Associative Composition Algebra/Transcendental paradigm

In the ancient geometry of the circle, the relation between angle above the horizon and altitude of a star is called the sine function. Though a part of the nature of a circle, it is not an algebraic function in the sense of being expressible as a finite sequence of additions, subtractions, multiplications, divisions, powers or roots. Thus it is called a transcendental function. The invitation of this function "sine" into the polite company of algebraic manipulations required a fundamental innovation: a constant "base" b > 0 raised to a variable exponent: This innovation was used by Leonard Euler in his Introduction to the Analysis of the Infinite (1748). But the research that led to this development is due to a Jesuit trio: Gregoire de Saint-Vincent, A. A. de Sarasa, and Marin Mersenne working in the previous century. They tackled the ancient problem of quadrature of the hyperbola, a prominent issue since Archimedes had shown the quadrature of the parabola millennia before.

Angle size and sector area are the same when the conic radius is √2. This diagram illustrates the circular and hyperbolic functions based on sector areas u.

Consider first a unit square area. Then consider those rectangles that have the same area as the square. If x and y are the sides of such a rectangle, the graph of y = 1/x represents the rectangles {(0,0), (x,0), (0, 1/x), (x, 1/x)}. The rectangle with x = y = 1 is the unit square s. Now suppose b > a > 1 and (b, 1/b) is the corner of a rectangle h while (a, 1/a) is the corner of a rectangle g. The rectangles can be viewed as squeezed forms of one another: h is s squeezed by a, g is s squeezed by b. Then h can be re-inflated to s by a−1, so that g is obtained from h by squeezing with a−1b.

The squeezing operation corresponds to a positive real number p > 0. The squeeze is generally viewed with p > 1 so that x expands and y contracts, all the while area is preserved. This property of preservation of area, called equi-areal mapping, brings squeezing into contact with translations, rotations, and shear mappings which share the property. The squeeze is not part of classical kinematics but appears in special relativity as a re-linearization of velocity after the finitude of the speed of light exposes the non-linearity of classical velocity addition with vectors.

Note the axis of symmetry L: x = y of the hyperbola xy = 1. A point (x, 1/x) on the hyperbola determines a hyperbolic sector S(1,x) delimited by L, the hyperbola, and the line from (0,0) to the point. The perpendicular projection from the point to L establishes the hyperbolic sine, sinh v, where v is the area of the sector S(1,x), commonly called the hyperbolic angle. The foot of the projection determines cosh v by the length of the diagonal from (0,0) to the foot. In accord with a circle of area 2π, the sinh and cosh are normalized by a factor of √2.

Infinite series edit

Leonard Euler provided infinite series as an access point to the transcendental functions:


Look what happens when f is required to be its own derivative: the derivative of the nth term is


Given a0,   where n! is the factorial. Take a0 = 1. Now   which Euler computed to be 2.71828… and is now designated mathematical constant e.

The function f is known as the exponential function


Euler also broke the sum into even and odd terms:  , where cosh takes the even terms, sinh the odd. The following lemma will be needed later.


proof: The odd terms of e−x turn negative, so they cancel in

  and add in  

Now squaring and adding in the right hand side of the lemma yields  

Exercises edit

  1. Use Euler's formula to show sine and cosine have alternating series.
  2. Using infinite series, show  
  3. Exchange triangles of area 1/2 to show a hyperbolic sector has the same area as a dented trapezoid under the hyperbola and against its asymptote.
  4. What are the merits and demerits of calling a squeeze mapping a "hyperbolic rotation" ?

Sample edit

For a sample of AC algebra, the following is offered: Let A = (R2, xy) be the real plane with quadratic form xy. Further, let A be equipped with component-wise addition and multiplication, making it a real algebra. Denote N(x,y) = xy in this case. Then


Thus N is said to compose over the multiplication in A, and A might be called a composition algebra. However, in this text, AC algebras have an involution called a conjugation, written x*, used to define N by N(x) = x x*. Nevertheless, the algebra A constructed above is very closely related to split-binarions described in the next chapter. The split-binarions are a normalized form of A, where the multiplicative identity is a unit distance from the origin, and it has some formal correspondences with the complex field C. In A, the quadratic form can be interpreted as a weight, so that a transformation leaving it invariant is an isobaric transformation, a name used in 1999 by Peter Olver (Classical Invariant Theory, page 217) to describe a squeeze mapping.

Categorical treatment edit

In his essay "Cayley Algebras" included in a larger work published by American Mathematical Society (ISBN 978-0-8218-4459-5), Guy Roos presents a sequence of exercises which give a categorical expression of composition algebras. Say that A is a composition algebra over field K, so that it has a norm n: A → K and for every a, b in A, n(ab) = n(a) n(b). Composition algebras are sometimes non-commutative, so A is presumed to have non-commutative multiplication though the addition operation is commutative. The exercises culminate in showing a composition algebra is an alternative algebra. This property is connected with the associativity proposition a(bc)=(ab)c with three universal quantifiers, except that in alternative algebras only two quantifiers hold, meaning that a=b or b=c in the expression. In particular, when   then the algebra is alternative.

Roos exercises edit

Definition: (a:b) = n(a+b) – n(a) – n(b)

  • 2(ab:ab) = (a:a)(b:b)
  • (ac:ad) = n(a)(c:d)
  • (ac:bc) = (a:b) n(c)
  • (ac:bd) + (ad:bc) = (a:b)(c:d)
  • (aa:d) + n(a)(d:1) = (a:1)(a:d)
  • (aa – (a:1)a + n(a)1 : d) = 0
  • aa – (a:1)a + n(a) = 0

Definition: The trace of an element a is t(a) = (a:1)

As a unital algebra, 1 is in A, and A has a basis as a linear space. Write e = 1e as the basis element associated with the multiplicative identity, 1 or one (in oral communication).

Definition: The conjugate of a is a* = (a:e)e – a.

  • (a*)* = a and n(a*) = n(a)
  • a + a* = t(a)
  • n(a) = a a*
  • (a:b) = (a*:b*)
  • (ac:d) + (ad:c) = ((a:1)c : d)
  • (ad:c) + (a*c : d)
  • (da:c) = (ca*:d)
  • (ax:y) = (x:a*y) and (xa:y) = (x:ya*)
  • (ab:1) = (a:b*) = (ba:1) so that t(ab) = t(ba)
  • t((ab)c) = (ab:c*) = (a:c* b*) = (ca :b*) = t((ca)b)
  • t((ab)c) = t(ca)b) = t(bc)a) = t(a(bc))
  • (ab)* = b*a*
    • For every c, ((ab)* :c) = (ab:c*) = (ca:b*) = (c:b*a*)
  • (a:b)c = b*(ac) + a*(bc)
    • For every d, (a:b)(c:d) = (b*(ac):d) + (a*(bc): d)
  • (a:b)c = (ca)b* + (cb)a*
  • n(a)c = a*(ac) = (ca)a*
  • (a*a)c = a*(ac) and a+a* in K implies a2c = a(ac)
  • (ca)a = ca2

Introduction · Binarions