Extended Grassmann and Clifford Algebras

Abstract

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 Cℓ p,q can be embedded in Cℓ p+1,q+1 , 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.

Non-Associativity in the Clifford Bundle on the Parallelizable Torsion 7-Sphere

“…The products here introduced are immediate generalization of the results in, e.g., Cederwall, Bengtsson, Rooman, Preitschopf, Brink [4,6,13], as well as other ones obtained by Toppan, Günaydin, Lukierski, Ketov, de Wit, Nicolai, Gursey, and others [7,14,15]. Finally, objects described here provide immediate generalizations of the instanton Hopf fibration and Lounesto spinor field classification [10] as well as generalizations of Clifford algebras [16] and the Lounesto spinor field classification in eight dimensions [17].…”

Lineshapes and artifacts in Multidimensional Fourier Transform of arbitrary sampled NMR data sets

“…It should be pointed out that, as an extension of complex 1D FT, ND case of MFT requires hypercomplex notation to properly describe quadrature. In the general N-dimensional case MFT quadrature could be shortly described using commutative Clifford Fourier Transform [37][38][39].…”

Clifford Algebra Cℓ 3(ℂ) for Applications to Field Theories

“…Then we will prove that such an algebra is sufficient to obtain all the algebraic properties of the 3-dimensional and 4-dimensional multivectorial calculus initiated by Grassmann [17,18]. We will also see the 3-dimensional and 4-dimensional multivectorial analysis as a mathematical opportunity of this algebra.…”

“…A similar description exists with the exterior algebra (or Grassmann algebra) involving n-forms, i.e. n sorts of entities [17,18]. Moreover, true vectors and pseudo vectors are fully regrouped in our presentation.…”