We present the deep connections among (Anti) de Sitter geometry, and complex conformal gravity-Maxwell theory, stemming directly from a gauge theory of gravity based on the complex Clifford algebra Cl(4, C). This is attained by simply promoting the de (Anti) Sitter algebras so(4, 1), so(3, 2) to the real Clifford algebras Cl(4, 1, R), Cl(3, 2, R), respectively. This interplay between gauge theories

In this paper, we consider the polynomial Dirac equation \( \left( D^m+\sum _{i=0}^{m-1}a_iD^i\right) u=0,\ (a_i\in {\mathbb {C}})\), where D is the Dirac operator in \({\mathbb {R}}^n\). We introduce a method of using series to represent explicit solutions of the polynomial Dirac equations.

Extended gamma matrix Clifford–Dirac and SO(1,9) algebras in the terms of \(8 \times 8\) matrices have been considered. The 256-dimensional gamma matrix representation of Clifford algebra for 8-component Dirac equation is suggested. Two isomorphic realizations \(\textit{C}\ell ^{\texttt {R}}\)(0,8) and \(\textit{C}\ell ^{\texttt {R}}\)(1,7) are considered. The corresponding gamma matrix representations

I introduce a notion of quaternionic regularity using techniques based on hypertwined analysis, a refined version of general hypercomplex theory. In the quaternionic and biquaternionic cases, I show that hypertwined holomorphic (regular) functions admit a decomposition in a hypertwined sum of regular functions in certain subalgebras. The hypertwined quaternionic regularity lies in between slice regularity

We give estimates of the Cauchy transform in Lebesgue integral norms in Clifford analysis framework which are the generalizations of Cauchy transform in complex plane, and mainly establish the \((L^{p}, L^{q})\)-boundedness of the Clifford Cauchy transform in Euclidean space \({\mathbb {R}^{n+1}}\) using the Clifford algebra and the Hardy–Littlewood maximal function. Furthermore, we prove Hedberg estimate

The Cayley–Dickson algebra has long been a challenge due to the lack of an explicit multiplication table. Despite being constructible through inductive construction, its explicit structure has remained elusive until now. In this article, we propose a solution to this long-standing problem by revealing the Cayley–Dickson algebra as a twisted group algebra with an explicit twist function \(\sigma (A

Let K be a field of characteristic other than 2, and let \(\mathcal {A}_n\) be the algebra deduced from \(\mathcal {A}_1=K\) by n successive Cayley–Dickson processes. Thus \(\mathcal {A}_n\) is provided with a natural basis \((f_E)\) indexed by the subsets E of \(\{1,2,\ldots ,n\}\). Two questions have motivated this paper. If a subalgebra of dimension 4 in \(\mathcal {A}_n\) is spanned by 4 elements

Linear coupled matrix equations are widely utilized in applications, including stability analysis of control systems and robust control. In this paper, we establish the necessary and sufficient conditions for the consistency of the system of linear coupled matrix equations and derive an expression of the corresponding general solution (where it is solvable) over quaternion. Additionally, we investigate

In this paper, we study the bicomplex version of weighted Bergman spaces and the composition operators acting on them. We also investigate the Bergman kernel, duality properties and Berezin transform. This paper is essentially based on the work of Zhu (Operator Theory in Function Spaces of Math. Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence, 2007).

The aim of this work is to show that given \(u\in {\mathbb {H}}{\setminus }{\mathbb {R}}\), there exists a differential operator \(G^{-u}\) whose solutions expand in quaternionic power series expansion \( \sum _{n=0}^\infty (x-u)^n a_n\) in a neighborhood of \(u\in {\mathbb {H}}\). This paper also presents Stokes and Borel-Pompeiu formulas induced by \(G^{-u}\) and as consequence we give some versions

In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension

In most cases, the Lipschitz monoid \(\textrm{Lip}(V,Q)\) is the multiplicative monoid (or semi-group) generated in the Clifford algebra \(\textrm{Cl}(V,Q)\) by the vectors of V. But the elements of \(\textrm{Lip}(V,Q)\) satisfy many other characteristic properties, very different from one another, which may as well be used as definitions of \(\textrm{Lip}(V,Q)\). The present work proposes several

The Riemann curvature tensor is constructed using the Clifford-Dirac geometric algebra and division-algebraic operator structure.

Mean value theorems appear as fundamental tools in the analysis of harmonic functions and elliptic partial differential equations. In the present paper, we establish their bicomplex analogs for bicomplex harmonic and strongly harmonic functions with bicomplex values. Their analytical converse as well as geometrical converse characterizing open idempotent discus are also discussed.

In this paper, we explore the field propagator with a structure that is general enough to encompas both the case of newly-defined mass-dimension 1 fermions and spin-1/2 bosons. The method we employ is to define a map between spinors of different Lounesto classes, and then write the propagator in terms of the corresponding dual structures.

