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	SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation

Edit detail for Snake Relation revision 2 of 11

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Editor: Bill Page Time: 2011/04/22 07:58:53 GMT-7
Note: dimension

added:

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

changed:
-d1:=(I*Ω)/(U*I);
-d2:=(Ω*I)/(I*U);
Iright:=(I*Ω)/(U*I);
Ileft:=(Ω*I)/(I*U);

changed:
-eq1:=equate(d1,I);
-eq2:=equate(d2,I);
eq1:=equate(Iright,I);
eq2:=equate(Ileft,I);

added:
\end{axiom}

This is equivalent to a matrix inverse (transposed!)
\begin{axiom}
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)
mU:=transpose inverse map(retract,Um)
Ωm:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)

changed:
-inverse map(retract,Um)
test(Ω=Ωm)

changed:
-Check "dimension": It depends on parameters!
Check that the snake relation holds

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

added:
The quantity "dimension" depends on $U$!

\begin{axiom}
d:𝐋:=

       Ω      /
       U    

\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

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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))

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)

$\label{eq8}{{u_{1, \: 1}}\ {|_{1 \ 1}}}+{{u_{1, \: 2}}\ {|_{1 \ 2}}}+{{u_{2, \: 1}}\ {|_{2 \ 1}}}+{{u_{2, \: 2}}\ {|_{2 \ 2}}}$

(8)

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

axiom

Iright:=(I*Ω)/(U*I);

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

axiom

Ileft:=(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(Iright,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(Ileft,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))

$\label{eq9}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 2}}}$

(9)

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

axiom

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

$\label{eq10}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}& -{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}} \ -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}$

(10)

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

This is equivalent to a matrix inverse (transposed!)

axiom

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

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

(11)

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

axiom

mU:=transpose inverse map(retract,Um)

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

(12)

Type: Matrix(Expression(Integer))

axiom

Ωm:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)

$\label{eq13}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 2}}}$

(13)

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

axiom

-- compare
test(Ω=Ωm)

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

(14)

Type: Boolean

Check that the snake relation holds

axiom

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

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

(15)

Type: Boolean

axiom

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

$\label{eq16} \mbox{\rm true}$

(16)

Type: Boolean

The quantity "dimension" depends on !

axiom

d:𝐋:=

       Ω      /
       U

$\label{eq17}{{2 \ {u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{2, \: 1}}^2}-{{u^{1, \: 2}}^2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}$

(17)

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