We describe in explicit detail the rich relationship between division algebras, split composition algebras, triality, Clifford algebras, spinors, triality Lie algebras, generalized reflections, exceptional magic square Lie algebras, triality eigenspaces, 3-symmetric spaces, Vinberg theta algebras, and Exceptional Unification in particle physics.
A simple algorithm for the inverse of a non-singular 6D multivector is presented. It is indifferent to signature (including degeneracy). A simpler algorithm is unlikely. A 4D algorithm of the same form is also presented. The formulas here are unlike the grade-involution-based inverse algorithms from which they were derived; instead, they rescale the scalar component. This algorithm requires two fewer multivector multiplications than its antecedent. A general expression for an annihilating polynomial of any 6D self-reverse multivector is given, with coefficients formed solely from the scalar parts of powers of the multivector. Opportunities for optimization are analyzed, and a strategy is presented that can reduce the number of real multiplications required to form the inverse to fewer than are required to multiply two arbitrary multivectors. Python code is presented, with additional code and documentation available in a public repository (see Ref. [7]).
This paper explores tachyonic, open bosonic string theory in which fermions are not strings, but rather reside on the D-brane boundary of open strings. The open-string tachyon is identified with a Higgs field transforming in the adjoint representation. The fermions of a generation of the standard model, including both internal and Weyl degrees of freedom, occupy a spinor representation of Spin(11,1) , and the Brauer-Weyl algebra of tensor products of the spinors is the Clifford algebra Cl(11,1) in 11+1 dimensions. The Chan-Paton condition that open strings transform in a tensor representation of the fermions at its endpoints requires the gauge group to be the group G^2(10) = U(16) generated by multivectors of grade 2 (mod 4) of the transverse Clifford algebra Cl(10) . The U(1) factor of U(16) generates a transverse spatial rotation of the 3+1 large spacetime dimensions, leaving as internal gauge group the rank 15 subgroup SU(16) . The 11+1 dimensions of the fermionic D-brane and the 15 dimensions of the bosonic torus combine to a total of 12+15 = 27 dimensions, one more than the 26 dimensions required by bosonic string theory. Grand symmetry breaking breaks SU(16) to the rank 14 group SU(8)_R ×SU(8)_L , yielding a spacetime of 12+14 = 26 dimensions, as required by bosonic string theory. After Pati-Salam symmetry breaking to the standard model, the 10 compact dimensions of the fermionic D-brane become the product of a weak Calabi-Yau 2-fold and a colour Calabi-Yau 3-fold.
We derive Hamilton’s equations in the language of geometric algebras from the general equations that describe dynamics on Lie groups, determined by an arbitrary Hamiltonian. We will then demonstrate how these equations can be utilised to describe the rigid body motion in projective and conformal geometric algebra.
In this paper, we present and study the ascent S-spectrum (ASS) and descent S-spectrum (DSS) of a bounded right linear operator T defined on a two-sided quaternionic Banach space X, as well as, the essential ascent S-spectrum (EASS) and essential descent S-spectrum (EDSS). On the one hand, we establish a connection between ASS, DSS, EASS, and EDSS with the usual ascent, descent, essential ascent, and essential descent spectra, respectively. Furthermore, we show that, under certain conditions, the spectral mapping theorem holds for these spectra for any intrinsic slice hyperholomorphic function f defined on σ _S(T). On the other hand, we prove some analogous properties known in the complex framework for their quaternionic counterparts. In particular, if s is an arbitrary element of the essential ascent S-spectrum (respectively, the essential descent S-spectrum), then the pseudo-resolvent Q_q(T) of T is semi-regular and upper semi-Fredholm (respectively, semi-regular and lower semi-Fredholm) for all q in 𝒱∖ [s], where [s] is the 2-dimensional sphere associated to s and 𝒱 is a neighborhood of s. As a result, the ASS, DSS, EASS, and EDSS are compact subsets of the S-spectrum. Moreover, this work introduces the concept of spherical poles of the S-resolvent, the quaternionic analog of the resolvent poles in the complex frame. We investigate the relationships between these poles and the ascent/descent S-spectra (ASS/DSS) and their essential counterparts (EASS/EDSS). Finally, we establish that the ASS, DSS, and EASS of the product operator TS (excluding zero) coincide with the corresponding spectra of ST (excluding zero).
We establish some L^2 -type Heisenberg–Pauli–Weyl uncertainty inequalities for the continuous wavelet transform and the Radon transform on the quaternionic Heisenberg group ℋ_q . The proofs are mainly based on some Gaussian-type estimates for the heat kernel associated with the sub-Lapalacain ℒ on ℋ_q .
