Search

In continuation of the work in Leventides and Petroulakis (Adv Appl Clifford Algebras 27:1503–1515, 2016), Leventides et al. (J Optim Theory Appl 169(1):1–16, 2016), which defines extremal varieties in P ⋀ 2 R n , we define a more general concept of extremal varieties of the real Grassmannian G p R n in P ⋀ p R n . This concept is based on the minimization of the sums of squares of the quadratic Plücker relations defining the Grassmannian variety as well as the reverse maximisation problem. Such extremal problems define a set of Grassmannian inequalities on the set of Grassmann matrices, which are essential for the definition of the Grassmann variety and its dual extremal variety. We define and prove these inequalities for a general Grassmannian and we apply the existing results, in the cases ∧ 2 R 2 n and ∧ n R 2 n . The resulting extremal varieties underline the fact which was demonstrated in Leventides et al. (2016, Linear Algebra Appl 461:139–162, 2014), that such varieties are represented by multi-vectors that acquire the property of a unique singular value with total multiplicity. Crucial to these inequalities are the numbers M n , p , which are calculated within the cases mentioned above. [ABSTRACT FROM AUTHOR]

A symmetric n × n matrix X is completely positive semidefinite (cpsd) if there exist d × d positive semidefinite matrices { P i } i = 1 n (for some d ∈ N ) such that X i j = Tr (P i P j) , for all i , j ∈ { 1 , ... , n }. The cpsd − rank of a cpsd matrix is the smallest d ∈ N for which such a representation is possible. It was shown independently in Prakash A, Sikora J, Varvitsiotis A, et al. [Completely positive semidefinite rank. 2016 Apr. arXiv:1604.07199] and Gribling S, de Laat D, Laurent M. [Matrices with high completely positive semidefinite rank. Linear Algebra Appl. 2017 May;513:122–148] that there exist completely positive semidefinite matrices with sub-exponential cpsd-rank. Both proofs were obtained using fundamental results from the quantum information literature as a black-box. In this work we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank. For this, we introduce matrix valued Gram decompositions for correlation matrices and show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra. Lastly, we show that this fact underlies and generalizes Tsirelson's results concerning the structure of quantum representations for extremal quantum correlation matrices. [ABSTRACT FROM AUTHOR]

A probability space is a pair (A , ϕ) where A is an algebra and ϕ is a state on the algebra. In classical probability, A is the algebra of linear combinations of indicator functions on the sample space, and in quantum probability, A is the Heisenberg or Clifford algebra. However, other algebras are of interest in noncommutative probability. After a short review of the framework of classical and quantum probability, other noncommutative probability spaces are discussed, in particular those associated with noncommutative space-time. [ABSTRACT FROM AUTHOR]

Protein structure calculations using nuclear magnetic resonance (NMR) experiments are one of the most important applications of distance geometry. The chemistry of proteins and the NMR data allow us to define an atomic order, such that the distances related to the pairs of atoms {i−3,i},{i−2,i},{i−1,i} are available, and solve the problem iteratively using a combinatorial method, called branch‐and‐prune. The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i−3,i−2,i, with radius given by the atomic distances di−3,i,di−2,i,di−1,i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di−3,i may not be precise or even not be available. Using conformal Clifford algebra, in addition to take care of NMR uncertainties, which implies that we have to calculate sphere intersections considering that their centers and radius may not be fixed anymore, we consider a more flexible atomic order, where distances di−3,i are replaced by dj,i, where j⩽i−3. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We establish some properties of normed modules over Clifford algebras and develop an approach to the study of generally invertible differential operators in Sobolev–Clifford modules. A similar approach was earlier proposed by the author for Sobolev spaces. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

We study concepts of stability associated to a smooth complex curve together with a linear series on it. In particular we investigate the relation between stability of the asso-ciated dual span bundle and linear stability. Our results imply that stability of the dual span holds under a hypothesis related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability. Finally, using our results we obtain stable vector bundles of slope 3, and prove that they admit theta-divisors. [ABSTRACT FROM AUTHOR]

In this paper we prove some facts about the geometric description of these structures, especially the generic rank of multiffinor structures called almost Clifford geometries, i.e A-modules, where A is a Clifford algebra, particularly Clifford algebra Cl(0; 3); Cl(1; 2); Cl(2; 1) or Cl(3; 0). Finally, we focus on A-structures, where A is a Clifford algebra Cl(0; 3) and describe their generalized geodesics. [ABSTRACT FROM AUTHOR]

