Unitary Spaces on Clifford Algebras

For the complex Clifford algebra (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These representations take into account the structure of unitary space on Clifford algebra. The work of N.M. is supported in part by the Russian President’s grant NSh-6705.2006.1.     

Norms and Generating Functions in Clifford Algebras

Generating functions are commonly used in combinatorics to recover sequences from power series expansions. Convergence of formal power series in Clifford algebras of arbitrary signature is discussed. Given , powers of u are recovered by expanding (1 − tu)⁻¹ as a polynomial in t with Clifford-algebraic coefficients. It is clear that (1 − tu)(1 + tu + t ² u ² + ...) = 1, provided the sum (1 + tu + t ² u ² + ...) exists, in which case u ^m is the Cliffordalgebraic coefficient of t ^m in the series expansion of (1 − tu)⁻¹. In this paper, conditions on for the existence of (1 − tu)⁻¹ are given, and an explicit formulation of the generating function is obtained. Allowing A to be an m × m matrix with entries in , a “Clifford-Frobenius” norm of A is defined. Norm inequalities are then considered, and conditions for the existence of (I − tA)⁻¹ are determined. As an application, adjacency matrices for graphs are defined with vectors of as entries. For positive odd integer k > 3, k-cycles based at a fixed vertex of a graph are enumerated by considering the appropriate entry of A ^k . Moreover, k-cycles in finite graphs are enumerated and expected numbers of k-cycles in random graphs are obtained from the norm of the degree-2k part of tr(1 − tu)⁻¹. Unlike earlier work using commutative subalgebras of , this approach represents a “true” application of Clifford algebras to graph theory.      

Idempotent structure of Clifford algebras

Spinor spaces can be represented as minimal left ideals of Clifford algebras and they are generated by primitive idempotents. Primitive idempotents of the Clifford algebras R _{p, q} are shown to be products of mutually nonannihilating commuting idempotent % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaIXaaabaGaaGOmaaaaaaa!3DBD!\[{\textstyle{1 \over 2}}\]2}}\](1+e _T ), where the k=q–r _q–p basis elements e _T satisfy e _T ²=1. The lattice generated by a set of mutually annihilating primitive idempotents is examined. The final result characterizes all Clifford algebras R _{p, q} with an anti-involution such that each symmetric elements is either a nilpotent or then some right multiple of it is a nonzero symmetric idempotent. This happens when p+q<-3 and (p, q)(2, 1).     

Ideal Structure of Clifford Algebras

The structures of the ideals of Clifford algebras which can be both infinite dimensional and degenerate over the real numbers are investigated.      

Interior Multiplications and Deformations with Meson Algebras

A meson algebra is involved in the Duffin wave equation for mesons in the same way as a Clifford algebra is involved in the Dirac wave equation for electrons. Therefore meson algebras too should have geometrical properties after the manner of Grassmann. Actually it is possible to define interior multiplications with similar properties, and deformations too. Every meson algebra is a deformation of a neutral meson algebra, in the same way as (almost) every Clifford algebra is a deformation of an exterior algebra. Some applications follow: the PBW-property is proved for all meson algebras, the injectiveness of Jacobson’s diagonal morphism is proved with the minimal hypothesis, and the existence of Lipschitz monoids is established at least for meson algebras over fields.      

The Clifford algebra of differential forms

This paper reviews Clifford algebras in mathematics and in theoretical physics. In particular, the little-known differential form realization is constructed in detail for the four-dimensional Minkowski space. This setting is then used to describe spinors as differential forms, and to solve the Klein-Gordon and Kähler-Dirac equations. The approach of this paper, in obtaining the solutions directly in terms of differential forms, is much more elegant and concise than the traditional explicit matrix methods. A theorem given here differentiates between the two real forms of the Dirac algebra by showing that spin can be accommodated in only one of them.     

Classification of Extended Clifford Algebras

Considering tensor products of special commutative algebras and general real Clifford algebras, we arrive at extended Clifford algebras. We have found that there are five types of extended Clifford algebras. The class of extended Clifford algebras is closed with respect to the tensor product.     

