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	SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation

Edit detail for Snake Relation revision 1 of 11

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Editor: Bill Page Time: 2011/04/21 20:39:17 GMT-7
Note: dimension

changed:
-
Non-degeneracy of the pairing

Ref:

- http://mat.uab.es/~kock/TQFT.html

  Frobenius algebras and 2D topological quantum field theories

  Section 2.3.11, page 112.

  *Joachim Kock*

We need the Axiom LinearOperator library.
\begin{axiom}
)library MONAL PROP LIN
\end{axiom}

Use the following macros for convenient notation
\begin{axiom}
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
-- list
macro Ξ(f,i,n)==[f for i in n]
-- subscript
macro sb == subscript
\end{axiom}

𝐋 is the domain of 4-dimensional linear operators
\begin{axiom}
dim:=2
macro ℒ == List
macro ℚ == Expression Integer
𝐋 := LinearOperator(dim, OVAR [], ℚ)
𝐞:ℒ 𝐋      := basisVectors()
𝐝:ℒ 𝐋      := basisForms()
I:𝐋:=[1]   -- identity for composition
X:𝐋:=[2,1] -- twist
\end{axiom}

A scalar product (pairing) is denoted by
\begin{axiom}
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)
\end{axiom}

Co-pairing

Solve the "snake relation" as a system of linear equations.
\begin{axiom}
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
d1:=(I*Ω)/(U*I);
d2:=(Ω*I)/(I*U);
equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g);
eq1:=equate(d1,I);
eq2:=equate(d2,I);
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));
if #snake ~= 1 then error "no solution"
Ω:=eval(Ω,snake(1))
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)
-- compare
inverse map(retract,Um)
\end{axiom}

Check "dimension": It depends on parameters!
\begin{axiom}

d:𝐋:=
    (   Ω    )  /
    (   U    )

test
    (  I Ω   )  /
    (   U I  )  =  I

test
    (   Ω I  )  /
    (  I U   )  =  I

\end{axiom}

Non-degeneracy of the pairing

Ref:

http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories

Section 2.3.11, page 112.

Joachim Kock

We need the Axiom LinearOperator? library.

axiom

)library MONAL PROP LIN

   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/PROP.NRLIB/PROP
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

Use the following macros for convenient notation

axiom

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

Type: Void

axiom

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

Type: Void

axiom

-- subscript
macro sb == subscript

Type: Void

𝐋 is the domain of 4-dimensional linear operators

axiom

dim:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

macro ℒ == List

Type: Void

axiom

macro ℚ == Expression Integer

Type: Void

axiom

𝐋 := LinearOperator(dim, OVAR [], ℚ)

$\label{eq2}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{OrderedVariableList}\ } ([ ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisVectors()

$\label{eq3}\left[{|_{1}}, \:{|_{2}}\right]$

(3)

Type: List(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))

axiom

𝐝:ℒ 𝐋      := basisForms()

$\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}\right]$

(4)

Type: List(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))

axiom

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

$\label{eq5}{|_{1}^{1}}+{|_{2}^{2}}$

(5)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

X:𝐋:=[2,1] -- twist

$\label{eq6}{|_{1 \ 1}^{1 \ 1}}+{|_{2 \ 1}^{1 \ 2}}+{|_{1 \ 2}^{2 \ 1}}+{|_{2 \ 2}^{2 \ 2}}$

(6)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

A scalar product (pairing) is denoted by

axiom

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

$\label{eq7}{{u^{1, \: 1}}\ {|_{\ }^{1 \ 1}}}+{{u^{1, \: 2}}\ {|_{\ }^{1 \ 2}}}+{{u^{2, \: 1}}\ {|_{\ }^{2 \ 1}}}+{{u^{2, \: 2}}\ {|_{\ }^{2 \ 2}}}$

(7)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)

$\label{eq8}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{2, \: 1}} \ {u^{1, \: 2}}&{u^{2, \: 2}}$

(8)

Type: Matrix(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))

Co-pairing

Solve the "snake relation" as a system of linear equations.

axiom

Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)

$\label{eq9}{{u_{1, \: 1}}\ {|_{1 \ 1}}}+{{u_{1, \: 2}}\ {|_{1 \ 2}}}+{{u_{2, \: 1}}\ {|_{2 \ 1}}}+{{u_{2, \: 2}}\ {|_{2 \ 2}}}$

(9)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

d1:=(I*Ω)/(U*I);

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

d2:=(Ω*I)/(I*U);

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g);

Type: Void

axiom

eq1:=equate(d1,I);

axiom

Compiling function equate with type (LinearOperator(2,
      OrderedVariableList([]),Expression(Integer)),LinearOperator(2,
      OrderedVariableList([]),Expression(Integer))) -> List(Equation(
      Expression(Integer)))

Type: List(Equation(Expression(Integer)))

axiom

eq2:=equate(d2,I);

Type: List(Equation(Expression(Integer)))

axiom

snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));

Type: List(List(Equation(Expression(Integer))))

axiom

if #snake ~= 1 then error "no solution"

Type: Void

axiom

Ω:=eval(Ω,snake(1))

$\label{eq10}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 2}}}$

(10)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)

$\label{eq11}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}& -{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}} \ -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}$

(11)

Type: Matrix(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))

axiom

-- compare
inverse map(retract,Um)

$\label{eq12}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}& -{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}} \ -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}$

(12)

Type: Union(Matrix(Expression(Integer)),...)

Check "dimension": It depends on parameters!

axiom

d:𝐋:=
    (   Ω    )  /
    (   U    )

$\label{eq13}{{2 \ {u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{2, \: 1}}^2}-{{u^{1, \: 2}}^2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}$

(13)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

test
    (  I Ω   )  /
    (   U I  )  =  I

$\label{eq14} \mbox{\rm true}$

(14)

Type: Boolean

axiom

test
    (   Ω I  )  /
    (  I U   )  =  I

$\label{eq15} \mbox{\rm true}$

(15)

Type: Boolean