Associative Composition Algebra/Printable version

Associative Composition Algebra

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Introduction

An associative composition algebra, or AC algebra, (A, +, ×, *) is an associative algebra (A, +, ×) that is at the same time a composition algebra (A, *). In terms of the axioms of a mathematical structure, these algebras are characterized by

$\forall a,b,c\in A\ \ (ab)c=a(bc),$ and

$\forall a,b\in A\ \ (ab)^{*}(ab)=(aa^{*})(bb^{*}).$

A composition algebra is constructed as an algebra over a field F, and is equipped with a mapping $N:\ A\rightarrow F\ \ {\text{by}}\ \ a\mapsto aa^{*}.$ The axiom involving the conjugation (*) expresses N's group homomorphism property between the multiplicative groups of A and F.

Associative composition algebras come in three levels: unarion, binarion, and quaternion. The unarion level in this text will be either R, the real numbers, or C, the complex numbers. At the unarion level, the conjugation is the identity mapping, and $N(a)=a^{2}$ at this level.

Five additional associative composition algebras will be described in this wikibook: There is just one binarion and one quaternion AC algebra over C, but two of each over R. Two of the latter are division algebras and have the greatest literature. The extra two over R are split composition algebras; they possess null vectors $(aa^{*}=0).$

Associative Composition Algebras over R or C

Reflections in the lines of symmetry of a square generate the dihedral group of order 8.

• R = real numbers
• C = division binarions, also known as complex numbers
• D = split binarions, a.k.a. split-complex numbers
• T = bibinarions, a.k.a. bicomplex numbers, a.k.a. tessarines
• H = division quaternions, a.k.a. Hamilton’s real quaternions
• Q = split-quaternions, a.k.a. coquaternions
• B = biquaternions, a.k.a. complex quaternions

The terms tessarine and coquaternion were used by James Cockle writing in Philosophical Magazine, in the wake of Hamilton's lectures on H and B. The term binarion, an essential linguistic insertion, was used by Kevin McCrimmon in his book A Taste of Jordan Algebras(2004).

Any AC algebra may provide arguments to a linear fractional transformation, here called a homography as is traditional in projective geometry. The demonstration begins with Mobius transformations of division binarions and the construction of a cross-ratio homography. Three-dimensional kinematics is expressed with quaternion homographies. Cosmological symmetry expressed by conformal mapping is described with biquaternion homography.

In an AC algebra A, $\ \{x\in A:x\ =\ x^{*}\}$ is the field of scalars, either R or C in this text. In the case of R, it is the real line embedded in A. For an element $x\in A,\ \ (x+x^{*})/2$ is the scalar part of x, and for real algebras it is the real part of x.

Each algebra A has a bilinear form on $N(x+y)-N(x)-N(y)$ written

$\langle x,\ y\rangle \ :={\tfrac {1}{2}}(\ (xy^{*})+(x^{*}y)\ ).$

Given $y\in A,\ \{x:\ \langle x,y\rangle \ =\ 0\}$ is the set of elements orthogonal to y, and given

$a,b\in A,\ \ \{x:(x-a)(x-a)^{*}=bb^{*}\}$ is a quadratic set in A.

The distinction between these two types of sets is reduced with Möbius transformations and later in the chapter Homographies by embedding A in its projective line.

The following lemma uses complex numbers C to prepare one of the approaches to Q:

Lemma: If two lines are inclined by θ radians, then the composition of reflections in these lines is a rotation of 2θ radians.

Algebraic proof: Lines L and M intersect at X, which is taken as (0,0) ∈ C, where L is aligned with the real axis. Reflection in L is complex conjugation. M passes through $e^{i\theta }$ (say), and reflection in M comes by rotating it to L, then conjugating, and rotating back to the original position of L:

$z\mapsto (e^{-i\theta }z)^{*}e^{i\theta }\ =\ e^{i\theta }z^{*}e^{i\theta }\ =\ e^{2i\theta }z^{*}.$

Reflection in L goes first, $z\mapsto z^{*}.$ Then reflection in M is

$z^{*}\mapsto e^{2i\theta }z^{**}\ =\ e^{2i\theta }z.$ The composition is $z\mapsto e^{2i\theta }z,$ a rotation of twice the angle of inclination.

