This site contains some papers by Diane G. Demers on the orientation congruent algebra and the native exterior calculus of twisted differential forms.
The orientation congruent algebra (or "OC algebra") is a nonassociative, Clifford-like algebra which is essential to the computation of the exterior and Clifford products of twisted multivectors and multicovectors in their "native" or natural representation. (Twisted objects are also commonly known à la de Rham as "impair," in French, or "odd," in English.) In their native representation twisted objects are directly decomposable into an "orientation," which acts as a generalized, geometric sign, and a "gauge" or "content," which acts as a generalized, geometric magnitude.
Just as Cartan's exterior calculus is based the exterior algerbra, the native exterior calculus is based on the OC algebra. The native exterior calculus was developed to generalize Cartan's exterior calculus and treat twisted differential forms in their natural, native representation.
The author hopes that, by clearing up misunderstandings, this natural and general formulation of twisted objects will powerfully resolve smoldering controversies in the mathematical and physical literature. We aim to put an end to confusing arbitrary or manual sign choices with this general framework.
In addition, the foundation of the orientation congruent algebra lies within some new algebraic theory. Thus, the standard ("structurally-elliptic") Clifford algebra is the dual of the "structurally-hyperbolic" orientation congruent algebra. This Clifford algebra dualization naturally leads to the dualization of the standard (structurally-elliptic) Cayley-Dickson algebras to the structurally-hyperbolic Cayley-Dickson algebras.
The multiplication table for the orientation congruent algebra OC3 (or OC3,0) as a PNG graphic:
The multiplication table for the orientation congruent algebra OC5 (or OC5,0) as an Adobe Acrobat PDF file (49,525 bytes), version 1.3, June 23, 2005: OC50MultTab013.pdf
The OC2 algebra is isomorphic to the hyperbolic quaternions of Alexander MacFarlane. MacFarlane published a paper "Hyperbolic Quaternions," (1989-1900, Proc. Royal Society of Edinburgh 23:169-181) on his discovery of "a system of quaternions complementary to that of Hamilton, which is capable of expressing trigonometry on the surface of the equilateral hyperboloids." According to this cache of the brief biographical essay on MacFarlane, Emil Borel later observed that the 3D hyperboloid model of hyperbolic space describes the kinematic velocity space of special relativity. Also see the Wikipedia article: Hyperbolic Quaternions.
The series of orientation congruent algebras OCn starting with n = 1 and increasing are isomorphic to the the double or split-complex numbers for n = 1, the hyperbolic quaternions for n = 2, and their generalizations to higher dimensions for n > 2. These generalizations are part of the sequence of "structurally-hyperbolic" Cayley-Dickson algebras that are dual in a natural sense to the usual Cayley-Dickson algebras, which may then be called "structurally-elliptic." For more on this see below under the heading "Structurally-Hyperbolic Algebras."
Title:
The Orientation Congruent Algebra: A Nonassociative Clifford-Like Algebra
Abstract.
The correlated grade form of twisted blades (twisted simple multivectors)
faithfully renders in symbols their native geometric structure. The discovery
of this paper's nonassociative Clifford-like algebra was driven by trying to calculate
exterior products of straight and twisted multivectors directly in a basis of
this form. The key was found to be the orientation congruent (OC) algebra. This
paper is being published electronically in about ten sections, each offered as soon
as written. In this first section we axiomatize the orientation congruent algebra
by generators and relations. The next section derives the sign factor function sigma and proves that the Clifford product times it is the multiplication of an explicitly Clifford-like algebra isomorphic to the orientation congruent algebra. Later
sections are planned to show how to calculate the OC product in Mathematica
and Clical; to define the orientation congruent contraction operators, deduce
their properties, derive other expressions for them, and use them to compute the
OC product within the exterior algebra using a modified Cartan decomposition
formula; to develop the algebra's product sequence graph with labeled edges; to
derive a predictor of a null associator as a function of the grades of the three
elements in it; to prove the associomediative property of the algebra's counit; to develop matrix representations (under a nonassociative matrix product) of the orientation congruent product; and to discuss the motivating application per se and as inspiration for the first set of axioms.
This paper is available for downloading or reading online. It will be split by section into separate PDF files. Each file will be offered as soon as it is written. As the paper approaches completion the finished sections may be combined into one file. Revisions may occur so be sure to get the latest versions of each section.
Section 1: An Axiom System for the OC Algebra
Version 1.3, June 27, 2005: Adobe Acrobat PDF file (295,623 bytes),
OriCon013ch01.pdf
Note: Contrary to the above plan, this paper will not be posted section-by-section and may possibly not be developed further beyond version 1.5 below. But some essential material from this paper will incorporated into the newer paper on the native exterior calculus
Title:
The Orientation Congruent Algebra
Part I: A Nonassociative Clifford-Like Algebra
Abstract. Similar to Version 1.3.
Table of Contents
Complete paper: The Orientation Congruent Algebra: A Nonassociative Clifford-Like Algebra
Version 1.5, December 4, 2005: Adobe Acrobat PDF file (865,403 bytes),
OriCon015ch01-09.pdf
Note: There are few errors in this draft; but none of them is fatal to the thrust of this research. An errata sheet or a revised draft paper will eventually be posted here.
The orientation congruent algebra is also a member of the large class of algebras known as noncommutative Jordan algebras. I have proved this by a brute force case-by-case examination using Wolfram Research's Mathematica software. Although the code executes in a reasonable time, it would be good to have an analytical proof. The Mathematica code and an explanation of it are available upon request.
An intriguing aspect of the OC algebra: in contrast to the Clifford algebra, it provides a simple and natural representation for the Hodge star dual operator * (unfortunately rendered here in superscript rather than baseline position) and its inverse as
*u = \Omega^{-1} @ u,Here u is an arbitrary differential form, \Omega is the volume form, @ is the OC product (@ is a short ASCII substitute for my convention, the AMS LaTeX operator symbol \circledcirc), and the inverse of \Omega (written above as \Omega^{-1}) is taken with respect to the OC product. Note that, for pseudo-Riemannian metrics the definition of the Hodge star and its inverse may need to be interchanged depending on which of the two possible conventions is observed.
*^{-1} u = u @ \Omega.
The text file linked here on the motivating application contains a vector space foretaste of the native exterior calculus. The terminology and square bracket notation used here will be revised in the draft paper on the native exterior calculus.
Text file: Motivate.txt
I am currently writing a paper on the native exterior calculus of twisted differential forms. As mentioned above in the introduction to this website, the orientation congruent algebra is integral to developing the native exterior calculus. This paper, "Exterior Calculus in the Image of Odd Forms with the Orientation Congruent Algebra," will treat two problems in electrodynamics as examples.
An abstract of this draft paper: ExtCalc.pdf.
Per the above introduction to this website, "Structurally-Hyperbolic Algebras Dual to the Cayley-Dickson and Clifford Algebras or Nested Snakes Bite Their Tails" is the title of another pending paper.
An abstract of this draft paper:
Nested_080715_abs.pdf.
The latest draft of this paper from July 15, 2008:
Nested_080715.pdf.
All Breakout™ puzzles, physics and math papers on this Web site are the work of Diane G. Demers (felicity@sdf.lonestar.org) and are copyrighted (©) material. All rights are reserved. The Breakout™ name and logo are trademarks (™) owned by Diane G. Demers.