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]}]


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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