Anassociative 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

${\displaystyle \mathrm{\forall }a,b,c\in A(ab)c=a(bc),}$ and

${\displaystyle \mathrm{\forall }a,b\in A(ab{)}^{\ast }(ab)=(a{a}^{\ast })(b{b}^{\ast }).}$

A composition algebra is constructed as analgebra over a fieldF, and is equipped with a mapping${\displaystyle N:A\to F\text{by}a\mapsto a{a}^{\ast }.}$ The axiom involving the conjugation (*) expresses N'sgroup homomorphism property between the multiplicative groups ofA andF.

Associative composition algebras come in three levels: unarion, binarion, and quaternion. The unarion level in this text will be either R, thereal numbers, or C, thecomplex numbers. At the unarion level, the conjugation is the identity mapping, and${\displaystyle N(a)=a^2}$ 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 possessnull vectors${\displaystyle (a{a}^{\ast }=0).}$

Associative Composition Algebras over R or C

• 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 termstessarine andcoquaternion were used by James Cockle writing inPhilosophical Magazine, in the wake of Hamilton's lectures on H and B. The termbinarion, an essential linguistic insertion, was used by Kevin McCrimmon in his bookA Taste of Jordan Algebras(2004).

Any AC algebra may provide arguments to a linear fractional transformation, here called ahomography 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,${\displaystyle \{x\in A:x=x^{\ast }\}}$ is the field ofscalars, either R or C in this text. In the case of R, it is thereal line embedded in A. For an element${\displaystyle x\in A,(x+x^{\ast })/2}$ is thescalar part ofx, and for real algebras it is thereal part ofx.

Each algebra A has abilinear form on${\displaystyle N(x+y)-N(x)-N(y)}$ written

${\displaystyle ⟨x,y⟩:=\frac{1}{2}((x{y}^{\ast })+(x^{\ast }y)).}$

Given${\displaystyle y\in A,\{x:⟨x,y⟩=0\}}$ is the set of elementsorthogonal to y, and given

${\displaystyle a,b\in A,\{x:(x-a)(x-a{)}^{\ast }=b{b}^{\ast }\}}$ is aquadratic set in A.

The distinction between these two types of sets is reduced with Möbius transformations and later in the chapterHomographies 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${\displaystyle e^{i\theta }}$ (say), and reflection in M comes by rotating it to L, then conjugating, and rotating back to the original position of L:

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

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

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

Transcendental paradigm →

• Book:Associative Composition Algebra