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

   CartesianTensor is now explicitly exposed in frame initial 
   CartesianTensor will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN
   Arity is now explicitly exposed in frame initial 
   Arity will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY
   ClosedMonoidal is now explicitly exposed in frame initial 
   ClosedMonoidal will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL
   ClosedProp is now explicitly exposed in frame initial 
   ClosedProp will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP
   ClosedLinearOperator is now explicitly exposed in frame initial 
   ClosedLinearOperator will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP
   CaleyDickson is now explicitly exposed in frame initial 
   CaleyDickson will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY

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)

(2)

𝐞:ℒ 𝐋      := basisOut()

(3)

𝐝:ℒ 𝐋      := basisIn()

(4)

I:𝐋:=[1]   -- identity for composition

(5)

X:𝐋:=[2,1] -- twist

(6)

V:𝐋:=ev(1) -- evaluation

(7)

Λ:𝐋:=co(1) -- co-evaluation

(8)

!:𝐋:=[-1]  -- color change 1 -> 1*

(9)

$:𝐋:=dagger ! -- 1* -> 1

(10)

J:𝐋:=$/!

(11)

test(!/$=I)

(12)

equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);

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])

(13)

QQ := ℂ(ℚ,'i,q);

Basis: Each B.i is a complex number

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

   There are 1 exposed and 0 unexposed library operations named hyper 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op hyper
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named hyper
      with argument type(s) 
                          List(Expression(Integer))

      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.

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY.fasl" {100289FA53}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.

Multiplication of arbitrary quaternions and

a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

(14)

b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

(15)

(a,b)/Y!

   There are 15 exposed and 15 unexposed library operations named / 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op /
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named / 
      with argument type(s) 
 Tuple(ClosedLinearOperator(OrderedVariableList([1,2]),Expression(Integer)))
                                Variable(Y!)

      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) 
    ClosedLinearOperator(OrderedVariableList([1,2]),Expression(Integer))

      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)

(16)

arity U

(17)

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:

(18)

(three-point function) is zero.

How should we color this?