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: $y=b^{x}.$ 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

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

$f(x)\ =\ \sum _{n=0}^{\infty }a_{n}x^{n}\ .$

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

$n\ a_{n}\ x^{n-1},\ {\text{so}}\ f\ =\ f^{\prime }\ {\text{implies}}\ a_{n-1}\ =\ na_{n}\ {\text{and}}\ \ a_{n}=a_{n-1}/n.$

Given a₀, $a_{n}\ =\ a_{0}/n!$ where n! is the factorial. Take a₀ = 1. Now $f(1)=\sum _{n=0}^{\infty }1/n!$ which Euler computed to be 2.71828… and is now designated mathematical constant e.

The function f is known as the exponential function

$f(x)\ =\ \exp(x)=e^{x}\ =\ \sum _{n=0}^{\infty }x^{n}/n!.$

Euler also broke the sum into even and odd terms: $e^{x}\ =\ \cosh x+\sinh x,$, where cosh takes the even terms, sinh the odd. The following lemma will be needed later.

Lemma: $\cosh 2x=\cosh ^{2}x+\sinh ^{2}x.$

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

$\cosh x=(e^{x}+e^{-x})/2$ and add in $\sinh x=(e^{x}-e^{-x})/2.$

Now squaring and adding in the right hand side of the lemma yields $(e^{2x}+e^{-2x})/2\ =\ \cosh 2x.$

Exercises

1. Use Euler's formula to show sine and cosine have alternating series.
2. Using infinite series, show $\ \forall x\in R,\ \cos(ix)=\cosh x\ {\text{and}}\ \sin(ix)=i\sinh x.$
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

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

$N(x_{1},y_{1})N(x_{2},y_{2})=(x_{1}y_{1})(x_{2}y_{2})=x_{1}x_{2}y_{1}y_{2}=N((x_{1},y_{1})(x_{2},y_{2})).$

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

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 $ab^{2}=(ab)b\ \ {\text{and}}\ \ a^{2}b=(ab)b,$ then the algebra is alternative.

Roos exercises

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

Binarions

The division binarions C is the field of complex numbers:

$z=x+yi,\ \ i^{2}=-1,\ \ x,y\in R,\quad z^{*}=x-yi.$

Several academic journals and many university texts are dedicated to the function theory of C, for example the Wikibook Complex Analysis.

Division binarions

With $z=x+iy,\ \ zz^{*}=x^{2}+y^{2}=N(z)$ is the square of the Euclidean distance from 0 to z. Furthermore, the vectors from 0 to w and z in C are perpendicular, z ⊥ w, when $zw^{*}+wz^{*}=0.$ These features make C an ideal vehicle for display of topics in Euclidean geometry.

Proposition: The diagonals of a w:rhombus are perpendicular.

proof: The four binarions $\{0,z,w,z+w\},\ \ zz^{*}=ww^{*}$ form a rhombus. One diagonal is z + w and the other is parallel to z – w. They are perpendicular because

$(z+w)(z-w)^{*}+(z-w)(z+w)^{*}=0.$.

Proposition: A Euclidean plane isometry is either a translation z → z + t or a rotation such as $z\rightarrow \omega z,\quad \omega \omega ^{*}=1.$

Note that a rotation about p is obtained by arithmetic with $z\mapsto \omega (z-p)+p=\omega z+p(1-\omega )$

where the last expression shows the mapping equivalent to rotation at 0 and a translation. Therefore, given direct isometry $z\mapsto \omega z+a,$ one can solve $p(1-\omega )=a$ to obtain $p=a/(1-\omega )$ as the center for an equivalent rotation, provided that $\omega \neq 1$, that is, provided the direct isometry is not a pure translation.

Möbius transformations

Möbius transformations act on the projective line over division binarions. The points on this line use projective coordinates: (a,b)~(c,d) if there is a non-zero u such that ua=c and ub=d. This binary relation ~ on pairs of division binarions is an equivalence relation, where an equivalence class is written [a:b] for any pair (a,b) in the class. No point in the projective line corresponds to (0,0).

