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last edited 13 years ago by Bill Page |
Edit detail for TwistedSnakeRelation revision 4 of 4
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Editor: Bill Page
Time: 2011/05/09 08:31:55 GMT-7 |
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| Note: better graphic | ||
added:
or equivalently
$$
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$$
Non-degeneracy of the pairing (snake relation)
(0.0,-1.2)(0.8,-1.2)(0.8,-0.4)
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or equivalently
(5.8,-0.9)(5.0,-0.9)(5.0,-0.1)
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\psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1)
\psbezier[linewidth=0.04](0.0,-0.1)(0.0,0.7)(0.8,0.7)(0.8,-0.1)
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}](images/2886468678326544459-16.0px.png "\scalebox{1} % Change this value to rescale the drawing.
{
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}") |
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
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 some convenient notation
fricas
macro Σ(x,i, n)==reduce(+, [x for i in n])
Type: Void
fricas
macro Ξ(f,i, n)==[f for i in n]
Type: Void
fricas
macro sb == subscript
Type: Void
fricas
macro sp == superscript
Type: Void
Let 𝐋 be the domain of 2-dimensional linear operators
fricas
dim:=2
 | (1) |
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := LinearOperator(OVAR ['1,'2], ℚ)
![\label{eq2}\hbox{\axiomType{LinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, 2 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))](images/6430769357992199278-16.0px.png "\label{eq2}\hbox{\axiomType{LinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, 2 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))") | (2) |
Type: Type
fricas
𝐞:ℒ 𝐋 := basisOut()
![\label{eq3}\left[{|_{1}}, \:{|_{2}}\right]](images/3103936605298351352-16.0px.png "\label{eq3}\left[{|_{1}}, \:{|_{2}}\right]") | (3) |
fricas
𝐝:ℒ 𝐋 := basisIn()
![\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}\right]](images/6443194698690308847-16.0px.png "\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}\right]") | (4) |
fricas
I:𝐋:=[1] -- identity for composition
 | (5) |
fricas
X:𝐋:=[2,1] -- twist
 | (6) |
Pairing
A scalar product (pairing) is represented by
fricas
U:=Σ(Σ(sp('u,
 | (7) |
In general we do not require that it be symmetric.
Co-pairing
Solve the "twisted snake relation" as a system of linear equations.
fricas
Ω:𝐋:=Σ(Σ(sb('u,
 | (8) |
fricas
Í := ( I Ω ) / ( I X ) / ( U I )
![\label{eq9}\begin{array}{@{}l}
\displaystyle
{{\left({{u^{1, \: 2}}\ {u_{1, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{1}}}+
\
\
\displaystyle
{{\left({{u^{1, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{2, \: 1}}}\right)}\ {|_{2}^{1}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{1, \: 2}}}+{{u^{2, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{2}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{2, \: 1}}\ {u_{2, \: 1}}}\right)}\ {|_{2}^{2}}}](images/8249647437031589122-16.0px.png "\label{eq9}\begin{array}{@{}l}
\displaystyle
{{\left({{u^{1, \: 2}}\ {u_{1, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{1}}}+
\
\
\displaystyle
{{\left({{u^{1, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{2, \: 1}}}\right)}\ {|_{2}^{1}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{1, \: 2}}}+{{u^{2, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{2}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{2, \: 1}}\ {u_{2, \: 1}}}\right)}\ {|_{2}^{2}}}") | (9) |
fricas
Ì:= ( Ω I ) / ( X I ) / ( I U )
![\label{eq10}\begin{array}{@{}l}
\displaystyle
{{\left({{u^{2, \: 1}}\ {u_{2, \: 1}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{1}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 1}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 2}}}\right)}\ {|_{2}^{1}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{2, \: 1}}}+{{u^{1, \: 2}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{2}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 2}}\ {u_{1, \: 2}}}\right)}\ {|_{2}^{2}}}](images/724613474392670189-16.0px.png "\label{eq10}\begin{array}{@{}l}
\displaystyle
{{\left({{u^{2, \: 1}}\ {u_{2, \: 1}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{1}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 1}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 2}}}\right)}\ {|_{2}^{1}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{2, \: 1}}}+{{u^{1, \: 2}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{2}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 2}}\ {u_{1, \: 2}}}\right)}\ {|_{2}^{2}}}") | (10) |
fricas
