CayleyDickson.jl

This module provides arithmetic for Cayley-Dickson constructs.

There is an abstract type:

Construct{T <: Number} <: Number

and three concrete types:

Elliptic{T <: Number} <: Construct{T}
Hyperbolic{T <: Number} <: Construct{T}
Parabolic{T <: Number} <: Construct{T}

Each of the concrete types is a pair of elements. These concrete types can be used and combined in a recursive way.

Basic arithmetic operations are implemented:

(+), (-), (*), (/), (\), inv, conj

A set of aliases are provided for convenience. There are three 2-dimensional algebras:

Binion{T <: Real} = Elliptic{T}
SplitBinion{T <: Real} = Hyperbolic{T}
Exo1Real{T <: Real} = Parabolic{T}

A Binion is a complex number (a somewhat janky version of the built-in Complex). A SplitBinion is a split-complex number. An Exo1Real is also known as a dual number. The prefix "exo" is used due to a relation with exterior algebras. The Exo1Real type is related to the multivector algebra of 1 unit 1-blade.

All 2-dimensional types have a multiplication operation that is commutative, associative, alternative, and flexible. They are all composition algebras.

There are five 4-dimensional algebras:

Quaternion{T <: Real} = Elliptic{Binion{T}}
SplitQuaternion{T <: Real} = Hyperbolic{Binion{T}}
Exo1Binion{T <: Real} = Parabolic{Binion{T}}
Exo1SplitBinion{T <: Real} = Parabolic{SplitBinion{T}}
Exo2Real{T <: Real} = Parabolic{Exo1Real{T}}

A Quaternion is a traditional Hamilton quaternion. Note that unlike most conventions, the units for a quaternion in this module are i, j, and ij (not k). A SplitQuaternion is a split-quaternion. An Exo2Real is not a hyper-dual number, since multiplication is non-commutative. The Exo2Real is related to the multivector algebra of 2 orthonormal 1-blades. The Exo1Binion and Exo1SplitBinion are not related to dual complex numbers or dual split-complex numbers, since the imaginary/split-imaginary units anti-commute with the parabolic unit.

All 4-dimensional types have a multiplication operation that is non-commutative, associative, alternative, and flexible. They are all composition algebras.

There are seven 8-dimensional types:

Octonion{T <: Real} = Elliptic{Quaternion{T}}
SplitOctonion{T <: Real} = Hyperbolic{Quaternion{T}}
Exo1Quaternion{T <: Real} = Parabolic{Quaternion{T}}
Exo1SplitQuaternion{T <: Real} = Parabolic{SplitQuaternion{T}}
Exo2Binion{T <: Real} = Parabolic{Exo1Binion{T}}
Exo2SplitBinion{T <: Real} = Parabolic{Exo1SplitBinion{T}}
Exo3Real{T <: Real} = Parabolic{Exo2Real{T}}

An Octonion is a traditional octonion. A SplitOctonion is a split-octonion. The Exo3Real type is related to the multivector algebra of 3 orthonormal 1-blades.

All 8-dimensional types have a multiplication operation that is non-commutative, non-associative, alternative, and flexible. They are all composition algebras.

There are nine 16-dimensional types:

Sedenion{T <: Real} = Elliptic{Octonion{T}}
SplitSedenion{T <: Real} = Hyperbolic{Octonion{T}} # Not implemented yet
Exo1Octonion{T <: Real} = Parabolic{Octonion{T}} # Not implemented yet
Exo1SplitOctonion{T <: Real} = Parabolic{SplitOctonion{T}} # Not implemented yet
Exo2Quaternion{T <: Real} = Parabolic{Exo1Quaternion{T}} # Not implemented yet
Exo2SplitQuaternion{T <: Real} = Parabolic{Exo1SplitQuaternion{T}} # Not implemented yet
Exo3Binion{T <: Real} = Parabolic{Exo2Binion{T}} # Not implemented yet
Exo3SplitBinion{T <: Real} = Parabolic{Exo2SplitBinion{T}} # Not implemented yet
Exo4Real{T <: Real} = Parabolic{Exo3Real{T}}

A Sedenion is a traditional sedenion. The Exo4Real type is related to the multivector algebra of 4 orthonormal 1-blades.

All 16-dimensional types have a multiplication operation that is non-commutative, non-associative, non-alternative, and flexible. None are composition algebras.

TO-DO:

• Documentation.
• Maybe make this a submodule of Pairs.jl, along with Plexifications.jl.