rt.representation theory – Can a non-distributive algebraic system be represented as matrices? Answer

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rt.representation theory – Can a non-distributive algebraic system be represented as matrices?

For instance, can the following 4-dimensional “number system” (which I would call “anti-split numbers”) be represented as matrices? It is commutative and associative but not distributive.

  • Numbers are represented by pairs $(a,b)$with $(1,0)$ being multiplicative unity and $(0,0)$ being zero.

  • The unit curve (lemniscate) has the following equation: $\left(x^2+y^2\right)^2=x^2-y^2$ (with the last sign changed to “+” the system becomes usual complex numbers).

  • Argument of a number is twice the area under unit curve between radius-vector of the number and $x$ axes: $\arg (a,b)=\frac{ab}{a^2+b^2}$

  • Modulus of the number is the ratio of its radius-vector and a radius-vector of a point on the unit curve that lays in the same direction: $M(a,b)=\frac{a^2+b^2}{\sqrt{2 a^2-\left(a^2+b^2\right)}}$

  • Angle (direction) of the number is $\phi (z)=\arctan \left(\frac{2 \arg z}{\sqrt{1-4 (\arg z)^2}+1}\right)$

  • Coordinates can be obtained from argument and modulus: $z=(M(z) \cos(\phi(z)) \sqrt{\cos^2(\phi(z))-\sin^2(\phi(z))},M(z) \ sin(\phi(z)) \sqrt{\cos^2(\phi(z))-\sin^2(\phi(z))})$

  • When multiplying the numbers, arguments are added up while moduluses get multiplied.

  • The components $(a,b)$ are allowed to be complex so to make the system closed under multiplication under the above formulas.

The following Mathematica code defines multiplication function:

arg[a_, b_] := (a b)/(a^2 + b^2)
mod[a_, b_] := (a^2 + b^2)/Sqrt[2 a^2 - (a^2 + b^2)]
\[Phi][A_] := ArcTan[(2 A)/(1 + Sqrt[1 - 4 A^2])] // FullSimplify
angle[A_] := \[Phi][A]
X[m_, A_] := m Cos[angle[A]] Sqrt[Cos[angle[A]]^2 - Sin[angle[A]]^2]
Y[m_, A_] := m Sin[angle[A]] Sqrt[Cos[angle[A]]^2 - Sin[angle[A]]^2]
Multiply[{a1_, b1_}, {a2_, b2_}] := {X[mod[a1, b1] mod[a2, b2], 
   arg[a1, b1] + arg[a2, b2]], 
  Y[mod[a1, b1] mod[a2, b2], arg[a1, b1] + arg[a2, b2]]}

The following code visualizes multiplication (both the terms and product should have real components to be visible on the plot):

p1 := {2, 1/2}
p2 := {1, 1/4}
p := Multiply[p1, p2] // N
ContourPlot[{(x^2 + y^2)^2 == x^2 - y^2, (x^2 + y^2)^2 == 
   y^2 - x^2}, {x, -4, 4}, {y, -4, 4}, Axes -> True, 
 AxesLabel -> Automatic, Frame -> None, 
 Epilog -> {Gray, Point[p1], Blue, Point[p2], Red, Point[p]}]

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The plots above show the red dot, a product of gray and blue dots.

Algebraic properties.

  • The system is commutative and associative, yet not distributive

  • The system has infinity divisors, for instance, $(\pm a,\pm a)$. These elements have infinite modulus. Whether their square should be considered infinity or improper element with infinite modulus and finite argument depends on the chosen compactification.

  • Some identities: $(0,1)=(-i,0)$, $(0,i)=(-1,0)$, $(0,1)^2=(-1,0)$.

  • The system possibly has zero divisors, such as $(a,i)$ as they have zero modulus. But I’m not sure.


That said, I wonder whether it is possible to represent such a system with matrices? Maybe with additional tricks but in a way that would be possible to generalize application of analytic functions to matrices to these numbers as well?

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