Clifford algebras and geometric algebra

The geometric algebra as defined by D. Hestenes is compared with a constructive definition of Clifford algebras. Both approaches are discussed and the equivalence between a finite geometric algebra and the universal Clifford algebra R _{p, q} is shown. Also an intermediate way to construct Clifford algebras is sketched. This attempt to conciliate two separated approaches may be useful taking into account the recognized importance of Clifford algebras in theoretical and applied physics.     

Isometric actions on the four dimensional Minkowski spacetime

In this paper we give representations of all connected Lie groups acting isometrically on the four dimensional Minkowski spacetime, up to conjugacy within the full isometry group of the space. For each obtained group, we study its induced orbits. Then we classify the Lie groups up to orbit equivalence.     

A Clifford algebra approach to real simple Lie algebras,II: The algebras with root system BCr

Received: 14 September 2001 / in final form: 28 April 2002 // Published online: 20 March 2003     

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.     

The structure of norm Clifford algebras

Orthomodular Hilbertian spaces are infinite-dimensional inner product spaces (E, 〈·, ·〉) with the rare property that to every orthogonally closed subspace U ? E there is an orthogonal projection from E onto U. These spaces, discovered about 30 years ago, are constructed over certain non-Archimedeanly valued, complete fields and are endowed with a non-Archimedean norm derived from the inner product. In a previous work [KELLER, H. A.—OCHSENIUS, H.: On the Clifford algebra of orthomodular spaces over Krull valued fields. In: Contemp. Math. 508, Amer. Math. Soc., Providence, RI, 2010, pp. 73–87] we described the construction of a new object, called the norm Clifford algebra C?(E) associated to E. It can be considered a counterpart of the well-established Clifford algebra of a finite dimensional quadratic space. In contrast to the classical case, C?(E) allows to represent infinite products of reflections by inner automorphisms. It is a significant step towards a better understanding of the group of isometries, which in infinite dimension is complex and hard to grasp. In the present paper we are concerned with the inner structure of these new algebras. We first give a canonical representation of the elements, and we prove that C? is always central. Then we focus on an outstanding special case in which C? is shown to be a division ring. Moreover, in that special case we completely describe the ideals of the corresponding valuation ring $\mathcal{A}$ . It turns out, rather unexpectedly, that every left-ideal and every right-ideal of $\mathcal{A}$ is in fact bilateral.     

The cross ratio and Clifford algebras

Motivated by recent quaternionic approach of Bobenko and Pinkall to the complex cross ratio we presenta simple method to eva]uate the cross ratio in the Euclidean space? ⁿ identifying the space with vectors generating the Clifford algebraC(n). We apply the Clifford cross ratio to describe discrete analogues of orthogonal nets in? ⁿ .     

The Clifford algebra of a metabolic space

The authors thank the Department of Mathematics of the University of Pisa for the hospitality accorded during the preparation of this paper.     

Isomorphism groups in Clifford algebras

A set of anticommuting multivectors in Clifford algebras can be taken as orthonormal basis set. The Clifford algebra generated by this basis is isomorphic to the original algebra. The non linear transformations between orthonormal basis sets form a group. In the four dimensionnal case six sets of five anticommuting multivectors are found. These sets yield 30 matrices defining basis sets. These matrices are representatives of left cosets, members of these cosets are related by permutation of rows. From the equivalence of all basis sets of multivectors it can be concluded that there is no canonical set of basis vectors in Clifford algebras.     

Minkowski Dimension and Cauchy Transform in Clifford Analysis

In the present paper we provide some conditions of a geometrical character for continuous extendibility of the Clifford–Cauchy transform to the boundary of a domain in the Euclidean space of higher dimensions if its density satisfies a H?lder condition. The criterion obtained in this work is an extension to a very general class of domains of a result, which has already become classical, obtained by Viorel Iftimie, who proved in 1965, for the case of a domain with compact Liapunov boundary, that the Clifford–Cauchy transform has H?lder–continuous limit values for any H?lder–continuous density. Received: August 15, 2006. Accepted: November 2, 2006.     

Between quantum Virasoro algebra $$\mathcal{L}_c $$ and generalized Clifford algebras

In this paper we construct the quantum Virasoro algebra generators in terms of operators of the generalized Clifford algebras C_n^k. Precisely, we show that can be embedded into generalized Clifford algebras. Junior Associate at The Abdus Salam ICTP, Trieste, Italy.     

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].     

The alternative Clifford algebra of a ternary quadratic form

We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion ring whose center is the ring of polynomials in one variable over the field of definition.     

The arens algebras of a nonassociative algebra

The Arens construction of dii algebra in the biduai bpcice of an algebra has provided in functional analysis a useful method for constructing an algebra containing a rich supply of idempotcnts As idera-potents are an Important too] in the study of non-associative algebras, we begin in this paper the study of the algebraic biduai of a nonassociative algebra. In particular, we obtain a partial spectral decomposition for a transcendental element of a power associative algebra by analyzing the biduai of the associative subaigebra it generates.     

Complexes of Dirac operators in Clifford algebras

In this paper we study complexes of k Dirac operators (or variations of Dirac operators) in the real or complex Clifford algebras i.e. complexes in which the first map is induced by the matrix where is the Dirac operator with respect to the variable . In particular we prove that, if , the complex in the case of 3 operators can be described in terms of relations coming from the so called radial algebra. Moreover we show that if the dimension m is less than 2k-1, then the Fischer decomposition does not hold. Received: 2 February 2000; in final form: 20 June 2000 / Published online: 25 June 2001     

Classification of Lie algebras of specific type in complexified Clifford algebras

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.     

Groups of spacetime transformations and symmetries of four-dimensional spacetime. I

The relativistic 4-interval (X-X ₍₀₎ ²=s ₂ ⁽⁰⁾ is interpreted as a 4-hyperboloid of radiuss ₍₀₎ and center at the pointX ⁽⁰⁾ that is formed by particles radiated isotropically from its center with velocities 0<1 whose positions in 4d spacetime are fixed at a proper times _(0)/c that is the same for all of them. Therefore, the 4-hyperboloid can be regarded as a mathematical model of an isotropically radiating source, and all transformations of the spacetime variables that leave its equation invariant have a physical meaning and determine the symmetry properties of 4d spacetime. These transformations form the group of motions of a rotating 4-hyperboloid. For constant radiuss ₍₀₎=const, its configuration space is the 8-dimensional bundleR(1,3)=R(1,3) (1,3), and the minimal group of motions isK=P O(1,3). It is shown that the well-known groupsP andO(1,3) are defined, respectively, only on the baseR(1,3) and only on the fiber (1,3) of the spaceR(1,3) and that the symmetry properties of 4d spacetime introduced by them are incomplete. The groupK extends the isotropy property of 4d spacetime to moving frames of reference. The group of spacetime transformations is extended to the case ofN bundles. It is shown that the new interpretation of the 4-interval makes it necessary to assume that the radiuss ₍₀₎ is variable. The groups of motion of a 4-hyperboloid of variable radius are constructed in the second part of the paper. They introduce new symmetry properties of 4d spacetime.D. V. Efremov Institute of Electrophysical Apparatus. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 458–475, September, 1994.