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Non-Abelian Vector multiplication "pseudo-powers".

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Roger Beresford

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Aug 10, 2006, 11:39:17 AM8/10/06

Background.
Associative (and a few alternative) Cayley tables provide
"symmetry-conserving" vector division algebras (I call them "Hoops".
See http://library.wolfram.com/infocenter/MathSource/6198/, [1] which
is intended to be comprehensible to non-Mathematica users.) The
conserved symmetries (I call them "sizes") are the real factors of the
determinant of the table mapped with the vector. Abelian tables with
some quadratic sizes have polar duals with radii (square roots of the
quadratic sizes) that multiply, and associated angles that add, on
vector multiplication. This allows the definition of powers and roots.
Sizes are raised to the power, and angles are multiplied by the power,
before reverting to the Cartesian (vector) form.
Appendix 1 demonstrates the C4, Davenport and C3C2 algebras as
output from a Mathematica session. Using an arbitrary integer vector A,
functions from [1] are used to give the polar form "pA", show reversion
to A, and calculate A*A and the square-root of A*A, recovering A
correctly. [1] includes 25 Abelian hoop algebras for which
(A^(1/p))^p==A, even for complex A and p (not demonstrated here!)
(Contact me directly if you wish to see the Mathematica instructions
that I used.)
Each quadratic size releases a degree of freedom; this is taken up
by an angle, which is a hidden variable for the Cartesian form.
Quadratic sizes can often be written as sums of squared "fragments"
such as (a-c)^2 in the C4 example. Sums of 2 or 3 squares have polar
forms related to those in the examples. (Differences of 2 squares give
hyperbolic duals related to Study numbers.)
Hoops may be relevant to particle physics, with abelian
multiplication "ejecting remainders" to conserve symmetry, just as
particle decays and interactions eject other particles.

Problem.
Non-Abelian tables have repeated roots [J.McKay, SMR 2/2/2002] and
these often define sizes with more than three squared "fragments". I
cannot formulate duals with conserved sizes and additive angles; too
many degrees of freedom are released by a repeated root. I speculate
that non-abelian powers may introduce uncertainty via their degrees of
freedom.
Polar forms that calculate "pseudo-powers" and "pseudo-roots" can be
defined by splitting the quadratic sizes into sets of 2 or 3 squares,
and defining non-conserved radii. These behave correctly (provided that
angles do not exceed 2Pi), insofar as (A^p)^(1/p)==A; but A^2 differs
from A.A. Appendix 2 demonstrates this for the quaternion and D3
algebras.

"Determinants" for Cayley tables rather than Matrices.
Whilst the determinant is a function of an array viewed as a matrix,
and abelian arithmetic is employed in its calculation, the hoop tables
are not matrices but are Cayley Algebra Tables. I speculate that
another function (?cadet?) may exist, involving non-abelian arithmetic.
This might replace each repeated size by two conserved sizes, allowing
true powers to be calculated, with the uncertainty expressed in terms
involving ab-ba. (Hamilton developed quaternions when he wrote (a-b)^2
as a^2 - a b - b a + b^2 .)
As the non-associative Octonions and Split-Octonions are
conservative hoops, "cadet" might even involve non-associative
arithmetic. Re-inforcing my point about matrices, it is well known that
Octonions cannot be described by matrix multiplication without
extending the definition of matrices. Dave Rusin has recently shown, in
the SMS "Sqrt discepancy" thread, that the matrix determinant can only
have trivial extensions.
The table for a non-abelian algebra, such as D3, can be expressed as
directions di; each entry is the product of two directions, dij=di**dj.
The elements are expressed as directions di and associated coefficients
ci, so a vector A is d1c1+d2c2+d3c3+d4c4+d5c5+d6c6. The mapped table
entries are dij cij. Only the (abelian) cij are used when the matrix
determinant is calculated, the dij are ignored and the result is a
scalar. I have failed to formulate "cadet" as a procedure that includes
the (non-abelian) multiplication of the directions dij. I have also
failed to factorize the polynomial created from the D3 table using
non-commutative multiplication for d4, d5 & d6. D3 can be created from
relators a^3=b^2=1, with the single non-commutative relation ba->aab;
this has not helped me.