Previous work using GAALOP to generate symbolically optimised 3D Characteristic Multivector (CM) rotor code showed that CM-based registration can achieve runtimes comparable to standard SVD-based methods for Absolute Orientation and ICP, while retaining the accuracy behaviour reported for CM in earlier studies. In that work it was also observed that naive CM implementations based on generic multivector libraries tend to run slower than both SVD-based solvers and specialised CM code. In this paper we give an implementation level analysis of the computational cost of the CM rotor providing an intrinsic cost baseline that helps explain these empirical differences and that can guide the design and evaluation of efficient CM implementations. Although the CM construction applies in any dimension and signature with a non-degenerate metric and can be used with non-orthonormal frames, we specialise to the Euclidean orthonormal case 𝒞ℓ (n,0) where n is the space dimension to make storage, constants and operation counts explicit. Our central idea is a stage-by-stage model in which each term is assigned to the stage determined by the largest frame index it contains. At stage i only coefficients indexed by even subsets of {1,… ,i} can change, so the main counts depend only on the dimension and not on the data. We derive closed form expressions for arithmetic and memory costs and compare two update strategies: an in-place scheme that writes each contribution directly to the stored even coefficients and a split scheme that collects all contributions for a stage in a temporary table and then commits them together. The main counts grow in direct proportion to the size N of the even subspace ( N=2^n-1 in 𝒞ℓ (n,0) ) with normalization contributing the same order but with a smaller constant. A complete 3D worked example fixes the constants and validates the model, and the resulting formulas provide a cost baseline for specialised implementations of the CM rotor that do not rely on generic multivector operations.
We present a compact mnemonic device for computing the product of two (split) octonions written in Cayley–Dickson form q+ℓ p with q,p∈ℍ . The rule appears as a simple (R+L) pattern of right-ordered and left-ordered (quaternionic) products inside a 2× 2 quaternionic matrix model. The pattern extends verbatim to all algebras in the Cayley–Dickson tower, providing an efficient computational tool in non-associative settings. To our knowledge, this explicit “ (R+L) ” mnemonic does not appear in the classical literature on octonions or composition algebras.
This paper explores the application of geometric algebra to Galilean spacetime and its physical implications. We introduce the Galilean spacetime algebra (GSTA), a five-dimensional conformal geometric algebra (CGA) generated by a specific metric, and demonstrate its utility in representing special Galilean transformations, rotations, and boosts. The general form of special Galilean transformations within the GSTA is derived, demonstrating their preservation. While the tensor formulation of Galilean electromagnetism is well-established, our work offers a fresh insight by deriving it from a geometric algebra perspective, utilizing the GSTA, and demonstrates how it seamlessly reduces to the familiar Maxwell equations in the non-relativistic limit. A significant aspect of this research is the introduction of Galilean spinors as elements of the minimal left ideals of the GSTA. We illustrate how these spinors can be utilized to construct the Lévy-Leblond equation for a free electron, along with its corresponding matrix representation. Furthermore, we establish a connection between the GSTA and the four-component dual numbers introduced by Majernik, suggesting pathways for developing a covariant formulation of Newtonian gravity. This work not only clarifies the geometric interpretation of Galilean symmetries but also opens avenues for future research in non-relativistic physics, highlighting the advantages of using CGA in this context.
Dual quaternion has a wide range of applications in various fields, and the study of its matrix theory has become a hot topic in recent years. In the theoretical study of dual quaternion matrices, the LU decomposition plays an important role. However, due to the non-commutativity of dual quaternions, the calculation of LU decomposition becomes difficult. In this paper, by means of the quaternion representation of the dual quaternion matrices given by semi-tensor product of matrices and the complex representation of the quaternion matrices, we give the complex representation of the dual quaternion matrices and its properties. The complex representation we proposed greatly facilitates the simplification of computational processes. And using these properties, we propose a fast and efficient complex structure-preserving algorithm for LU decomposition of dual quaternion matrices. The algorithm avoids the complexity of dual quaternion operations (essentially, quaternion operations). In order to ensure the stability of the algorithm, we further give a partial pivoting dual quaternion LU decomposition algorithm. Based on LU decomposition, we also present a complex structure-preserving algorithm for LDL^H decomposition of dual quaternion Hermitian matrices. In addition, we illustrate the effectiveness of the complex structure-preserving algorithms through numerical experiments. Finally, we give the application of dual quaternion matrix LU decomposition in color image authentication and kinematic linear equations.
The k-Cauchy–Fueter complex is the quaternionic counterpart of the Cauchy–Riemann complex in several complex variables, which plays a fundamental role in quaternionic analysis. In this work, we investigate the regularity of solutions to the non-homogeneous k-Cauchy–Fueter equation. Based on the Bourgain–Brezis inequalities, we extend the Limiting Sobolev inequalities to k-Cauchy–Fueter complex. In particular, we get the Gagliardo–Nirenberg inequality for the k-CF operator. In certain cases, the Hardy space ℋ^1 is used in place of L^1.
Let X be a two-sided Banach quaternionic space and A, C, B, D: X → X be the right bounded linear operators satisfying operator equation set A C D=D B D and D B A=A C A. In this paper, we generalize Jacobson’s Lemma and investigate the common properties of (A C)^2-2 Re(q) A C+|q|^2 I and (B D)^2-2 Re(q) B D+|q|^2 I where I stands for the identity operator on X and non-zero quaternion q. In particular, we show that σ _*^S(A C) \{0}=σ _*^S(B D) \{0}, where σ _*^S(.) is a distinguished part of the spherical spectrum.
