Refs:

• http://en.wikipedia.org/wiki/Braided_monoidal_category
• http://en.wikipedia.org/wiki/Strongly_ribbon_category

We need the Axiom LinearOperator library.

(1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY

   >> System error:
   The value
  15684
is not of type
  LIST

Use the following macros for convenient notation

-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])

-- list
macro Ξ(f,i,n)==[f for i in n]

-- subscript and superscripts
macro sb == subscript

macro sp == superscript

𝐋 is the domain of 2-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

dim:=2

(1)

macro ℒ == List

macro ℂ == CaleyDickson

macro ℚ == Expression Integer

𝐋 := ClosedLinearOperator(OVAR ['1,'2], ℚ)

   There are no library operations named ClosedLinearOperator 
      Use HyperDoc Browse or issue
                        )what op ClosedLinearOperator
      to learn if there is any operation containing " 
      ClosedLinearOperator " in its name.

   Cannot find a definition or applicable library operation named 
      ClosedLinearOperator with argument type(s) 
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

We want to be able to implement linear operators with two "colors" like the following:

An example starting with Complex Algebra

The basis consists of the real and imaginary units. We use complex multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Split-complex can be specified by Caley-Dickson parameter (q0 = -1)

--q:=1  -- split-complex
q:=sp('i,[2])

(2)

QQ := ℂ(ℚ,'i,q);

   There are no library operations named CaleyDickson 
      Use HyperDoc Browse or issue
                            )what op CaleyDickson
      to learn if there is any operation containing " CaleyDickson " in
      its name.

   Cannot find a definition or applicable library operation named 
      CaleyDickson with argument type(s) 
                                    Type
                                 Variable(i)
                                   Symbol

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Basis: Each B.i is a complex number

B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)

   QQ is not a valid type.

Multiplication of arbitrary quaternions and

a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

   There are no library operations named 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
   Cannot find a definition or applicable library operation named 𝐞 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

   Cannot find a definition or applicable library operation named 𝐞 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Multiplication is Associative

test(
  (  Y! J ) / _
  (   Y   ) = _
  (  I Y  ) / _
  (   Y!  ) )

   There are no library operations named Y! 
      Use HyperDoc Browse or issue
                                 )what op Y!
      to learn if there is any operation containing " Y! " in its name.

   Cannot find a definition or applicable library operation named Y! 
      with argument type(s) 
                                 Variable(J)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

A scalar product is denoted by the (2,0)-tensor

U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*($/𝐝.j), i,1..dim), j,1..dim)

   There are no library operations named 𝐝 
      Use HyperDoc Browse or issue
                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.
   Cannot find a definition or applicable library operation named 𝐝 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named 𝐝 
      Use HyperDoc Browse or issue
                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.

   Cannot find a definition or applicable library operation named 𝐝 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Definition 1

We say that the scalar product is associative if the tensor equation holds:

    Y   =   Y
     U     U

In other words, if the (3,0)-tensor:

(3)

(three-point function) is zero.

How should we color this?