As a projective linear transformation, a Möbius transformation may be written

$[z,w]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=[az+bw,\ cz+dw].$

The point [z,0] = [1,0] corresponds to the point at infinity with respect to the rest of C, which is represented by [z,w]=[zw⁻¹, 1].

Exercise: Show that the Möbius transformation takes infinity to a/c.

For other points, let w = 1 so

$[z,1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=[az+b,\ cz+d]=\left[{\frac {az+b}{cz+d}},1\right]$ when z ≠ −d/c.

To avoid singular transformations, ad −bc is taken to be non-zero. Particular transformations coming under the Möbius umbrella include:

• Case 1: b=c=0, d=1. a>1 magnification, 0<a<1 contraction, a=−1 reflection in 0, aa*=1 rotation
• Case 2: c=0, a=d=1, b=t translation of division binarion plane by t in C
• Case 3: a=d=0, b=c=1 multiplicative inversion of C extended to 0 and infinity.

Note that the rotation z to uz leaves [0,1] and [1,0] fixed. Furthermore, any two distinct points p and q can be placed to these polar opposites by

$[z,1]{\begin{pmatrix}1&1\\-q&-p\end{pmatrix}}=[z-q,\ z-p].$

The image of a third point r cannot be [1,1] since p ≠ q, but r can be moved there: Let $w={\frac {r-p}{r-q}}.$ Then

$[r,1]{\begin{pmatrix}w&1\\-wq&-p\end{pmatrix}}=[rw-wq,\ r-p]=[1,1].$

Thus the constructed transformation maps p,q,r to infinity, 0, 1 respectively. Applied to a fourth division binarion z, the image is the cross ratio [z,p,q,r].

Exercises:

1. Construct the transformation taking −i to [1,0], 0 to [i,1] and i to [1,1]. What is the image of the unit disk zz* < 1 under this transformation?
2. Two points in the projective line are fixed with a rotation. Show that a Möbius transformation that leaves three points fixed must be the identity mapping.
3. If μ is a circle or line and g is a Möbius transformation, then μg is a circle or line.
4. If the image of z under the cross ratio transformation of p, q, and r is a real number, then the four points z, p, q, and r lie on a circle or line.

Hamilton

Nearly two centuries ago (in 1834 and 1837) William Rowan Hamilton wrote on algebraic couples as he explored the formation of division binarions from real numbers. See his essays Theory of Conjugate Functions or Algebraic Couples edited by David R. Wilkins. Hamilton wrote the product of a couple as

$(a_{1},\ a_{2})(b_{1},\ b_{2})=(a_{1}b_{1}-a_{2}b_{2},\ a_{1}b_{2}+a_{2}b_{1}).$

This is equation 37, from Transactions of the Royal Irish Academy, volume 17, page 93 of Wilkins text.

This approach to establishing the field of division binarions ("complex numbers") was taken by w:Raymond Wilder in 1965 in Introduction to the Foundations of Mathematics, second edition, page 62. The same author, in 1981, identified the use of "ordered pairs (a,b) of real numbers, a and b along with rules for operating with such pairs" as a forced origin of a new concept, by Hamillton, in his Mathematics as a Cultural System (page 33).

Euclid contradicted

There are more parallel lines in hyperbolic geometry than in Euclidean geometry.

The parallel postulate of Euclid has been phrased as a unique parallel existing off a given line and passing through a given point. The geometry of w:Lobachevski provides an infinite number of lines through a point, not a given line, and parallel to it. In the illustration, consider the given blue arc, and the pink arc illustrating an arc not intersecting the blue arc.

The division binarions enable a model of a hyperbolic plane in the unit disk. The geodesics in this model are circular arcs that meet the unit circle at right angles. The motions of the model are Mobius transformations that preserve the disk. These transformations are represented by homographies with $uu^{*}-vv^{*}=1.$

Indeed, for a point [z : 1] in the projective line over division binarions, the action of SU(1,1) is given by

${\bigl [}\;z:\;1\;{\bigr ]}\,{\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}=[\;u\,z+v^{*}:\,v\,z+u^{*}\;]\,=\,\left[\;{\frac {uz+v^{*}}{vz+u^{*}}}:\,1\;\right]$

since in projective coordinates $(\;u\,z+v^{*}:\;v\,z+u^{*}\;)\thicksim \left(\;{\frac {\,u\,z+v^{*}\,}{v\,z+u^{*}}}:\;1\;\right)~.$