Given a fixed binary form f(u,v) of degree d over a field k, the associated Clifford algebra is the k-algebra Cf=k{u,v}/I, where I is the two-sided ideal generated by elements of the form (αu+βv)d−f(α,β) with α and β arbitrary elements in k. All representations of Cf have dimensions that are multiples of d, and occur in families. In this article, we construct fine moduli spaces U=Uf,r for the irreducible rd-dimensional representations of Cf for each r≥2. Our construction starts with the projective curve defined by the equation wd=f(u,v), and produces Uf,r as a quasiprojective variety in the moduli space of stable vector bundles over C with rank r and degree dr=r(d+g−1), where g denotes the genus of C. [ABSTRACT FROM PUBLISHER]

The article discusses the structure of the exterior polymetric algebras which can be used in examining fermions in the presence of a gravity field. It describes the Clifford multiplication using rules based on the standard axioms of an associative algebra to calculate the products of any elements of the vector space, and explores the different properties of Clifford algebras. Moreover, it demonstrates how the property of relation can be inverted to express the products of generators of the Clifford algebra in terms of the exterior products of such generators.

Abstract: We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type. [ABSTRACT FROM AUTHOR]

We study special linear systems called ‘very special’ whose dimension does not satisfy a Clifford-type inequality given by Huisman. We classify all these very special linear systems when they are compounded of an involution. Examples of very special linear systems that are simple are also given. [ABSTRACT FROM PUBLISHER]

Introducing Clifford algebra as the mathematical foundation, a unified spatio-temporal data model and hierarchical spatio-temporal index are constructed by linking basic data objects, like pointclouds and Spatio-Temporal Hyper Cubes of different dimensions, within the multivector structure of Clifford algebra. The transformation from geographic space into homogeneous and conformal space means that geometric, metric and many other kinds of operators of Clifford algebra can be implemented and we then design the shortest path, high-dimensional Voronoi and unified spatial-temporal process analyses with spacetime algebra. Tests with real world data suggest these traditional GIS analysis algorithms can be extended and constructed under Clifford Algebra framework, which can accommodate multiple dimensions. The prototype software system CAUSTA (Clifford Algebra based Unified Spatial-Temporal Analysis) provides a useful tool for investigating and modeling the distribution characteristics and dynamic process of complex geographical phenomena under the unified spatio-temporal structure. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named “alphabetic (re)presentations.” The Clifford algebra generators are expressed as m-letter words written with a three-character or a four-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end, we briefly discuss the motivations of this work and outline some possible applications. [ABSTRACT FROM AUTHOR]

Abstract: Clifford theory of finite groups is generalized to association schemes. It shows a relation between irreducible complex characters of a scheme and a strongly normal closed subset of the scheme. The restriction of an irreducible character of a scheme to a strongly normal closed subset contains conjugate characters with same multiplicities. Moreover some strong relations are obtained. [Copyright &y& Elsevier]

This is an introduction to Clifford theory of induced representations from normal subgroups of finite groups. As an application, a complete explicit description of the irreducible representations of a class of metacyclic groups and of the groups with a subgroup N of index two (in terms of the representations of N) is given. [ABSTRACT FROM AUTHOR]

A regular orthogroup S with the property that D e = R e or D e = L e for any idempotent e∈ S is called a WLR-regular orthogroup. In this paper, we give constructions of such semigroups in terms of spined products of left and right regular orthogroups with respect to Clifford semigroups. WLR-cryptogroups and its special cases are also investigated. [ABSTRACT FROM AUTHOR]

One of the problems concerning entanglement witnesses (EWs) is the construction of them by a given set of operators. Here several multi-qubit EWs called stabilizer EWs are constructed by using the stabilizer operators of some given multi-qubit states such as GHZ, cluster and exceptional states. The general approach to manipulate the multi-qubit stabilizer EWs by exact(approximate) linear programming (LP) method is described and it is shown that the Clifford group play a crucial role in finding the hyper-planes encircling the feasible region. The optimality, decomposability and non-decomposability of constructed stabilizer EWs are discussed. [ABSTRACT FROM AUTHOR]

We obtain some estimates of the codimensions and PI-exponent of identities on the Clifford algebras associated to a quadratic form of a given rank, find a hook where the cocharacters of Clifford algebras lie, exhibit an identity that holds on the Clifford algebras of a given rank, and describe the multilinear identities of minimal degree for the Clifford algebras of rank 1. [ABSTRACT FROM AUTHOR]

