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)

(14)

-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

(15)

-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real(x/y),M)

-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ

Compiling function S with type CaleyDickson(Expression(Integer),i,i[
      ;2]) -> Matrix(Expression(Integer))

(16)

-- structure constants form a tensor operator
Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*($/𝐝.k), i,1..dim), j,1..dim), k,1..dim)

(17)

arity Y

(18)

Y! := (I,!)/Y

(19)

matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y!, i,1..dim), j,1..dim)

(20)

Multiplication of arbitrary quaternions and

a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

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b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

(22)

(a,b)/Y!

(23)

Multiplication is Associative

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

(24)

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)

(25)

arity U

(26)

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:

(27)

How should we color this?