Writing $\;suv+{\overline {suv}}=2\,\Re {\mathord {\bigl (}}\,suv\,{\bigr )}\;,$ division-binarion arithmetic shows

${\bigl |}u\,z+v^{*}{\bigr |}^{2}=S+z\,z^{*}\quad {\text{ and }}\quad {\bigl |}v\,z+u^{*}{\bigr |}^{2}=S+1~,$

where $~S=v\,v^{*}\left(z\,z^{*}+1\right)+2\,\Re {\mathord {\bigl (}}\,uvz\,{\bigr )}~.$ Therefore, $~z\,z^{*}<1\implies {\bigl |}uz+v^{*}{\bigr |}<{\bigl |}\,v\,z+u^{*}\,{\bigr |}~$ so that their ratio lies in the open disk.

Bibinarions

A construction of a doubled algebra was initiated by L. E. Dickson and recounted by A. A. Albert. The method presumes an algebra with conjugate * and produces one of double the dimension and a new conjugation: (u, v)* = (u*, −v) where u* denotes the original conjugation. The new algebra has products given by

$(u,v)\times (w,x)=(uw-vx^{*},ux+vw^{*}).$

Starting with a field and its conjugation, a sequence of algebras can be so constructed. The division binarions arises when the starting field is the real numbers $\mathbb {R} .$ Since the reals have no conjugation, the identity is substituted, and the conjugation arises as above. Continuing with the construction, using the binarion conjugate, quaternions are obtained as will be seen. However, the binarion conjugation may be forgotten (identity substituted), and bibinarions produced according to the Dickson/Albert method given above.

A bibinarion is then a pair of division binarions (u, v) with conjugation (u, v)* = (u, −v). The norm of a bibinarion is then

$(u,v)(u,v)^{*}=(u,v)(u,-v)=(u^{2}-v(-v),\ u(-v)+vu)=(u^{2}+v^{2},\ 0).$

Notice that this norm is a division binarion, and is not the kind of norm that produces a metric.

Furthermore, with i² = −1 in C, a bibinarion (u, i u) has zero norm. Such an element is called a null vector. The bibinarions form a split algebra since some elements are null vectors.

The product of two bibinarions is commutative since the generating conjugation is the identity. Most remarkably, there is bibinarion j = (0, i) with j² = (0, i)² = (−i², 0) = +1. The two-dimensional subalgebra of bibinarions on basis {1, j } is called split binarions.

History

The idea of an algebra with two imaginary units that commute was considered in mid-19th century Britain. Hamilton used a commuting h with his biquaternions. James Cockle saw that the square of the product hi of imaginary units was plus one, thus creating "a new imaginary in algebra" as he wrote in Philosophical Magazine in 1848. His use of the letter j,   j² = +1, has been widely adopted. Although Hamilton provided a vocabulary of vector operations (including the del operator), these explorations preceded set theory, group theory, and the unfolding of mathematical notation. With 1 on the real axis, the two imaginary units h and i, and their product hi, Cockle's commutative algebra T (tessarines) has a real basis of four elements. By the end of the 19th century tessarines and quaternions were referred to as hypercomplex numbers.

In 1892 Corrado Segre introduced bicomplex numbers in Mathematische Annalen (v 40: 455 to 67).

The division binarion basis of this algebra is used in the Dickson-style construction of biquaternions.

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

Quaternions

William Rowan Hamilton's real quaternions H and biquaternions B are constructed from pairs of division binarions or bibinarions, respectively. These operations are defined:

${\text{Multiplication:}}\ (w,x)(y,z)=(wy-xz^{*},\ wz+xy^{*}),$

${\text{Conjugation:}}\ (w,x)^{*}=(w^{*},-x),$

${\text{so that}}\ N(w,x)=(w,x)(w^{*},-x^{*})=ww^{*}+xx^{*}=N(w)+N(x).$

A third quaternion algebra Q = split-quaternions is a variant of H and a subalgebra of B. The following chapter explores split-quaternions through exercises.

