help links subscribe changes refresh edit

	SandBox Quaternion Algebra is Frobenius in Many Ways
SandBoxPauliAlgebra ←	↑

Snake Relation

Edit detail for Snake Relation revision 5 of 11

	1 2 3 4 5 6 7 8 9 10 11
Editor: Bill Page Time: 2011/04/22 15:50:05 GMT-7
Note: assume U symmetric

added:

Repeat the calculation, assuming that U is symmetric.

\begin{axiom}
U:=eval(U,sp('u,[2,1])=sp('u,[1,2]))
\end{axiom}

\begin{axiom}
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)
mU:=transpose inverse map(retract,Um)
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)
\end{axiom}

Check that the snake relation holds
\begin{axiom}

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

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

\end{axiom}

Dimension

  This quantity no longer depends on $U$!

\begin{axiom}

d:=
    Ω /
    U

\end{axiom}

This one apparently does not.
\begin{axiom}
d':=
     Ω /
     X /
     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

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.82)(5.22,0.82) \psbezier[linewidth=0.04](0.0,0.0)(0.0,-0.8)(0.6,-0.8)(0.6,0.0) \psbezier[linewidth=0.04](0.6,0.0)(0.6,0.8)(1.2,0.8)(1.2,0.0) \psline[linewidth=0.04cm](0.0,0.0)(0.0,0.6) \psline[linewidth=0.04cm](1.2,0.0)(1.2,-0.6) \psline[linewidth=0.04cm](2.6,0.6)(2.6,-0.6) \psbezier[linewidth=0.04](5.2,0.0)(5.2,-0.8)(4.6,-0.8)(4.6,0.0) \psbezier[linewidth=0.04](4.6,0.0)(4.6,0.8)(4.0,0.8)(4.0,0.0) \psline[linewidth=0.04cm](5.2,0.0)(5.2,0.6) \psline[linewidth=0.04cm](4.0,0.0)(4.0,-0.6) \usefont{T1}{ptm}{m}{n} \rput(1.8948437,0.105){=} \usefont{T1}{ptm}{m}{n} \rput(3.2948437,0.105){=} \end{pspicture} }$

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))

A scalar product (pairing) is denoted 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))

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 !

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 one apparently does not.

axiom

d':=
     Ω /
     X /
     U

$\label{eq20}2$

(20)

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

Repeat the calculation, assuming that U is symmetric.

axiom

U:=eval(U,sp('u,[2,1])=sp('u,[1,2]))

$\label{eq21}{{u^{1, \: 1}}\ {|_{\ }^{1 \ 1}}}+{{u^{1, \: 2}}\ {|_{\ }^{1 \ 2}}}+{{u^{1, \: 2}}\ {|_{\ }^{2 \ 1}}}+{{u^{2, \: 2}}\ {|_{\ }^{2 \ 2}}}$

(21)

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

axiom

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

$\label{eq22}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{1, \: 2}}&{u^{2, \: 2}}$

(22)

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

axiom

mU:=transpose inverse map(retract,Um)

$\label{eq23}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}}& -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}} \ -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}}$

(23)

Type: Matrix(Expression(Integer))

axiom

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

$\label{eq24}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}}\ {|_{1 \ 1}}}-{{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}^2}}}\ {|_{2 \ 2}}}$

(24)

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

Check that the snake relation holds

axiom

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

$\label{eq25} \mbox{\rm true}$

(25)

Type: Boolean

axiom

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

$\label{eq26} \mbox{\rm true}$

(26)

Type: Boolean

Dimension

This quantity no longer depends on !

axiom

d:=
    Ω /
    U

$\label{eq27}2$

(27)

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

This one apparently does not.

axiom

d':=
     Ω /
     X /
     U

$\label{eq28}2$

(28)

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