Re: Question: SPACE of super-complex numbers

From: Roger Beresford (mail_at_beresford22.freeserve.co.uk)
Date: 08/22/04 

Date: 22 Aug 2004 01:19:15 -0700

"Robert J. Kolker" <robert_kolker@hotmail.com> wrote in message news:<otHVc.48521$mD.21020@attbi_s02>...
SNIP
> There are no 3 D division algebras over the reals. You have to go from
> complex number to quaternions.
>
> All of these manipulations were tried back in the middle of the 19-th
> century. Google on the history of quaternions and octernians. Also
> Google division algebras. Having a quotient is a strong constraint.

Moufang loops (all groups and Octonions) provide that constraint. They
are "mxm" Cayley multiplication tables with left and right
multiplicative inverses for every element. They multiply and divide
sets of unsigned coefficients A, and have "Frobenius conservation",
with Detm[A]Detm[B]=Detm[AB], where Detm is the determinant of the
multiplication table mapped with the coefficients. They become
algebras when another operation (generalized negation) "collapses" an
"mxm" table (iff it has r-fold symmetry) to an "(m/r)x(m/r)" table,
and the "m" unsigned coefficients to "m/r" signed coefficients. Their
multiplicative inverses Ai have Detm[A] as divisors; if this
factorises (into conserved "sizes") the inverse splits into partial
fractions. Division by zero occurs if any size becomes zero; it can be
avoided by working in a constrained sub-algebra (renormalization).
Real algebras have r=2; the four "real division algebras without
divisors of zero" (R, C, H, O) conserve the sum of their squared
elements. As they are monosized, they cannot renormalize. They do not
have (non-trivial) real divisors of zero because the sum of squares is
only zero in the {0,0...} case.
Every Group and Octonion defines a Hoop algebra [1] over the real
(r=2), terplex (r=3), complex (r=4), (etc.) numbers. Most of these are
"partial fraction division algebras"; Clifford, Davenport,
Pauli-sigma, etc, algebras are Hoops; Wedge (exterior) and Lie
algebras are obtained by constraining particular hoops.
There are many ways to go to multiple dimensional algebras, and some
of them are interesting, despite the mathematicians horror of
"division by zero".
Roger Beresford.
[1]http://library.Wolfram.com/infocenter/Mathsource/4894
"It was the secrets of heaven and earth that I desired to learn."
(Mary Shelley).

Relevant Pages

  • Re: Question: SPACE of super-complex numbers
    ... >> There are no 3 D division algebras over the reals. ... Having a quotient is a strong constraint. ... Division by zero occurs if any size becomes zero; ...
    (sci.math)
  • Re: 3 dimensional numbers !
    ... For associative algebras, the definition can be simplified as follows: ... The best-known examples of associative division algebras are the ... the reals themselves, ...
    (sci.math)
  • Re: Non-associative algebras
    ... >field theories based on non-assciative algebras. ... >is now being done on the use of octonions and other Cayley-Dickson ... book "On Quaternions and Octonions: Their Geometry, ... use of all four normed division algebras: ...
    (sci.physics.research)
  • Re: "A better understanding of n / 0" - plz comment on this
    ... each characteristic is a dimension, and a combination of dimensions is ... Quaternion and 8-element Octonion algebras were obtained from the ... These determinants are the divisors of the ... elements, and so they are only zero in the trivial case, hence ...
    (sci.math)
  • Re: "A better understanding of n / 0" - plz comment on this
    ... each characteristic is a dimension, and a combination of dimensions is ... Quaternion and 8-element Octonion algebras were obtained from the ... These determinants are the divisors of the ... elements, and so they are only zero in the trivial case, hence ...
    (sci.math)