H and B were both described by W. R. Hamilton in his Lectures on Quaternions (1853). AC algebra Q was described by James Cockle and called coquaternions. For a time H, B, and Q had special profiles in their use as AC algebra, but matrix rings were exploited in the twentieth century to provide linear representations for them, and thus absorb them into the larger study of linear algebra. Indeed, Q is ring isomorphic to M(2,R), the 2 × 2 real matrices, and B is ring isomorphic to M(2,C), the 2 x 2 complex matrices. Representation of H uses the context in B. In the Linear Algebra the idea of composition is visible with the determinant of a matrix, which has a similar property.

Division quaternions

In the notation of Hamilton, with $w=a+bi,\ z=c+di,$ (w,z) is written a + bi + cj + dk, where the products $ij=k=-ji,\ jk=i=-kj,\ ki=j=-ik$ can be confirmed, and noted for anticommutativity. The set {i, j, k} has been taken as the basis of space in presentations of kinematics, mechanics, and physical science.

Furthermore, $i^{2}=j^{2}=k^{2}=-1.$ In fact,

$(q\in H\land q^{2}=-1)\equiv (q=xi+yj+zk\land x^{2}+y^{2}+z^{2}=1),$

so there is a sphere S² of imaginary units in H.

Say that u is one of them, then the complex arithmetic of Euler's formula gives $e^{au}=\cos a+u\sin a.$ In the quaternion context, e^au is a versor, and versors are the points of elliptic space, a geometry entirely devoted to rotations. W. K. Clifford was an exponent of elliptic geometry, and much more, until his flame was extinguished at age 34.

For vectors in V ⊂ H, anticommutivity means perpendicularity:

$\forall a,b\in H\ \ ab+ba=0\equiv (a,b\in V\land a\perp b).$

Lemma: if a and b are square roots of minus one and a ⊥ b, then aba = b.

proof: $0=a(ab+ba)=a^{2}b+aba=-b+aba.$

Lemma: Under the same hypothesis, a ⊥ ab and b ⊥ ab.

proof:$(a\perp ab):\ \ a(ab)+(ab)a=-b+aba=0.$

Let u = exp(θ r) be a versor. There is a group action on H determined by u:

Conjugation of a vector by a versor

Suppose a pair (a,b) in HxH, not both zero, and a pair (c,d) are related by a non-zero quaternion q through qa=c and qb=d. The relation is denoted (a,b) ~ (c,d). It is an equivalence relation and HxH/~ is a quaternion projective line. The homographies of this projective line are given by matrices from M(2,H). For example,

$(q,1){\begin{pmatrix}u&0\\0&u\end{pmatrix}}=(qu,u)\thicksim (u^{-1}qu,1).$

The equivalence class for (a,b) is written [a,b]. The mapping $q\mapsto u^{-1}qu$ is called the conjugation of q by u, conventionally taken as a versor. The real part of q is invariant under the conjugation, but it applies to the vector part. The following quaternion arithmetic computation shows that the vector is rotated about the axis of the versor, and by twice its angle:

Note that $u=e^{r\theta }$ commutes with all elements in the plane $\{x+yr:x,y\in R\}.$ Select s from the great circle on S² that is perpendicular to r. Then rsr = s by the first lemma. Now compute the conjugate of s by u:

$u^{-1}su\ =\ (\cos \theta -r\sin \theta )s(\cos \theta +r\sin \theta )$

$=\ (\cos \theta s\ -rs\sin \theta )(\cos \theta +r\sin \theta )$

$=\ (\cos ^{2}\theta -\sin ^{2}\theta )s+(2\sin \theta \cos \theta )sr$

$=\ s\cos 2\theta +sr\sin 2\theta ,$ which is rotation by 2 theta in the (s, sr) plane.

Screw displacement

Linear fractional transformations with quaternions can be demonstrated by considering a kinematic exercise: Given a rotation about the i axis by 2 θ (inner automorphism with versor exp(θ i)) and a desired translation in the j-k plane, find the position of the axis parallel to the i axis where the rotation effects the translation.

