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.     

Extended Grassmann and Clifford Algebras

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras can be embedded in which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota’s bracket, relating it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms. R. da Rocha is supported by CAPES     

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.     

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.     

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.      

Rigidity of Clifford Minimal Hypersurfaces

In this paper, we prove some rigidity theorems for Clifford minimal hypersurfaces in a unit sphere.     

Rigidity of Clifford Minimal Hypersurfaces

In this paper, we prove some rigidity theorems for Clifford minimal hypersurfaces in a unit sphere.Received March 18, 2002; in revised form December 25, 2002 Published online October 15, 2003     

Explicit Matrix Realization of Clifford Algebras

In this paper, we construct associative subalgebras ${{L_{2}}{n}(\mathbb{R})}$ of the real ${2^{n} \times 2^{n}}$ matrix algebra ${{M_{2}}{n}(\mathbb{R})}$ , which is isomorphic to the real Clifford algebra ${C \ell_{0},n}$ for every ${n \in N}$ .     

Spectral Characterizations of Cliford Minimal Hypersurfaces

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Representations of Clifford Algebras with Hyperbolic Numbers

The representations of Clifford algebras and their involutions and anti-involutions are fully investigated since decades. However, these representations do sometimes not comply with usual conventions within physics. A few simple examples are presented, which point out that the hyperbolic numbers can close this gap.      

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.      

Reductions in Computational Complexity Using Clifford Algebras

A number of combinatorial problems are treated using properties of abelian null-square-generated and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λ^k, where Λ is an appropriate nilpotent adjacency matrix, the k-cycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class P. The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and #P-complete to class P in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph.     

Clifford Algebras of Forms of Higher Degrees

The aim of this lecture is to introduce Clifford algebras of polynomial forms of higher degrees. We recall that these algebras are in general of infinite dimension, and we give a basis depending on a given basis of the underlying vector space. We then show that, though they contain large free associative algebras, we may construct finite dimensional representations of these algebras, also called linearizations of the polynomial form. If the polynomial form is, in a certain sense, non degenerate, the dimensions of these representations are multiples of the degree of the form. In the end, we recall some results known for the special case of a binary cubic form with at least one simple zero, when explicit computations can be done: the Clifford algebra is an Azumaya algebra of rank 9 over its center, which is the algebra of functions over a cubic curve depending on the given cubic form.     

Metric and Involution Scores of Clifford Algebras

The notion of a metric score is introduced as a separate tally of the total number of standard (Grassmann) basis elements spanning a Clifford algebra \({C\ell_{p,q}}\) that square to + 1 and ?1. A closed-form expression is derived for any given vector space dimension n = p+q. This is then generalized to reversion and Clifford-conjugation. A central application is that two real Clifford algebras are isomorphic if and only if their metric scores are identical.