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# Edit detail for Snake Relation revision 1 of 11

 1 2 3 4 5 6 7 8 9 10 11 Editor: Bill Page Time: 2011/04/21 20:39:17 GMT-7 Note: dimension

changed:
-
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*

We need the Axiom LinearOperator library.
\begin{axiom}
)library MONAL PROP LIN
\end{axiom}

Use the following macros for convenient notation
\begin{axiom}
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
-- list
macro Ξ(f,i,n)==[f for i in n]
-- subscript
macro sb == subscript
\end{axiom}

𝐋 is the domain of 4-dimensional linear operators
\begin{axiom}
dim:=2
macro ℒ == List
macro ℚ == Expression Integer
𝐋 := LinearOperator(dim, OVAR [], ℚ)
𝐞:ℒ 𝐋      := basisVectors()
𝐝:ℒ 𝐋      := basisForms()
I:𝐋:=[1]   -- identity for composition
X:𝐋:=[2,1] -- twist
\end{axiom}

A scalar product (pairing) is denoted by
\begin{axiom}
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)
\end{axiom}

Co-pairing

Solve the "snake relation" as a system of linear equations.
\begin{axiom}
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
d1:=(I*Ω)/(U*I);
d2:=(Ω*I)/(I*U);
equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g);
eq1:=equate(d1,I);
eq2:=equate(d2,I);
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));
if #snake ~= 1 then error "no solution"
Ω:=eval(Ω,snake(1))
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)
-- compare
inverse map(retract,Um)
\end{axiom}

Check "dimension": It depends on parameters!
\begin{axiom}

d:𝐋:=
(   Ω    )  /
(   U    )

test
(  I Ω   )  /
(   U I  )  =  I

test
(   Ω I  )  /
(  I U   )  =  I

\end{axiom}



Non-degeneracy of the pairing

Ref:

We need 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

Use the following macros for convenient notation

axiom
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
axiom
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
axiom
-- subscript
macro sb == subscript
Type: Void

𝐋 is the domain of 4-dimensional linear operators

axiom
dim:=2
 (1)
Type: PositiveInteger?
axiom
macro ℒ == List
Type: Void
axiom
macro ℚ == Expression Integer
Type: Void
axiom
𝐋 := LinearOperator(dim, OVAR [], ℚ)
 (2)
Type: Type
axiom
𝐞:ℒ 𝐋      := basisVectors()
 (3)
Type: List(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))
axiom
𝐝:ℒ 𝐋      := basisForms()
 (4)
Type: List(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))
axiom
I:𝐋:=[1]   -- identity for composition
 (5)
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))
axiom
X:𝐋:=[2,1] -- twist
 (6)
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

A scalar product (pairing) is denoted by

axiom
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
 (7)
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))
axiom
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)
 (8)
Type: Matrix(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))

Co-pairing

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

axiom
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
 (9)
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))
axiom
d1:=(I*Ω)/(U*I);
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))
axiom
d2:=(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(d1,I);
axiom
Compiling function equate with type (LinearOperator(2,
OrderedVariableList([]),Expression(Integer)),LinearOperator(2,
OrderedVariableList([]),Expression(Integer))) -> List(Equation(
Expression(Integer)))
Type: List(Equation(Expression(Integer)))
axiom
eq2:=equate(d2,I);
Type: List(Equation(Expression(Integer)))
axiom
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('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))
 (10)
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))
axiom
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)
 (11)
Type: Matrix(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))
axiom
-- compare
inverse map(retract,Um)
 (12)
Type: Union(Matrix(Expression(Integer)),...)

Check "dimension": It depends on parameters!

axiom
d:𝐋:=
(   Ω    )  /
(   U    )
 (13)
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))
axiom
test
(  I Ω   )  /
(   U I  )  =  I
 (14)
Type: Boolean
axiom
test
(   Ω I  )  /
(  I U   )  =  I
 (15)
Type: Boolean