The problem can formulated in terms of t = xj + yk, and the transformation first drawing t back to the origin, then preforming the rotation before restoring the position of t:

$[q,1]{\begin{pmatrix}1&0\\-t&1\end{pmatrix}}{\begin{pmatrix}u&0\\0&u\end{pmatrix}}{\begin{pmatrix}1&0\\t&1\end{pmatrix}}={\begin{pmatrix}u&0\\z&u\end{pmatrix}},$

where $z=tu-ut=2\sin \theta (xk-yj).$

Presume the desired translation is in the j direction at distance a, so the desired image of 0 is aj:

$[q,1]{\begin{pmatrix}u&0\\uaj&u\end{pmatrix}}=[qu+uaj,u]=[u^{-1}qu+aj,1].$

Set z = uaj and compare j and k coordinates. The k component equation leads to x = a/2 and shows that t must lie on the perpendicular bisector of the segment from 0 to aj (so the radii to 0 and aj are the same). The j component equation leads to $y=-{\tfrac {1}{2}}\cot \theta$ which corresponds to the right triangle with y on the bisector and a/2 as the opposite side, giving $\tan \theta =a/2y.$

The idea of a rotation providing translation by moving the axis of rotation appropriately was described by Mozzi in 1763 and Chasles in 1830, and is considered a feature of Euclidean motions and kinematics. The proposition is stated as the sufficiency of screw displacements to effect the Euclidean group of proper isometries. Screw displacements are rotations in 3-space, possessing an axis of rotation, and the screw motion includes a translation along the axis of rotation. Notice of the sufficiency is attributed variously to Mozzi and Chasles.

Biquaternions

The AC algebra (B, +, x, * ) has conjugation

$(w+xi+yj+zk)^{*}=w-xi-yj-yk,\quad w,x,y,z\in C.$

In biquaternions a new imaginary unit h commutes with all the other imaginary units i, j, k, including all r satisfying r² = − 1. For example, the division binarion w = a + b h, a,b in R.

Suppose now the complex conjugation is invoked:${\bar {w}}=a-bh,\ \ h^{2}=-1.$ as second involution, denoted by an overbar:

${\bar {q}}={\bar {w}}+{\bar {x}}i+{\bar {y}}j+{\bar {z}}k\ \ {\text{and}}\ \ M=\{q\in B:q^{*}={\bar {q}}\}.$

The two involutions agree on $M=\{q=a+bhi+chj+dhk:\ a,b,c,d\in \mathbb {R} \}.$

This four-dimensional subspace M was exploited by Ludwik Silberstein (1914) and Cornelius Lanczos (1949) to exhibit a mathematical model of spacetime with speed of light set to one, and admitting Lorentz transformations as conjugation of an event by a versor or hyperbolic versor.

In B, for each square root of −1, r ∈ S², (hr)² = +1. Then the plane

$J=\{x+y(hr)\in B:x,y\in R\}$ is a split binarion algebra with (x + y(hr))* = x - y(hr). In particular

$u=\exp(ahr)\ =\ \cosh a+hr\sinh a,$ with hyperbolic angle a, is a hyperbola in the plane of R and hr.

Hyperbolic rotation, or squeeze, can be obtained by conjugation with u. Using r and s ∈ S² ⊂ H as above, then $f(s)=u^{-1}su=$

$=\ (\cosh a-hr\sinh a)s(\cosh a+hr\sinh a)$

$=\ (s\cosh a\ -hrs\sinh a)(\cosh a+hr\sinh a)$

$=\ (\cosh ^{2}a+\sinh ^{2}a)s+(2\sinh a\cosh a)\ hsr$

$=\ s\cosh 2a\ +hsr\sinh 2a\ ,$

which is hyperbolic rotation of s by a hyperbolic angle 2a in the (s, hsr) plane. The real vector s, outside of M, is found to have a component (sinh a) hsr ∈ M after f.

Exercises

1. Let f be a mapping on B given by f(s) = v s v, where v = exp(a hr). Show that r ⊥ s implies f(s) = s.

2. Show f(e^{b hr}) = exp((2a + b) hr).

3. Interpret f as a mapping on M. Hint: Use terminology of special relativity.

Split-quaternions

The imaginary units v such that v ² = −1 lie on a two-sheeted hyperboloid in split quaternions

There are at least three portals leading to split quaternions: the dihedral group of a square, matrix products in M(2,R), and the modified Cayley-Dickson construction. The work of Max Zorn on split octonions showed the necessity of including split real AC algebras in the aufbau of the category.

The development through the dihedral group was started with a lemma in the Introduction, and is completed with exercises below.

Or one can start with a basis {1, i, j, k} taken from M(2,R), where the identity matrix is one, ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ is j, ${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$ is i, and ${\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ is k. Some practice with matrix multiplication shows they are anticommutative like division quaternions, but some products differ:

j² = +1 = k²,   j k = − i .

Then the real AC algebra of split-quaternions uses coefficients w, x, y, z ∈ R to express an element, its conjugate, and the quadratic form N:

$q=w+xi+yj+zk,\quad q^{*}=w-xi-yj-zk,\quad N(q)=w^{2}+x^{2}-y^{2}-z^{2}.$

Exercises

1. What are the involutions on a square ?

2. As reflections, what is the angle of incidence of the axes of reflection ?

3. The composition of these reflections has what angle of rotation ?

Computations

Insight into the structure and dynamics of split-quaternions is available through elementary computational exercises. These exercises use j² = +1 = k² and jk = −i, contrary to the quaternion group, which is expressed with the same letters i, j, k, but which here refer to the dihedral group of a square instead.

1. For r = j cos θ + k sin θ, show that r² = +1 = −r r*.
2. Compute ir .
3. Recall that <q, t> = (q t* + t q*)/2. Show <q, t> = real part of q t*
4. Definition: q and t are orthogonal when <q, t> = 0.
5. Show that for any theta, r and ir are orthogonal.
6. Let p = i sinh a + r cosh a. Show that p² = +1 for any a and r.
7. Let v = i cosh a + r sinh a. Show that v² = −1.
8. For a given a and r, show that p and v are orthogonal.
9. Let m = p exp(bp) = sinh b + p cosh b. Show that m m* = −1.
10. Let w = exp(bp) = cosh b + p sinh b. Show that m is orthogonal to w.
11. Show that m is orthogonal to v.
12. For any θ, a, and b defining r, p, w, v, and m, the set {m, w, v, ir} is an orthonormal basis.
13. If u is a unit, <qu, tu> = uu* <q, t>.

Homographies

The concept of homography has already been introduced as Möbius transformations on the projective binarion line. In fact, the concept has been extended to screw displacements using quaternions, where non-commutativity of the algebra has been accommodated.

Since the associative property is a requisite of a mathematical group, this text has required AC algebras to have it so that the algebras have a multiplicative group. Furthermore, associativity, and the fact that multiplication distributes over addition, are used in the following application of matrix multiplication:

Proposition: On an associative composition algebra, the homography ${\begin{pmatrix}p&q\\r&s\end{pmatrix}}$ is well-defined on the projective line.

With u taken from the group of units of A, (ua, ub) are the homogeneous coordinates of a point in the projective line P(A). One writes :$(a,b)\sim (ua,ub)$ and ~ is an equivalence relation on A x A; for instance, it is a transitive relation because of associativity.

$(ua,\ ub){\begin{pmatrix}p&q\\r&s\end{pmatrix}}=((ua)p+(ub)r,(ua)q+(ub)s)$

$=(u(ap)+u(br),u(aq)+u(bs))=(u(ap+br),u(aq+bs)).$

These equalities, involving the matrix product on the right, show that the result of the matrix transformation does not depend on the representative (a,b) from an equivalence class of the relation.

The condition $aA+bA=A$ requires the pair (a,b) to be sufficient to generate A: they must not both lie in a proper subalgebra. The projective line is

$P(A)=\{U[a:b]:aA+bA=A\},$ where U[a: b] represents the equivalence class of (a, b).

Embedding, far and near translations

A canonical embedding of A into P(A) is given by

$E:\ A\rightarrow P(A)\ {\text{by}}\ a\mapsto U[a:\ 1].$

If ab = 1, then $U[a:\ 1]\sim U[1:\ b]$ since a ∈ U. For such a,

$E(a){\begin{pmatrix}0&1\\1&0\end{pmatrix}}\ =\ E(a^{-1}),$

showing that ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ moves the elements of U ⊂A to the equivalence class of U[a⁻¹: 1], thus extending the multiplicative inverse map to P(A).

$U[0:\ 1]{\begin{pmatrix}0&1\\1&0\end{pmatrix}}=U[1:\ 0]$

is referred to as a point at infinity, but unless A is a division algebra, it is not the only element of $P(A)\backslash E(A).$

For a positive real number p, the action of ${\begin{pmatrix}p&0\\0&1\end{pmatrix}}$ on E(A) agrees with the dilation a → pa acting in A. Furthermore, inner automorphisms are extended by homographies:

$U[a:\ 1]{\begin{pmatrix}u&0\\0&u\end{pmatrix}}\ =\ U[au:\ u]\ \sim \ U[u^{-1}au:\ 1].$

With the canonical embedding z → [z : 1 ] of A into P(A), the transformation ${\begin{pmatrix}1&0\\t&1\end{pmatrix}}$ is a near translation. The other embedding z → [ 1 : z ] takes the origin to [ 1 : 0 ], sometimes written $\infty$ since it is a point at infinity with respect to the canonical embedding. The transformation

$[z:1]{\begin{pmatrix}1&t\\0&1\end{pmatrix}}=[z:zt+1]$ is a far translation because of its effect on the second embedding:

$[1:z]{\begin{pmatrix}1&t\\0&1\end{pmatrix}}=[1:z+t].$

Proposition : Suppose ${\begin{pmatrix}a&c\\b&1\end{pmatrix}}\in SL(2,A).$ Then the matrix is a product of a far translation with a near translation.

proof: 1 = a − bc implies a = 1 + bc. Then

${\begin{pmatrix}1&c\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\b&1\end{pmatrix}}={\begin{pmatrix}1+bc&c\\b&1\end{pmatrix}}={\begin{pmatrix}a&c\\b&1\end{pmatrix}}.$

Conformal spacetime transformations

In 1910 reference was made to "conformal transformations of spacetime" by Harry Bateman and Ebenezer Cunningham, though the method of description was by differential geometry of transformations respecting Maxwell's equations of electromagnetism. Using M ⊂ B to represent spacetime, and the AC algebra B for homographies on P(B) to represent transformations, a general conformal transformation can be written:

$g={\begin{pmatrix}pu&b\\a&v\end{pmatrix}}.$ The more commonly noted subgroups are an affine group (b = 0), the Poincaré group (p = 1 and b = 0), and the Lorentz group (p = 1 and a = b = 0).

There are 15 degrees of freedom in g: p is one, a and b contribute four each, while u and v contribute six.

In particular, ${\begin{pmatrix}u&0\\0&u\end{pmatrix}}$ with u = exp(a r) generates the orthogonal group O(3) in the Lorentz group, and

${\begin{pmatrix}v&0\\0&1/v\end{pmatrix}}$ with v = exp(b hr) generates the boosts, per the exercises in the last chapter.

Exercises

1. Find the coordinates of elements of the projective line over the field with two elements.

2. For g extending translation, rotation, and inversion, find {x : xg = x}, the fixed point set of g.

Cross ratio

On the real projective line the homography

${\begin{pmatrix}-1&1\\a&b\end{pmatrix}}$ takes [a: 1] to [0: 1] and [b: 1] to [1: 0].

The numbers between them have positive real values. The midpoint of the interval (a,b) goes to [1: 1].

For commutative rings (binarions here), there is a cross ratio homography which maps a sufficiently distinct triple from the ring to the projective line over the Galois field Z/2Z, which is contained in the projective line over any ring. But in the non-commutative case (quaternions here), the homography which separates points p and q is only conditionally normalizable. Indeed, compose

${\begin{pmatrix}t-q&0\\0&t-p\end{pmatrix}}$ with the separation of p and q:

$[t:1]\rightarrow [t-p:\ t-q]\rightarrow [(t-p)(t-q):\ (t-q)(t-p)]\neq [1:1].$

Exercise: Show that $t(q-p)+(p-q)t+qp-pq=0$

is sufficient to provide a normalized homography mapping of {p, q, t} to {[0 :1], [1 :0], [1, 1] }.