equate(f,g)==map((x, y)+->(x=y), ravel f, ravel g);
Type: Void
fricas
eq1:=equate(Í,I)
fricas
Compiling function equate with type (LinearOperator(
OrderedVariableList([1,![\label{eq11}\begin{array}{@{}l}
\displaystyle
\left[{{{{u^{1, \: 2}}\ {u_{1, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{1, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{2, \: 1}}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{{u^{2, \: 2}}\ {u_{1, \: 2}}}+{{u^{2, \: 1}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{2, \: 1}}\ {u_{2, \: 1}}}}= 1}\right]](images/5250008307464236527-16.0px.png "\label{eq11}\begin{array}{@{}l}
\displaystyle
\left[{{{{u^{1, \: 2}}\ {u_{1, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{1, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{2, \: 1}}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{{u^{2, \: 2}}\ {u_{1, \: 2}}}+{{u^{2, \: 1}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{2, \: 1}}\ {u_{2, \: 1}}}}= 1}\right]") | (11) |
Type: List(Equation(Expression(Integer)))
fricas
eq2:=equate(Ì,I)
![\label{eq12}\begin{array}{@{}l}
\displaystyle
\left[{{{{u^{2, \: 1}}\ {u_{2, \: 1}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{2, \: 1}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 2}}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{{u^{2, \: 2}}\ {u_{2, \: 1}}}+{{u^{1, \: 2}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 2}}\ {u_{1, \: 2}}}}= 1}\right]](images/5203550341738782448-16.0px.png "\label{eq12}\begin{array}{@{}l}
\displaystyle
\left[{{{{u^{2, \: 1}}\ {u_{2, \: 1}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{2, \: 1}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 2}}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{{{u^{2, \: 2}}\ {u_{2, \: 1}}}+{{u^{1, \: 2}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 2}}\ {u_{1, \: 2}}}}= 1}\right]") | (12) |
Type: List(Equation(Expression(Integer)))
fricas
snake:=solve(concat(eq1,eq2), concat Ξ(Ξ(sb('u, [i, j]), i, 1..dim), j, 1..dim));
Type: List(List(Equation(Expression(Integer))))
fricas
if #snake ~= 1 then error "no solution"
Type: Void
fricas
Ω:=eval(Ω,snake(1))
![\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^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}-
\
\
\displaystyle
{{{u^{1, \: 2}}\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}}}](images/6529015669216114030-16.0px.png "\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^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}-
\
\
\displaystyle
{{{u^{1, \: 2}}\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) |
fricas
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j),i, 1..dim), j, 1..dim)
![\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}}}}}](images/6358362493493517664-16.0px.png "\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) |
This is equivalent to a matrix inverse (transposed!)
fricas
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U,i, 1..dim), j, 1..dim)
![\label{eq15}\left[
\begin{array}{cc}
{u^{1, \: 1}}&{u^{2, \: 1}}
\
{u^{1, \: 2}}&{u^{2, \: 2}}](images/2588446488723942802-16.0px.png "\label{eq15}\left[
\begin{array}{cc}
{u^{1, \: 1}}&{u^{2, \: 1}}
\
{u^{1, \: 2}}&{u^{2, \: 2}}") | (15) |
fricas
mU:=inverse map(retract,Um)
![\label{eq16}\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}}}}}](images/4645034059072279958-16.0px.png "\label{eq16}\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}}}}}") | (16) |
Type: Union(Matrix(Expression(Integer)),
fricas
Ωm:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i, 1..dim), j, 1..dim)
![\label{eq17}\begin{array}{@{}l}
\displaystyle
{{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}-
\
\
\displaystyle
{{{u^{1, \: 2}}\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}}}](images/3529119345871485550-16.0px.png "\label{eq17}\begin{array}{@{}l}
\displaystyle
{{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}-
\
\
\displaystyle
{{{u^{1, \: 2}}\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}}}") | (17) |
fricas
-- compare test(Ω=Ωm)
 | (18) |
Type: Boolean
Check that the twisted snake relation holds
fricas
test ( I Ω ) / ( I X ) / ( U I ) = I
 | (19) |
Type: Boolean
fricas
test ( Ω I ) / ( X I ) / ( I U ) = I
 | (20) |
Type: Boolean
Dimension
Since the "snake" is twisted, dimension is as expected.
(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}
}](images/4347138082113371837-16.0px.png "\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}
}") |
fricas
d:=
Ω /
U
 | (21) |
This "twisted dimension " depends on 
!
(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}
}](images/8439666260330341149-16.0px.png "\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}
}") |
fricas
d':=
Ω /
X /
U
 | (22) |