We introduce the Cayley–Dickson Fourier transform (CDFT), a novel framework for harmonic analysis of functions valued in the non-associative Cayley–Dickson algebras 𝒞_m . The central challenge lies in the failure of associativity and alternativity for m ⩾ 4 , which obstructs classical Fourier analytic methods. To overcome this, we develop a two-stage approach: first, we construct the transform on real-valued Schwartz-type spaces, establishing continuity, inversion, and isometric properties; second, we extend the theory to fully 𝒞_m -valued functions by leveraging intrinsic algebraic structures, such as slice-wise multiplicativity and weak commutativity. Key innovations include a modified duality between differentiation and multiplication, governed by twisted sign involutions that precisely compensate for non-associative distortions, and a restricted convolution theorem for Gaussian-type functions that exploits the real scalar structure of their transforms. We prove that the CDFT admits an explicit inverse via symmetrization over coordinate reflections, acts isometrically on L^2(ℝ^m, 𝒞_m) , and exhibits a period-four symmetry that generalizes classical Fourier periodicity. These results collectively establish the CDFT as a rigorous and structurally faithful extension of Fourier analysis to the full Cayley–Dickson hierarchy.
The Dirac equation is mapped to a first order differential equation for complex quaternionic functions to benefit from potential theory and quaternionic analysis. This method provides solutions which do not depend on particular matrix representations for Clifford algebras. We additionally deduce zero-mode solutions for the Dirac–Weyl equation, when non-vanishing vector potentials are presupposed.
In this paper, we study an effective algorithm for the Schur decomposition of a quaternion matrix. Firstly, we give a complex structure-preserving algorithm for the Hessenberg decomposition of a quaternion matrix by Householder transformation. Secondly, we implement the implicit double shift QR strategy on the obtained Hessenberg matrix, and design the corresponding complex structure-preserving algorithm. Moreover, the effectiveness of the newly proposed algorithm is verified by numerical experiments. At last, the proposed algorithm is used to deal with a blind color image watermarking problem.
Let V be a vector space of finite dimension over a field K, and Q a quadratic form on V. A bilinear form compatible with Q is a bilinear form φ defined on any subspace S of V such that φ (s,s)=Q(s) for all s∈ S . The bilinear forms compatible with Q, together with an exceptional empty element, constitute an associative and unital monoid Cbf(V,Q) . In the first part of this work, the main purpose is a surjective homomorphism from the Lipschitz monoid Lip(V,Q) onto this monoid Cbf(V,Q) . In the second part, V is provided with an alternating bilinear form Ω , and some analogous properties are established for the monoid of bilinear forms compatible with Ω . When K is the field of real numbers, the controversy about an eventual Lipschitz monoid for Ω is recalled.
We consider a bicomplex analogue of the Landau Hamiltonian defined via its idempotent representation as a couple of the classical Landau Hamiltonians on two separate complex discs. We provide a complete characterization of its L^2 -eigenspaces when acting on the so-called bicomplex p-Hilbert space, which next employed to explore the common eigenfunction problem associated with the magnetic bc-Laplacian and its † -conjugate. The corresponding eigenspaces give rise to the polyanalytic version of the bicomplex Bergman spaces for which we provide the explicit expressions for their reproducing kernels.
We present the ways of constructing special subsets of conics present in the pencils of conics using Geometric Algebra for Conics (GAC). In particular, we offer geometrically oriented approaches to obtain the line-pairs and generalised parabolas that can be found in the pencils, by applying the tools of GAC and the classical theory of projective conics. In addition, we also describe the construction of a conic passing through five points and, throughout the work, we demonstrate the usage of points at infinity in the mentioned problems as well. The text is accompanied by examples with corresponding figures and includes a partial classification of some of the cases one may encounter in the topic.
We recall the Lounesto classification of 1/2-spin spinor fields, based on the vanishing of spinorial bilinear quantities: the classes are the regular spinor fields (i.e. the Dirac field), as well as singular spinor fields, also known as flag-dipole spinor fields, admitting two limiting sub-classes, given by the dipole spinors (i.e. the Weyl spinor) and the flagpole spinors (i.e. the Majorana spinor). We discuss each class in terms of its representatives, with particular emphasis upon the flag-dipole spinor fields.
This paper extends the concepts of weighted Drazin-star (WDS) and weighted star-Drazin (WSD) matrices to domain of quaternion matrices. We develop determinantal representations for these matrices, leveraging the theory of noncommutative row-column determinants, considering both general and Hermitian cases. As specific instances, we derive the determinantal representations of the complex WDS and WSD matrices by employing minors of appropriately constructed complex matrices. Furthermore, we investigate two-sided quaternion equations, along with one-sided particular types, where the unique solutions are expressed using WDS and WSD matrices. Explicit solutions for these quaternion matrix equations are obtained using Cramer-type methods. Finally, a numerical example is provided to confirm applicability and efficacy of our findings.