A novel Chern–Simons E8 gauge theory of gravity in D = 15 based on an octicE8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern–Simons E8 gauge theory. We review the construction showing why the ordinary 11D Chern–Simons gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E8 Yang–Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E8 gauge bundle formulation was instrumental in understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E8 gauge theory remains unknown. We hope that the Chern–Simons E8 gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction. [ABSTRACT FROM AUTHOR]

Isotopic liftings of algebraic structures are investigated in the context of Clifford algebras, where it is defined a new product involving an arbitrary, but fixed, element of the Clifford algebra. This element acts as the unit with respect to the introduced product, and is called isounit. We construct isotopies in both associative and non-associative arbitrary algebras, and examples of these constructions are exhibited using Clifford algebras, which although associative, can generate the octonionic, non-associative, algebra. The whole formalism is developed in a Clifford algebraic arena, giving also the necessary pre-requisites to introduce isotopies of the exterior algebra. The flavor hadronic symmetry of the six u, d, s, c, b, t quarks is shown to be exact, when the generators of the isotopic Lie algebra $\mathfrak {su}(6)$ are constructed, and the unit of the isotopic Clifford algebra is shown to be a function of the six quark masses. The limits constraining the parameters, that are entries of the representation of the isounit in the isotopic group SU(6), are based on the most recent limits imposed on quark masses. [ABSTRACT FROM AUTHOR]

The theory of functions with values in Clifford algebras shows a lot of analogies to the complex function theory in the complex one-dimensional case. The class of holomorphic functions is now the set of null solutions of a generalized Cauchy-Riemann system, the class of monogenic functions. Analogously to the complex case one can define a derivative of monogenic functions by applying the adjoint Cauchy-Riemann operator. The main goal of this article is to study whether the derivative can be "inverted" and how monogenic primitives of monogenic functions can be constructed. [ABSTRACT FROM AUTHOR]

Abstract: If V is a commutative algebraic group over a field k, is a commutative ring that acts on V, and is a finitely generated free -module with a right action of the absolute Galois group of k, then there is a commutative algebraic group over k, which is a twist of a power of V. These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties. [Copyright &y& Elsevier]

Abstract: We present an algebro-geometric technique for constructing regular Clifford algebras A of global dimension four with associated point scheme consisting of a prespecified finite number of points. In particular, if A has more than one point in its point scheme, then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors; these cubic divisors correspond to regular Clifford subalgebras of A of global dimension three. If A has exactly a finite number, n, of distinct points in its point scheme, then and all these possibilities occur. We also prove that if a regular Clifford algebra R of global dimension has exactly a finite number, n, of distinct isomorphism classes of point modules, then n is odd if and only if R is an Ore extension of a regular Clifford subalgebra of R of global dimension . [Copyright &y& Elsevier]

Abstract: Within the framework of constructions for quantifying entanglement, we build a natural scenario for the assembly of multipartite entanglement measures based on Hopf bundle-like mappings obtained through Clifford algebra representations. Then, given the non-factorizability of an arbitrary two-qubit density matrix, we give an alternate quantity that allows the construction of two types of entanglement measures based on their arithmetical and geometrical averages over all pairs of qubits in a register of size N, and thus fully characterize its degree and type of entanglement. We find that such an arithmetical average is both additive and strongly super additive. [Copyright &y& Elsevier]

Abstract: The well-known classification of the Clifford algebras Cl(r, s) leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For s ⩾1 representations of Cl(r, s) on R 2N are obtained from representations on R N by adding two new generators while in passing from a representation of Cl(p,0) on R N to a representation of Cl(r,0) on R 2N the number of generators that can be added is either 1, 2 or 4, according as the Clifford algebra represented on R N is of real, complex or quaternionic type. We have expressed canonical forms of these representations in terms of the complex and quaternionic structures in the half dimension and we obtained algorithms for transforming any given representation of Cl(r, s) to a canonical form. Our algorithm for the transformation of the representations of Cl(8d + c,0), c ⩽7 to canonical forms is based on finding an abelian subalgebra of Cl(8d + c,0) and its invariant subspace. Computer programs for determining explicitly the change of basis matrix for the transformation to canonical forms are given for lower dimensions. The construction of the change of basis matrices uniquely up to the commutant provides a constructive proof of the uniqueness properties of the representations and may have applications in computer graphics and robotics. [Copyright &y& Elsevier]

