Re: Multiplication and Division in Triplets; division by zero-sized vectors

From: Roger Beresford (mail_at_beresford22.freeserve.co.uk)
Date: 10/15/04 

Date: 15 Oct 2004 14:45:01 -0400

Robin Chapman <rjc@ivorynospamtower.freeserve.co.uk> wrote in message news:<cklh97$r3i$2@south.jnrs.ja.net>...
> Roger Beresford wrote:
>
> > "I can only add and subtract [triplets]." (W.R.Hamilton, speaking to
> > his son in the 1840's.)
snip
> > [Triplet] Multiplication uses the C3 group as a multiplication table.
>
> So this is the group algebra of C_3 over R or C.

Yes, and over Q, and over my "terplex", and over Octonions, because
the groups C3Q8, C3C3 and C3 composed with the Octonion loop can all
be "collapsed" to C3 as a "conservative Hoop algebra".
>
> > Conserved properties.
> > For any triplet X={x1,x2,x3}, the functions "Xs" and "Xp",
> > Xs={x1+x2+x3}, Xp={((x1-x2)^2 +(x2-x3)^2 +(x3-x1)^2)/2}, are conserved
> > properties (I call them "sizes") for triplet algebra, i.e. ABs =As Bs,
> > ABp =Ap Bp.
>
> If K is a field of characteristic zero, then the group algebra
> K C_3 is isomorphic to K x K(omega) or K x K x K according
> to whether K contains a root of x^2 + x + 1 = 0 or not. Here omega is
> a primitive cube root of unity. Your maps X |--> X_s and X |--> X_p
> are related to the projections of K C_3 onto the factors
> of this decomposition. In all cases X_s is the projection of
> X onto the first factor.
snip

 I agree, but the concept of a "field" has binary negation "built-in";
C3C3 involves "ternegation", providing a non-field algebra.
Incidentally, the "first factor" always disappears when a Moufang loop
is "collapsed" to a signed multiplication table.

snip
 
> Frobenius created the theory of group characters in his study
> of the factorization of the determinants of multiplication tables of
> arbitrary finite groups.

Frobenius showed that the determinant of the inverse of associative
multiplication tables is conserved. I have found that this extends to
Octonions and their compositions, but not to other square-associative
(alternative) tables. Tables with Moufang division properties conserve
the irreducible (linear, quadratic, cubic, etc) factors but not the
complex conjugate factors.

snip
> > If one of As and Ap is zero, A is a "zero-sized" triplet, and Ainv
> > appears to involve division by zero.
>
> "appears"? Such elements of the group ring are not invertible.

 I should have said "the expression for Ainv appears to involve
division by zero. However, this infinity can be 'factored out' ".

snip

> This seems to be analogous to the Moore-Penrose inverse of matrices.

Yes. I mentioned this in my (accidental) posting to this group dated
12 June 2001. M-P is formulated in terms of complex conjugates.
Express it as reals, and it involves sums of squares of square
sub-table determinants I have a generalized "det" that is not
resticted to complex numbers.

snip
> Then A A^+ is the idempotent generator of the ideal generated
> by A, and BA^+ is a solution of AX = B provided such a solution exists.
> (Your inverse BA_d lacks this property in general).

Non-trivial AX=B always has a solution in any conservative algebra
with more than one size, though it may involve intervals or
undetermined coefficients. Xinv splits into partial fractions and the
equations can be solved for the finite ones. I have not yet
generalized this approach, because I only recently discovered
"remainders".

> More generally one can consider a general group algebra C G
> over a finite group, and define a Moore-Penrose type "inverse"
> by decomposing CG as a product of matrix algebras and applying
> Moore-Penrose inversion in each. I was going to suggest this with
> C replaced by an arbitrary characteristic zero field K, but that
> would require Moore-Penrose inverses in matrix algebras over skew
> fields. I don't know if there are such things .... anyone?
> Of course there are no problems if, as here G is abelian,

Why bother? My engineer's approach is that I have already found a lot
of conservative non-abelian algebras, with division (and remainders in
some cases), simply(!) by finding the determinant factors and using
genTimes and genInverse. Using matrix formulations is going the long
way round.
AX=B may be a problem though, for non-abelians have repeated factors.

Thank you for your comments

Roger Beresford.

"A mathematician is a blind man in a dark room looking for a black cat
that is'nt there." (Charles Darwin.)

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