In this research, the Clifford–Fourier transform introduced by E. Hitzer, satisfies some uncertainty principles similar to the Euclidean Fourier transform. An analog of the Beurling–Hörmander’s theorem for the Clifford–Fourier transform is obtained. As a straightforward consequence of Beurling’s theorem, other versions of the uncertainty principle, such as the Hardy, Gelfand–Shilov and Cowling–Price

We give a formula for \(f(\eta ),\) where \(f:{\mathbb {C}} \rightarrow {\mathbb {C}}\) is a continuously differentiable function satisfying \(f(\bar{z}) = \overline{f(z)},\) and \(\eta \) is a dual quaternion. Note this formula is straightforward or well known if \(\eta \) is merely a dual number or a quaternion. If one is willing to prove the result only when f is a polynomial, then the methods of

We define a new \({{\mathbb {Z}}}_2\)-graded quantum (2+1)-space and show that the extended \({{\mathbb {Z}}}_2\)-graded algebra of polynomials on this \({{\mathbb {Z}}}_2\)-graded quantum space, denoted by \({\mathbb F}({{\mathbb {C}}}_q^{2\vert 1 })\), is a \({{\mathbb {Z}}}_2\)-graded Hopf algebra. We construct a right-covariant differential calculus on \({{\mathbb {F}}}({{\mathbb {C}}}_q^{2\vert

In this paper, we consider a family of the hypercomplex rings \({\mathscr {H}}=\left\{ {\mathbb {H}}_{t}\right\} _{t\in {\mathbb {R}}}\) scaled by \({\mathbb {R}}\), and the dynamical system of \({\mathbb {R}}\) acting on \({\mathscr {H}}\) via a certain action \(\theta \) of \({\mathbb {R}}\). i.e., we study an analysis on dynamical system induced by \({\mathscr {H}}\). In particular, we are interested

We study the inhomogeneous equation \({\text {curl}}\vec {w}+\lambda \vec {w}=\vec {g},\,\lambda \in {\mathbb {C}},\,\lambda \ne 0\) over unbounded domains in \({\mathbb {R}}^{3}\), with \(\vec {g}\) being an integrable function whose divergence is also integrable. Most of the results rely heavily on the “good enough” behavior near infinity of the \(\lambda \) Teodorescu transform, which is a classical

The Atiyah-Singer index formula for Dirac operators acting on the space of spinors put across a kind of topological invariant (\(\hat{{A}}\) genus) of a closed spin manifold \({{\mathcal {M}}}\), hence offering a bridge between geometric and analytical aspects of the original spin manifold. In this study, we prove the index theorem for a family of fractional Dirac operators in particular for complex

The polynomial null solutions of the k-hyperbolic Dirac operator are investigated by the \(\mathfrak {osp}(1|2)\) approach. These solutions are then utilized to construct the (fractional) Fourier transform associated to the k-hyperbolic Dirac operator. The resulting integral kernels are found to be a specific kind of Dunkl kernels. Additionally, we give tight uncertainty inequalities for three distinct

In this paper, we study the properties of the S-spectrum of right linear bounded operators on a right quaternionic Banach space. We prove some relations between the S-spectrum and most of its important parts; the approximate S-spectrum, the compression S-spectrum and the surjective S-spectrum. Among other results, we provide some properties of duality and orthogonality on a right quaternionic Banach

The “10-fold way” refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, \({\mathbb Z}_2\)-graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by

We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems \(\big (\)for short RHBVPs\(\big )\) for axially monogenic functions on Euclidean space \({\mathbb {R}}^{n+1},n\in {\mathbb {N}}\). With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on

The k-Cauchy–Fueter operator and the tangential k-Cauchy–Fueter operator are the quaternionic counterpart of Cauchy–Riemann operator and the tangential Cauchy–Riemann operator in the theory of several complex variables, respectively. In Wang (On the boundary complex of the k-Cauchy–Fueter complex, arXiv:2210.13656), Wang introduced the notion of right-type groups, which have the structure of nilpotent

We study the notions of centrally-extended higher \(*\)-derivations and centrally-extended generalized higher \(*\)-derivations. Both are shown to be additive in a \(*\)-ring without nonzero central ideals. Also, we prove that in semiprime \(*\)-rings with no nonzero central ideals, every centrally-extended (generalized) higher \(*\)-derivation is a (generalized) higher \(*\)-derivation.

The fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath \( \left( A\otimes H_{4}^{op}, H_{4}, \psi \right) \), where \(H_{4}\) is the Sweedler 4-dimensional

The present study is the first of its kind which aims to analyse the Clifford-valued functions by introducing the notion of a two-sided Clifford-valued linear canonical transform in \(L^2({\mathbb {R}}^n, C\ell _{0,n})\), which not only embodies the classical Clifford–Fourier transform, but also yields another new variant of Clifford transforms based on the fractional Clifford–Fourier transform. To

Complete hom-Lie superalgebras are considered and some equivalent conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra are established. In particular, the relation between decomposition and completeness for a hom-Lie superalgebra is described. Moreover, some conditions that the linear space of \(\alpha ^{s}\)-derivations of a hom-Lie superalgebra to be complete and simply complete

The Hayman Theorem of left-monogenic function in a Clifford Analysis Setting is established in this article. A few established conclusions regarding subharmonic functions in Euclidean half space are extended to Clifford half space.

In this paper the mean ergodic theorem in bicomplex Banach modules is studied. Under appropriate conditions of boundness for the iterates compositions of a bicomplex linear and bounded operator on a bicomplex Banach module, and of weak compactness of the average sequence on the idempotent components, analogous to that of the classical case on Banach spaces, strong convergence of the mean sequence is

In this research, we look at problems in the theory of approximation of functions in real Clifford algebras. We prove analogues of direct and inverse approximation theorems in terms of best approximations of functions with bounded spectrum and the moduli of smoothness of all orders constructed by the generalized Steklov operators.

We consider spinorial fields in polar form to deduce their respective tensorial connection in various physical situations: we show that in some cases the tensorial connection is a useful tool, instead in other cases it arises as a necessary object. The comparative analysis of the different cases possessing a tensorial connection is done, investigating the analogies between space-time structures. Eventual

In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator \(\mathcal S_a\) with the decomposition process $$\begin{aligned} f=e_a\langle f, e_a\rangle +B_{a}*\mathcal S_a f, \end{aligned}$$ where \(e_a\) denotes the slice normalized Szegö kernel and \( B_a \) the slice Blaschke factor. Iterating the above decomposition process, a corresponding

In every Clifford algebra \(\textrm{Cl}(V,Q)\) over a field, there is a Lip-schitz monoid (or semi-group) \(\textrm{Lip}(V,Q)\) that satisfies a lot of remarkable properties. In general, it is the multiplicative monoid generated by the vectors of the space V. Each of its two components is an irreducible algebraic submanifold. The present article is devoted to the equations that are satisifed by the

A novel invariant decomposition of diagonalizable \(n \times n\) matrices into n commuting matrices is presented. This decomposition is subsequently used to split the fundamental representation of \(\mathfrak {su}({3})\) Lie algebra elements into at most three commuting elements of \(\mathfrak {u}({3})\). As a result, the exponential of an \(\mathfrak {su}({3})\) Lie algebra element can be split into

In this paper, we consider the actions of affine Yangian and \(W_{1+\infty }\) algebra on three cases of symmetric functions. The first one is Schur functions of 2D Young diagrams. It is known that affine Yangian and \(W_{1+\infty }\) algebra can be represented by 1 Boson field with center 1 in this case. The second case is the symmetric functions \(Y_\lambda ({\mathbf{p}})\) of 2D Young diagrams which

In this paper, we prove the sharp Pitt’s inequality for a generalized Clifford-Fourier transform which is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. As an application, the Beckner’s logarithmic uncertainty principle for the Clifford-Fourier transform is established.

In this paper, we first define supersymmetric Schur Q-functions and give their vertex operators realization. By means of the vertex operator, we obtain a series of non-linear partial differential equations of infinite order, called the super BKP hierarchy and the super BKP hierarchy governs the supersymmetric Schur Q-functions as the tau functions. Moreover, we prove that supersymmetric Schur Q-functions

We generalize the space-time Fourier transform (SFT) (Hitzer in Adv Appl Clifford Algebras 17(3):497–517, 2007) to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3, 1)-valued signals over the domain of space-time (Minkowski space) \(\mathbb {R}^{3,1}.\) We establish how it can be computed in terms of the SFT,

In this note, we first derive a Schödinger uncertainty relation for any pair of quaternionic observables and a mixed state. Then, the Wigner–Yanase skew information is introduced in the quaternion setting. Based on the skew information, we establish a new quantum uncertainty inequality for the non-Hermitian quaternionic observables.

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras \(\mathcal {G}_{p,q}\) of vector space of dimension \(n=p+q\). We present basis-free formulas for all characteristic polynomial coefficients in the cases \(n\le 6\), alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations

This paper proposes to go beyond the Einstein General Relativity theory in a noncommutative geometric framework. As a first step, we rewrite the General Relativity theory intrinsically (coordinate-free formulation). As a second step, we rewrite the first and the second Bianchi identities, including torsion, within minimal algebraic hypotheses. Then, in order to extend the General Relativity theory