It is well known that starting with real structure, the Cayley–Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteraton process p → p + 2 → p + 4 → ⋯, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens." The present paper aims at an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. The proof of the Periodicity Theorem is given in the case where the index of the generator of the algebra in question exceeds the order of the initial algebra. In contrast to earlier results, the fractal bundle flower structure, in particular sepals, bisepals, perianths, and calyces are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. The same concerns canonical two-layer pairing of prianth sepals and the relationship of fractal bundles of algebraic structure with the Hurwitz problem. [ABSTRACT FROM AUTHOR]

This paper is an updated version of our former article (Sommen, F., 1998, Curved Radon transforms in Clifford analysis. In: Clifford Algebras and their Applications in Mathematical Physics, Fund. Theories Phys., Vol. 94 (Dordrecht: Kluwer Academic Publishers), pp. 369–381). First of all, we study weighted integrals of functions over general surfaces of higher codimension whereby the weights take values in a Clifford algebra. We also introduce and study multi-linear Grassmann and Clifford algebras and apply them to the multi-linear Radon transform. In cases of symmetric multi-linear functions (homogeneous polynomials), we obtain a Clifford analysis generalization of the generalized Radon transform investigated by V.P. Palamodov (Palamodov, V.P., 1994, Radon transformation on real algebraic varieties. In: Gindikin, S. and Michor, P. (Eds) 75 Years of Radon Transform, Lecture Notes in Mathematical Physics, Vol. IV (Boston: International Press), pp. 252–262). [ABSTRACT FROM AUTHOR]

In this article we establish the basis for a Clifford analysis over finite reflection groups. [ABSTRACT FROM AUTHOR]

Let Cℓn be the (universal) Clifford algebra generated by el, ⋯ en, satisfying eiej + ejei = -2δij, i, j = 1, ⋯ n. The Dirac operator in Cℓn is defined by D = ∑i=0nei(∂/∂xi), where e0 = 1. The modified Dirac operator is introduced in (Eriksson-Bique and Leutwiler 2000, Hypermonogenic functions. In: Clifford Algebras and their Applications in Mathematical Physics, Vol. 2 (Boston: Birkhäuser), pp. 287–302) by Mƒ = Dƒ + k(Q′ƒ/xn), where ′ denotes the main involution in Cℓn and Qƒ is given by the decomposition f(x) = Pƒ(x) + Qƒ(x)en with Pƒ(x), Qƒ(x) ∊ Cℓn-1. A continuously differentiable function f: Ω → Cℓn is called hypermonogenic in an open subset Ω of ℝn+1, if xn Mƒ(x) = 0, for all x ∊ Ω. Paravector-valued hyper-monogenic functions are called H-solutions, see (Leutwiler, 1992, Modified Clifford analysis, Complex Variables, 17, 153–171). The power function xm (m ∊ ℕ0) is an H-solution. We give a Cauchy-type formula for H-solutions. Furthermore we derive the equation D(Δ(n-2)/2) g) = 0 for the restriction g of the hypermonogenic function ƒ to the hyperplane xn = 0. This equation has been considered by Laville, Lehman and Ramadanoff, see (2004, Analytic Cliffordian functions, Annales Academiae Scientiarium Fennicae. Mathematica, 29(2), 251–268; 1998, Holomorphic Cliffordian functions, Advances in Applied Clifford Algebras, 8(2), 323–340) and in a different context by J. Ryan, see (1990, Iterated Dirac operators in Cn, Zeitschrift fur Analysis und ihre Anwendungen, 9(5), 385–401), and others. [ABSTRACT FROM AUTHOR]

Let Γ be an n-dimensional rectifiable ∣F∣-regular surface in ℝn+1. Let u be a continuous ℝ0, n- valued function on Γ, where ℝ0, n is the Clifford algebra associated with ℝn. Then we prove that the weighted Cliffordian Cauchy transform (Multiple line equation(s) cannot be represented in ASCII text) has continuous limit values on F if and only if the truncated integrals (Multiple line equation(s) cannot be represented in ASCII text) converge uniformly in Γ as ϵ → 0. [ABSTRACT FROM AUTHOR]

Abstract: Applying the techniques of an earlier paper with Frenkel, we develop a geometric realization of spin representations and Clifford algebras. In doing so, we give an explicit parametrization of the irreducible components of Nakajima varieties of type D in terms of Young diagrams. We explicitly compute the geometric action of the Lie algebra and are able to extend the geometric action to the entire Clifford algebra used in the classical construction of the spin representations. [Copyright &y& Elsevier]

