Associative Composition Algebra/Split-binarions

Planar algebra

The 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 + jy: x,y in R}. Component-wise addition and multiplication according to

(u+jv)(x+jy) = 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(aj): a in R} forms a one-parameter group:

exp(aj) exp(bj) = exp((a+b)j).

G ∪ −G is the unit hyperbola $x^{2}-y^{2}=1\ ,$ but parametrized with hyperbolic functions.

The conjugate hyperbola is jG ∪ −jG, also given as $\{x+jy:y^{2}-x^{2}=1\}.$

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:

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

Simulaneity

When 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 jg where g = exp(aj) and a is increasing. Thus the term squeeze mapping is applied when appropriate orientation is in place.

Area

Given that j² = +1, it follows that jⁿ is one (1) when n is even, and equals j when n is odd. Therefore

$\exp(aj)\ =\ \cosh a+j\sinh a$ as the powers of j separate the even and odd terms.

The variable a is a hyperbolic angle along a unit hyperbola $x^{2}-y^{2}=1.$ 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 $e^{aj}\ e^{bj}=e^{(a+b)j}$ 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

The group action of the right branch of the unit hyperbola on D corresponds to a squeeze mapping σ on R².

1. Matrix $S={\begin{pmatrix}1&1\\1&-1\end{pmatrix}}$ and σ is a squeeze mapping on R². Show that the matrix S provides a mapping that makes D and (R², 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(aj), 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 $\sum _{n=1}^{\infty }1/n\ ?$  Answer: no bound. Compare their geometry.

4. Draw the subgroup $\{pj^{n}e^{aj}:p>0,\ n=0,1,\ a\in R\}\subset U.$