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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:


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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 U!


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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.


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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 U!

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))