For any element a in a generalized 2 n –dimensional Clifford algebra $${\fancyscript C}$$ ℓ n ( $${\Bbb F}$$ ) over an arbitrary field $${\Bbb F}$$ of characteristic not equal to two, it is shown that there exits a universal invertible matrix P n over $${\fancyscript C}$$ ℓ n ( $${\Bbb F}$$ ) such that $$ P^{{ - 1}}_{n} D_{a} P_{n} = \phi {\left( a \right)} \in F^{{2^{n} \times 2^{n} }} $$ , where ϕ( a) is a matrix representation of a over and D a is a diagonal matrix consisting of a or its conjugate. [ABSTRACT FROM AUTHOR]

Clifford Theory concerns the representations over a field F of a finite group G and one of the normal subgroups N of G. It is often viewed as a series of reduction theorems. Clifford classes are certain equivalence classes of G/N-algebras over F. They have been used to describe the Schur indices of the irreducible characters of certain families of classical groups. In the present paper, we show how Clifford classes can also be used to explain and to reflect the reduction theorems of classical Clifford theory. We show that certain products of characters correspond to certain products of Clifford classes. We show that induction, restriction, inflation, and extension of the base field can all be defined naturally for Clifford classes. We prove that the properties of these elementary operations on Clifford classes imply all of the standard Clifford-theoretic reductions in the case when the field is ℂ. They also provide important information in the case when the field F is an arbitrary field of characteristic zero. [ABSTRACT FROM AUTHOR]

New higher–dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co–ordinates, the "finite part" distribution $$ Fpx^{\mu }_{ + } $$ on the real line and the generalized spherical means involving vector–valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector–valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self–contained, it offers an overview of the results found so far. [ABSTRACT FROM AUTHOR]

Given Clifford group operations and the ability to repeatedly prepare a single-qubit mixed state ρ, can one do universal quantum computation? We show a sharp threshold in the Hadamard “magic” direction of the Bloch sphere between those ρ allowing universal quantum computation, and those for which any calculation can be efficiently classically simulated. As a corollary, the ability to repeatedly prepare any pure state which is not a stabilizer state (e.g., any single-qubit pure state which is not a Pauli eigenstate), together with Clifford group operations, gives quantum universality. As motivation for this question, “magic state” distillation procedures can reduce the general fault-tolerance problem to that of performing fault-tolerant Clifford group circuits. [ABSTRACT FROM AUTHOR]

In general, Clifford algebras of quadratic forms are finite dimensional; therefore, their representations are easy to describe. However, for homogenous polynomial forms of degree d > 2, the situation is different because their Clifford algebras are infinite dimensional. In this article, we get a finite set of pairwise orthogonal idempotents of sum 1 in these algebras. This permits us to obtain interesting properties for d-dimensional representations of polynomial forms of degree d; for example, we show that the image C of the Clifford algebra by such representation is an endomorphism algebra of finitely generated projective Z(C)-module of d-rank, direct sum of finitely generated projective Z(C)-module of I-rank. Before establishing this, we give a new proof of the Poincaré-Birkhoff-Witt theorem for these algebras with the help of a general composition lemma. At the end of this work, we give a linearization of diagonal binary and ternary forms of degree d > 3. [ABSTRACT FROM AUTHOR]

We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper R and lower length λ scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl (2, 6, R) and complexified Clifford Cl C (4) algebra related to Twistors. A unified theory of all Noncommutative branes in Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization whose deformation parameter involves the lower/upper scale $$(\hbar \lambda / R)$$. Previous work led us to show from first principles why the observed value of the vacuum energy density (cosmological constant) is given by a geometric mean relationship $$\rho \sim L_{\rm Planck}^{-2}R^{-2} = L_{P}^{-4} (L_{\rm Planck} / R)^{2} \sim 10^{-122}M_{\rm Planck}^4$$, and can be obtained when the infrared scale R is set to be of the order of the present value of the Hubble radius. We proceed with an extensive review of Smith’s 8 D model based on the Clifford algebra Cl (1, 7) that reproduces at low energies the physics of the Standard Model and Gravity, including the derivation of all the coupling constants, particle masses, mixing angles, ....with high precision. Geometric actions are presented like the Clifford-Space extension of Maxwell’s Electrodynamics, and Brandt’s action related to the 8 D spacetime tangent-bundle involving coordinates and velocities (Finsler geometries). Finally we outline the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces (associated with the 8 D Phase space) as a Unified Field theory action candidate that should reproduce the physics of the Standard Model plus Gravity in the low energy limit. [ABSTRACT FROM AUTHOR]

