# Associative Composition Algebra/Split-binarions

## Planar algebra

editThe equation jj=1 expresses an *involution*, an operation that returns to the original upon iteration. When 1 is taken as the identity matrix, then the matrix equation mm = identity has many solutions (even in the 2x2 case), and such a solution is an *involutory matrix*.

The split binarions use this idea of extra solutions (beyond 1 and minus 1) to generate the set of numbers {*x* + j*y*: *x,y* in R}. Component-wise addition and multiplication according to

- (
*u*+j*v*)(*x*+j*y*) =*ux*+*yv*+ j(*uy*+*xv*)

make a 2-algebra here called **split-binarions**, described as *split-complex numbers* in the Encyclopedia, there also provided with a list of synonyms.

To describe the invertible elements of the split-binarion plane, the two lines *x*=*y* and *y*=−*x* must be scratched from the plane. Of the four quadrants so formed, the one containing 1+0j is the most important as the square of any unit is found in this quadrant. Within it the set

*G*= {exp(*a*j):*a*in R} forms a one-parameter group:- exp(
*a*j) exp(*b*j) = exp((*a*+*b*)j).

*G* ∪ −*G* is the unit hyperbola but parametrized with hyperbolic functions.

The conjugate hyperbola is j*G* ∪ −j*G*, also given as

In division binarions *perpendicularity* and *orthogonality* are synonyms, but in split-binarions orthogonality differs geometrically but is consistent algebraically: Two units *z* and *w* are **orthogonal** if the real part of *zw** = 0. The bilinear form says <z,w> = 0. For example, for any *g* in *G*,

- g(jg)* = −j gg* = −j exp(aj) exp(−aj) = − j, which has zero real part.

**Exercises**:

- Show that the group of units
*U*=*F*x*P*x*G*where*P*is the multiplicative group of positive real numbers and*F*= {j, −j, 1, −1}, the four-group. - Show that
*x*+ j*y*is in the quadrant of the identity if and only if*y*< |*x*|. - Show that the effect of multiplying by j is to flip the plane in the diagonal x=y.
- For
*g*= cosh*a*+ j sinh*a*, show that as*a*increases the orthogonal points*g*and j*g*converge toward the asymptote.

## Simulaneity

editWhen Hermann Minkowski was developing his model of the universe using the concept of a worldline for the track in time of something, he argued that the *simultaneous space* of the moving thing depends on its velocity. Thus simultaneity is relative to moving observers. The orthogonality in split-binarions corresponds to the relation between a velocity vector and its peculiar simultaneous space. The term *hyperbolic orthogonality* has been used to distinguish it from perpendicularity. The simultaneous space is called a *simultaneous hyperplane* since it is a three-dimensional subspace of Minkowski’s universe.

The elements of *G* can be used to form a group action on the plane. The effect is sometimes called a *hyperbolic rotation* since for any constant *k*, the hyperbola {u : u u* = *k* ≠ 0} is an invariant set under *u* -> *gu*. But the action does not mingle the quadrants, so the term *rotation* is not appropriate. Another effect is that the dimension perpendicular to *y*=*x* is squashed or squeezed, as evidenced by the converging orthogonal vectors *g* and j*g* where *g* = exp(*a*j) and *a* is increasing. Thus the term *squeeze mapping* is applied when appropriate orientation is in place.

## Area

editGiven that j^{2} = +1, it follows that j^{n} is one (1) when *n* is even, and equals j when *n* is odd. Therefore

- as the powers of j separate the even and odd terms.

The variable *a* is a hyperbolic angle along a unit hyperbola This configuration is a normalized form of the natural hyperbola, where now the multiplicative identity is a unit distance from the origin, so sector areas are **half** the angle sizes due to the normalization.

Instead of squeeze mappings preserving areas in sectors of the natural hyperbola, the multiplication in D does the squeezing. The re-linearization of velocity addition in special relativity uses the parametrization of the unit hyperbola in D. Indeed, if two rapidities *a* and *b* are added, the result is their sum according to in D.

The notion of orthogonality in D is arithmetically consistent with the condition in C, but expresses instead hyperbolic orthogonality, the relation of a worldline to its simultaneous hyperplane. Though only two-dimensional, the split binarions contribute to understanding special relativity.

### Exercises

edit1. Matrix and σ is a squeeze mapping on R^{2}. Show that the matrix *S* provides a mapping that makes D and (R^{2}, xy) isomorphic as rings and quadratic spaces, but that *S* is *not* an isometry on the real plane with Euclidean metric.

2. For *K* ⊂ D, area(*K*) finite, and any *a* in R with *u* = exp(*a*j), show that the area of {*u k* : *k* in K} equals the area of *K*.

3. What does the hyperbolic angle have in common with the harmonic series Answer: no bound. Compare their geometry.

4. Draw the subgroup