Questions.
(1) Is there anything published on conserved properties for non-Abelian
or nonassociative algebras?
(2) Can anyone see a way to handle non-Abelian powers and roots? As
they rotate and scale the polar forms, it may be possible to pre- and
post-multiply by a half angle.

Any comments or advice will be appreciated.

Roger Beresford.

Appendix 1. Roots & powers in C4, Davenport, C3C2 Algebras.

In the following demonstrations, a vector A is multiplied by itself to
give AA. Provided that the sizes are positive, Sqrt[AA] = A.

The C4 Table is {{1,2,3,4}, {2,3,4,1}, {3,4,1,2}, {4,1,2,3}}. It
describes complex algebra, with each complex number expressed as real
part = a-c, imaginary part = b-d.

C4 algebra. Polar form for {a,b,c,d} is
{a + b + c + d, a - b + c - d, ((a - c)^2 + (b - d)^2, ArcTan[a - c, b
- d]},
Vector reversion from {a,b,c,d} is
{a + b + 2 Sqrt[c]Cos[d], a - b + 2 Sqrt[c] Sin[d], a + b - 2 Sqrt[c]
Cos[d], a - b - 2 Sqrt[c] Sin[d]}/4,
If A= {3., -1.,1., 2.}, Polar A is", {5., 3., 13.,-0.98279},
A*A=", {6.,-2., 11., 10.}, Polar A*A is {25., 9., 169., -1.96559},
Note that 2 sizes are squared, and the angle is doubled.
The square root is correct AA^(1/2) = A ,{3.,-1.,1.,2.}

The Davenport Table is {{1,2,3,4}, {2,-1,4,-3}, {3,4,-1,-2},
{4,-3,-2,1}}.
It describes two complex planes intersecting at the origin.

Davenport algebra. Polar form for {a,b,c,d} is
{(b+c)^2+(a-d)^2, ArcTan[a-d,b+c], (b-c)^2+(a+d)^2,ArcTan[a+d,b-c]},
Vector reversion from {a,b,c,d} is
{Sqrt[a] Cos[b]+Sqrt[c] Cos[d], Sqrt[a] Sin[b]+Sqrt[c] Sin[d],
Sqrt[a] Sin[b]-Sqrt[c] Sin[d], -Sqrt[a] Cos[b]+Sqrt[c] Cos[d]}/2,
If A = {3.,-1.,1.,2.}, Polar A is {1.,0.,29.,-0.38051},
A*A= {11.,-10.,10.,10.} Polar A*A is ,{1.,0.,841.,-0.761013},
Note that 2 sizes are squared, and the angle is doubled,
The square root is correct Sqrt[AA] = A, {3.,-1.,1.,2.}

The C3C2 Table is {{1,2,3,4,5,6}, {2,3,1,5,6,4}, {3,1,2,6,4,5},
{4,5,6,1,2,3}, {5,6,4,2,3,1}, {6,4,5,3,1,2}}.
It is a new algebra that may be related to quarks.

C3C2 algebra. Polar form for {a,b,c,d,e,f} is {a + b + c + d + e + f,
a + b + c - d - e - f, ((a - b + d - e)^2 + (b - c + e - f)^2 + (-a +
c - d + f)^2)/2, ArcTan[2 a - b - c + 2 d - e - f, Sqrt[3](-b + c - e +
f)], ((a - b - d + e)^2 + (- a + c + d - f)^2 + ( b - c - e + f)^2)/2,
ArcTan[2 a - b - c - 2 d + e + f, Sqrt[3] (-b + c + e - f)]},
Vector reversion from {a,b,c,d,e,f} is
{a + b + 2 Sqrt[c] Cos[d] + 2 Sqrt[e] Cos[f],
a + b + 2 Sqrt[c] Cos[d + 2 Pi/3] + 2 Sqrt[e] Cos[f + 2 Pi/3],
a + b + 2 Sqrt[c] Cos[d - 2 Pi/3] + 2 Sqrt[e] Cos[f - 2 Pi/3],
a - b + 2 Sqrt[c] Cos[d] - 2 Sqrt[e] Cos[f],
a - b + 2 Sqrt[c] Cos[d + 2 Pi/3] - 2 Sqrt[e] Cos[f + 2 Pi/3],
a - b + 2 Sqrt[c] Cos[d - 2 Pi/3] - 2 Sqrt[e] Cos[f - 2 Pi/3]}/6,
If A = {3.,-1., 1., 0., 2.,-1.}, Polar A is", {4., 2., 7.,
0.33347,31.,.89102}
A*A = ", {3., -4., 11., 6.,10., -10.}, Polar A*A is", {16., 4., 49.,
-0.666946, 961.,1.78204},
Note that 3 sizes are squared, and the angles are doubled.
The Square root is correct Sqrt[AA] = A", {3., -1.1.0., 2.`, -1.}