Let k be a field, char k ? 2, F = k ( x ), D a biquaternion division algebra over k , and s an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form f ? I 2 ( F ) such that dim f = 8, [ C (f)] = [ D ], and f does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F ( D )/ F is not excellent. Also we prove that if A is a central simple K -algebra of degree 8 with an orthogonal involution s, then s is hyperbolic if and only if s K ( A ) is hyperbolic. Finally, let s be a decomposable orthogonal involution on the algebra M 2 m ( K ). In the case m = 5 we give another proof of the fact that s is a Pfister involution. If m = 2 n -2 - 2 and n = 5, we show that q s ? I n ( K ), where q s is a quadratic form corresponding to s. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms. [ABSTRACT FROM AUTHOR]

We describe maximal Clifford subsemigroups of the finite symmetric inverse semigroup In of the set Xn - {1,2, ..., n} and obtain their complete classification, determine the number and the order of such subsemigroups, give a criterion for two maximal Clifford subsemigroups of In to be isomorphic, and determine (up to isomorphism) the number of such subsemigroups. Further, both maximal ideal extensions and maximal nil extensions of any Clifford subsemigroup S of In are described. In particular, we show that S has a unique maximal ideal extension in In and, up to isomorphism, S has a unique maximal nil extension in In. [ABSTRACT FROM AUTHOR]

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli-Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner-Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space-time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated. [ABSTRACT FROM AUTHOR]

For finite groups G, it is shown that central simple G-algebras arise naturally from irreducible modules of twisted group algebras over fields of characteristic 0. This leads to an extension of Turull's Clifford theory with Schur indices that applies in the setting of projective representations of finite groups. This theory has the same properties as in the ordinary representation theory setting, perhaps the most interesting of which is a Schur index preserving bijection between certain sets of irreducible projective characters. [ABSTRACT FROM AUTHOR]

In this paper, over a field k, we give the structure theorem of the quantum double of a finite Clifford monoid through bicrossed products and quantum doubles of groups. By this result, it is shown that the quantum double of a finite Clifford monoid is semisimple (resp. von Neumann regular) if and only if the semigroup is a finite group and the characteristic p of k does not divide the order of this group. [ABSTRACT FROM AUTHOR]

The Schur multipliers of reflection groups were first constructed by I. Ihara and T. Yokonuma. Later, R. B. Howlett gave a more unified, but less concrete, construction for arbitrary Coxeter groups. In this paper, Clifford algebras are used to give a more unified approach than that originally used by I. Ihara and T. Yokonuma. [ABSTRACT FROM AUTHOR]

Regular semigroups S with the property eS c Se or Se c eS for all idempotents e ∈ S include all left and right Clifford semigroups. Characterizations of such semigroups are given and their structure investigated, in particular in terms of spined products of left and right Clifford semigroups with respect to Clifford semigroups. [ABSTRACT FROM AUTHOR]

The three-line theorem on the octonions is obtained, which generalizes the result of J. Peetre and P. Sjölin from the associative Clifford algebra to non-associative octonion algebra. [ABSTRACT FROM AUTHOR]

Clifford's algebra of the Minkowski space is constructed in the present paper. A method of solving the equations of motion within the formalism of this algebra is described. A solution of the equation of spin motion (the Bargmann–Michel–Telegdi equation) for the case of motion of a magnetic top in a constant uniform electromagnetic field is derived with the help of the algorithm constructed. [ABSTRACT FROM AUTHOR]

In this note, structures of null solutions of the polynomial Dirac operators D -λ D k , are studied, where D is the Dirac operator in is the identity operator. Explicit decompositions of null solutions of the polynomial Dirac operators in the respectively relevant subspaces are obtained which are used to derive their Taylor series expansions. The solutions of inhomogeneous equation p ( D ) f = g are discussed for a special class of R ( m )-valued continuous functions g . [ABSTRACT FROM AUTHOR]

Beginning in 1870 Charles Sanders Peirce published a series of papers on a “logic of relations,” which corresponded to a linear associative algebra. This algebra is related by a linear transformation to quaternions and thus to the C(3, 0) algebra of William Kingdon Clifford. This Clifford algebra contains the Pauli matrices and so constitutes an operator basis for the nonrelativistic quantum theory of spin one-half particles. A further unification is achieved by taking the wave functions themselves to be 2 × 2 matrices which are Peirce logical operators and also elements of the Clifford algebra. Thus we have discovered a direct path from the Peirce logic to quantum theory. A diagrammatic method follows from the Peirce/Clifford algebraic approach and is suitable for describing particle interactions. [ABSTRACT FROM AUTHOR]