Combinatorial aspects of Clifford algebra

Journal of Mathematical Physics, 2009

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 algebras generators are expressed as m-letter words written with a 3-character or a 4-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.

In this paper we focus on some combinatorial aspects of Clifford algebra and show how this algebra allows combinatorial theorems -like e.g. Sperner's lemma -to be "built into the algebraic background", and become part of the structure of the algebra itself. We also give an example of how cumbersome combinatorial proofs can be "mechanized" and carried out in a purely computational manner.

2004

CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C (B) -the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUMbased on Chevalley's recursive formula, and cmulRS -based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

Advances in Applied Clifford Algebras, 2014

We present different methods for symbolic computer algebra computations in higher dimensional (≥ 9) Clifford algebras using the CLIFFORD and Bigebra packages for Maple R . This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.

2018

Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprßigTrends Math, Birkhäuser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.

Advances in Applied Clifford Algebras, 2013

We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly demonstrate the benefit of developing problem solutions in a unified framework for algebra and geometry with the widest possible scope: from quantum computing and electromagnetism to satellite navigation, from neural computing to camera geometry, image processing, robotics and beyond.

Journal of Mathematical Physics, 2001

We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using such representations for their permutations obey Clifford statistics. The vectors supporting the Clifford algebras of permutations and rotations are plexors and spinors respectively. Physical spinors may actually be plexors describing quantum ensembles, not simple individuals. We use Clifford statistics to define quantum fields on a quantum space-time, and to formulate a quantum dynamics-field-spacetime unity that evades the compactification problem. The quantum bits of history regarded as a quantum computation seem to obey a Clifford statistics.

Linear Algebra and its Applications, 2015

In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial Φ(Z, X 1 , . . . , of degree d in n + 1 variables over some field F . We completely determine its structure in the following cases: n = 2 and d = 3 and either char (F ) = 3, 2 for some r, t, e ∈ F . Except for a few exceptions, this algebra is an Azumaya algebra of rank nine whose center is the coordinate ring of an affine elliptic curve. We also discuss representations of arbitrary generalized Clifford algebras assuming the base field F is algebraically closed of characteristic zero.

2008

Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl_{p, q} associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful "8-periodicity theorem" of Elie Cartan and Raoul Bott (with proofs).

Advances in Applied Clifford Algebras, 2011

The structure of a complex Clifford algebra is studied by direct sum decompositions into eigenspaces of specific linear operators.