Quantum Clifford Hopf gebra for quantum field theory

We give arguments for the necessity to employ quantum Clifford Hopf gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design of convolution Hopf gebras is given as a program for future research.     

Asymptotic Behavior in Lie Nilpotent Relatively Free Algebras and Extended Grassmann Algebras

The present paper is devoted to estimating the codimensions c_n for the variety of associative algebras given by the identity [x₁, …, x_l] = 0 for odd l. The characteristic of the ground field is different from 2 and 3. The construction of the so-called extended Grassmann algebra is applied very effectively.

     

Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics for standard pseudo-hermitian spaces H_p,q. Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of H_p,q. We study the topology of the projective quadrics and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2ⁿ⁻¹, 2ⁿ⁻¹). The spinorial group Spin U(p,q) is imbedded into SU(2ⁿ⁻¹, 2ⁿ⁻¹). At last, we study the natural imbeddings of the projective quadrics      

Algebras Generated by Scalar <Emphasis Type="Italic">K</Emphasis>-atic Forms and Their Linear Forms

A linear form with an N-elements basis set {e _i ; i = 1,...,N} generates an algebra which is that of multivectors, provided some commutation relation is defined to give a meaning to the outer product of the basis vectors. If, moreover, an inner product of sets of K basis vectors is also introduced, for a mapping producing a 0-form, a geometric algebra is obtained. The algebra has thus two basic numbers to define its dimension: the dimension N of the basis set and the dimension K of the number of elements to be multiplied together to obtain a scalar. If the dimension K refers to the order of the power of [e _i ] ^K to obtain the scalar we will say that we have a K-atic algebra, the best known example is when the scalar form is a quadratic expression; these algebras are said to have a metric which in general is either diagonal or at least symmetric. Otherwise if the dimension K refers again to the number of different basis vectors to be multiplied together in (with j ≠ i and in general all subindexes different) then we obtain a simplectic algebra where the best known case is also when K = 2 and the metric in this case is antisymmetric. In the present paper we define these sets of algebras, give the commutation relations for the algebras with a K-atic scalar form and relate the results to the best known examples of current use in the literature.     

Algèbre de Clifford symplectique. Revêtements du groupe symplectique. Indices de Maslov et spineurs symplectiques

Sans résumé

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This article is the developed text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

The Graded Structure of Nondegenerate Meson Algebras

Meson algebras are involved in the wave equation of meson particles in the same way as Clifford algebras are involved in the Dirac wave equation of electrons. Here we improve and generalize the information already obtained about their structure and their representations, when the symmetric bilinear form under consideration is nondegenerate. We emphasize their parity grading. We calculate the center of these meson algebras, and the center of their even subalgebra. Finally we show that every nondegenerate meson algebra over a field contains a group isomorphic to the group of automorphisms of the symmetric bilinear form.      

Clifford Product and Lorentzian Plane Displacements In 3-Dimensional Lorentzian Space

In this paper, by defining Clifford algebra product in 3-dimensional Lorentz space, L ³, it is shown that even Clifford algebra of L ³ corresponds to split quaternion algebra. Then, by using Lorentzian matrix multiplication, pole point of planar displacement in Lorentz plane L ² is obtained. In addition, by defining degenerate Lorentz scalar product for L ³ and by using the components of pole points of Lorentz plane displacement in particular split hypercomplex numbers, it is shown that the Lorentzian planar displacements can be represented as a special split quaternion which we call it Lorentzian planar split quaternion.      

Geršgorin type theorems for quaternionic matrices

This paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices.     

Circles and Clifford Algebras

Consider a smooth map of a neighborhood of the origin in a real vector space into a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra. We also describe a connection between our result and the Hurwitz–Radon theorem about sums of squares.     

An algorithm for the Cartan-Dieudonné theorem on generalized scalar product spaces

We present an algorithmic proof of the theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonné-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.