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	SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation

Edit detail for Snake Relation revision 8 of 11

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Editor: Bill Page Time: 2011/04/23 20:16:57 GMT-7
Note: Kock's notes and Barlett's thesis

added:
- Also related notes: http://mat.uab.es/~kock/TQFT/FS.pdf

  Section 2.2.9, page 23.


added:

- http://arxiv.org/abs/math/0512103

  Categorical Aspects of Topological Quantum Field Theories

  Section 2.3.3, page 27.

  *Bruce H. Bartlett*

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

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

and convenient notation

axiom

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

Type: Void

axiom

macro Ξ(f,i,n)==[f for i in n]

Type: Void

axiom

macro sb == subscript

Type: Void

axiom

macro sp == superscript

Type: Void

Let 𝐋 be the domain of 2-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))

Pairing

A scalar product (pairing) is represented by

axiom

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

In general we do not require that it be symmetric.

Co-pairing

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

axiom

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

axiom

Í:=(I*Ω)/(U*I);

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

axiom

Ì:=(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(Í,I)

axiom

Compiling function equate with type (LinearOperator(2,
      OrderedVariableList([]),Expression(Integer)),LinearOperator(2,
      OrderedVariableList([]),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)))

axiom

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)))

axiom

snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(sb('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))

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

(11)

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

axiom

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?(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)

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

(13)

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

axiom

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))

axiom

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

axiom

-- compare
test(Ω=Ωm)

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

(16)

Type: Boolean

Check that the snake relation holds

axiom

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

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

(17)

Type: Boolean

axiom

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

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

(18)

Type: Boolean

Dimension

This quantity depends on !

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(0.82,0.92) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1) \psbezier[linewidth=0.04](0.0,0.1)(0.0,0.9)(0.8,0.9)(0.8,0.1) \psline[linewidth=0.04cm](0.0,0.1)(0.0,-0.1) \psline[linewidth=0.04cm](0.8,0.1)(0.8,-0.1) \end{pspicture} }$

axiom

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

This "twisted" quantity does not.

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

axiom

d':=
     Ω /
     X /
     U

$\label{eq20}2$

(20)

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

Symmetric Pairing

Repeat the calculation, assuming that U is symmetric.

axiom

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

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

(21)

Type: List(Polynomial(Integer))

axiom

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

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

(22)

Type: List(Symbol)

axiom

eqs:=solve(sym,vars).1

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

(23)

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

axiom

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

axiom

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

axiom

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))

axiom

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

Check that the snake relation holds

axiom

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

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

(28)

Type: Boolean

axiom

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

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

(29)

Type: Boolean

These quantities no longer depends on !

axiom

d:=
    Ω /
    U

$\label{eq30}2$

(30)

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

axiom

d':=
     Ω /
     X /
     U

$\label{eq31}2$

(31)

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