right column space of \({\mathbf{A}}\) and the left row space of \({\mathbf{B}}\). Unique solution to the considered

This paper deals with the static Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} div(\Phi \overrightarrow{E})&{}=0,\\ \ curl\overrightarrow{E}&{}=0,\ (x_0,x_1,x_2)\in \mathbb {R}^3. \end{array} \right. \end{aligned}$$ The system is reformulated in quaternion analysis by Kravchenko in the form \(\mathcal {L}F=0\) with \(\mathcal {L}F=DF+F\alpha \). We consider special cases of the coefficient

In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space \(L^2({\mathbb {R}}^2,{\mathbb {H}})\), such as Parseval’s formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations