Associative Composition Algebra/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.