Appendix 2. Pseudo-Roots and Pseudo powers in Quaternion and D3
algebras.

In these demonstrations, even though the sizes of A are positive,
Sqrt[AA] is not A though the shapes are the same.

The Quaternion Table is {{1,2,3,4}, {2,-1,4,-3}, {3,-4,-1,2},
{4,3,-2,-1}}.
Complex quaternions (real a, pure imaginary b,c,d) have the Minkowski
metric.

Quaternion Algebra. Polar form for {a,b,c,d} is
{a^2+b^2+c^2+d^2, ArcTan[a,b], c^2+d^2, ArcTan[c,d]},
Vector reversion from {a,b,c,d} is
{Sqrt[a-c] Cos[b],Sqrt[a-c] Sin[b], Sqrt[c] Cos[d],Sqrt[c] Sin[d]},
If A = {1.,3.,4.,1.}, Polar A is {27.,1.24905,17.,0.244979},
tovec[toPolar[A]] = A, {1.,3.,4.,1.},
AA = {-25.,6.,8.,2.}, Polar AA is {729.,2.90605,68.,0.244979},
Note that only the first size is squared, and the angles are not
doubled,
rA2=Sqrt[AA] ≠ A,{0.508843,4.30057,2.85011,0.350864},
Polar rA2 is {27.,1.45302,8.24621,0.122489},
Only the size (27) matches A.

The D3 algebra is {{1,2,3,4,5,6}, {2,1,6,5,4,3}, {3,4,5,6,1,2},
{4,3,2,1,6,5}, {5,6,1,2,3,4}, {6,5,4,3,2,1}}. This is the non-abelian
twin of C3C2.

Dihedral D3 Algebra. Polar form for {a,b,c,d,e,f} is
{a+b+c+d+e+f, a-b+c-d+e-f,
((a-c)^2-(b-d)^2+(c-e)^2+(-a+e)^2-(d-f)^2-(-b+f)^2)/2,
ArcTan[2 a-c-e,Sqrt[3] (-c+e)],
((a-c)^2+(c-e)^2+(-a+e)^2)/2,
ArcTan[2 b-d-f,Sqrt[3] (-d+f)]},
Vector reversion from {a,b,c,d,e,f} is
{ (a+b+4 Sqrt[e] Cos[d]),
(a-b+4 Sqrt[e] Cos[f]),
(a+b+4 Sqrt[-c+e] Cos[d+2 Pi/3]),
(a-b+4 Sqrt[-c+ e] Cos[f+2 Pi/3]),
(a+b+4Sqrt[e] Cos[d-2Pi/3]),
(a-b+4 Sqrt[-c+e] Cos[f-2 Pi/3])/6},
If A ={3.,-1.,1.,0.,2.,-1.}, Polar A is
{4.,8.,2.,0.523599,3.,-2.0944},
A*A = {15.,-9.,11.,-6.,14.,-9.},Polar A*A
is,{16.,64.,4.,0.804634,13.,-2.0944},
Note that only the first 3 sizes are squared, and the angles are not
doubled,
The Square root is incorrect. Sqrt[AA]
={3.16481,-0.244299,0.98834,-0.244299,1.84685,-1.5114},
Polar rAA is 4.,8.,2.,0.402317,3.60555,-1.0472}
Only the first 3 sizes match A.

I apologize for any proof-reading errors. Tired old eyes!. R.

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