MathAction Snake Relation

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TwistedSnakeRelation

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.
Also related notes: http://mat.uab.es/~kock/TQFT/FS.pdf
Section 2.2.9, page 23.

Joachim Kock
http://arxiv.org/abs/math/0512103
Categorical Aspects of Topological Quantum Field Theories

Section 2.3.3, page 27.

Bruce H. Bartlett

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We use the Axiom LinearOperator library

fricas

)library CARTEN MONAL PROP LOP

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

and convenient notation

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

Type: Void

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macro Ξ(f,i,n)==[f for i in n]

Type: Void

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macro sb == subscript

Type: Void

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macro sp == superscript

Type: Void

Let 𝐋 be the domain of 2-dimensional linear operators

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dim:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

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macro ℒ == List

Type: Void

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macro ℚ == Expression Integer

Type: Void

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𝐋 := LinearOperator(OVAR ['1, '2], ℚ)

$\label{eq2}\hbox{\axiomType{LinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, 2 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Type

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𝐞:ℒ 𝐋      := basisOut()

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

(3)

Type: List(LinearOperator(OrderedVariableList([1,2]),Expression(Integer)))

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𝐝:ℒ 𝐋      := basisIn()

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

(4)

Type: List(LinearOperator(OrderedVariableList([1,2]),Expression(Integer)))

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

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

(5)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

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

Pairing

A scalar product (pairing) is represented by

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U:=Σ(Σ(sp('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(OrderedVariableList([1,2]),Expression(Integer))

In general we do not require that it be symmetric.

Co-pairing

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

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Ω:𝐋:=Σ(Σ(sb('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(OrderedVariableList([1,2]),Expression(Integer))

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Í:=(I*Ω)/(U*I);

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

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Ì:=(Ω*I)/(I*U);

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

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

Type: Void

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eq1:=equate(Í,I)

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Compiling function equate with type (LinearOperator(
      OrderedVariableList([1,2]),Expression(Integer)), LinearOperator(
      OrderedVariableList([1,2]),Expression(Integer))) -> List(Equation
      (Expression(Integer)))

$\label{eq9}\begin{array}{@{}l} \displaystyle \left[{{{{u^{1, \: 2}}\ {u_{2, \: 1}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{1, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 2}}}}= 0}, \: \right. \ \ \displaystyle \left.{{{{u^{2, \: 2}}\ {u_{2, \: 1}}}+{{u^{2, \: 1}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{2, \: 1}}\ {u_{1, \: 2}}}}= 1}\right]$

(9)

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

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eq2:=equate(Ì,I)

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{{{{u^{2, \: 1}}\ {u_{1, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{2, \: 1}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{2, \: 1}}}}= 0}, \: \right. \ \ \displaystyle \left.{{{{u^{2, \: 2}}\ {u_{1, \: 2}}}+{{u^{1, \: 2}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 2}}\ {u_{2, \: 1}}}}= 1}\right]$

(10)

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

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

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

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

Type: Void

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Ω:=eval(Ω,snake(1))

$\label{eq11}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 2}}}$

(11)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

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

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

(12)

Type: Matrix(LinearOperator(OrderedVariableList([1,2]),Expression(Integer)))

This is equivalent to a matrix inverse (transposed!)

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

$\label{eq13}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{2, \: 1}} \ {u^{1, \: 2}}&{u^{2, \: 2}}$

(13)

Type: Matrix(LinearOperator(OrderedVariableList([1,2]),Expression(Integer)))

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mU:=transpose inverse map(retract,Um)

$\label{eq14}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}& -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}} \ -{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}$

(14)

Type: Matrix(Expression(Integer))

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

$\label{eq15}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 2}}}$

(15)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

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-- compare
test(Ω=Ωm)

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

(16)

Type: Boolean

Check that the snake relation holds

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

$\label{eq17} \mbox{\rm true}$

(17)

Type: Boolean

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

$\label{eq18} \mbox{\rm true}$

(18)

Type: Boolean

Dimension

This quantity depends on !

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d:=
    Ω /
    U

$\label{eq19}{{2 \ {u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{2, \: 1}}^{2}}-{{u^{1, \: 2}}^{2}}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}$

(19)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

This "twisted" quantity does not.

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.22)(0.82,1.22) \psbezier[linewidth=0.04](0.0,-0.4)(0.0,-1.2)(0.8,-1.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.0,0.4)(0.0,1.2)(0.8,1.2)(0.8,0.4) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,0.4)(0.0,-0.2)(0.8,0.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.8,0.4)(0.8,-0.2)(0.0,0.2)(0.0,-0.4) \end{pspicture} }$

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d':=
     Ω /
     X /
     U

$\label{eq20}2$

(20)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

Symmetric Pairing

Repeat the calculation, assuming that U is symmetric.

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sym:=groebner ravel(U-X/U)

$\label{eq21}\left[{{u^{2, \: 1}}-{u^{1, \: 2}}}\right]$

(21)

Type: List(Polynomial(Integer))

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vars:=map(x+->kernels(x).1,sym)::List Symbol

$\label{eq22}\left[{u^{2, \: 1}}\right]$

(22)

Type: List(Symbol)

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eqs:=solve(sym,vars).1

$\label{eq23}\left[{{u^{2, \: 1}}={u^{1, \: 2}}}\right]$

(23)

Type: List(Equation(Fraction(Polynomial(Integer))))

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U:=eval(U,eqs)

$\label{eq24}{{u^{1, \: 1}}\ {|_{\ }^{1 \ 1}}}+{{u^{1, \: 2}}\ {|_{\ }^{1 \ 2}}}+{{u^{1, \: 2}}\ {|_{\ }^{2 \ 1}}}+{{u^{2, \: 2}}\ {|_{\ }^{2 \ 2}}}$

(24)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

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

$\label{eq25}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{1, \: 2}}&{u^{2, \: 2}}$

(25)

Type: Matrix(LinearOperator(OrderedVariableList([1,2]),Expression(Integer)))

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mU:=transpose inverse map(retract,Um)

$\label{eq26}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}}& -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}} \ -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}}$

(26)

Type: Matrix(Expression(Integer))

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

$\label{eq27}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}}\ {|_{1 \ 1}}}-{{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^{2}}}}\ {|_{2 \ 2}}}$

(27)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

Check that the snake relation holds

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

$\label{eq28} \mbox{\rm true}$

(28)

Type: Boolean

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

$\label{eq29} \mbox{\rm true}$

(29)

Type: Boolean

These quantities no longer depends on !

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d:=
    Ω /
    U

$\label{eq30}2$

(30)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

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d':=
     Ω /
     X /
     U

$\label{eq31}2$

(31)

Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

Twist dimension or twist snake? --Bill Page, Sun, 08 May 2011 14:16:39 -0700 reply

TwistedSnakeRelation

Subject: Be Bold !!

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