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# Edit detail for Snake Relation revision 4 of 11

 1 2 3 4 5 6 7 8 9 10 11 Editor: Bill Page Time: 2011/04/22 10:45:12 GMT-7 Note: twisted dimension

changed:
-d:𝐋:=
-
-       Ω      /
-       U

d:=
Ω /
U

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


Non-degeneracy of the pairing

Ref: 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 (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:=   -- 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(sp('u,[i,j])*𝐝.i*𝐝.j, i,1..dim), j,1..dim) (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) (8)
Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))
axiom
Í:=(I*Ω)/(U*I);
Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
Internal Error
The function contract with signature $(Integer)$(Integer)$(Integer) is missing from domain CartesianTensor12(Expression (Integer)) This is equivalent to a matrix inverse (transposed!) axiom Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim) Function: contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer)) Internal Error The function contract with signature$(Integer)$(Integer)$(Integer) is
missing from domain CartesianTensor12(Expression (Integer))

Check that the snake relation holds

axiom
test
(  I Ω   )  /
(   U I  )  =  I
Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
Internal Error
The function contract with signature $(Integer)$(Integer)$(Integer) is missing from domain CartesianTensor12(Expression (Integer)) ## Dimension This quantity depends on ! axiom d:= Ω / U Function: contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer)) Internal Error The function contract with signature$(Integer)$(Integer)$(Integer) is
missing from domain CartesianTensor12(Expression (Integer))

This one apparently does not.

axiom
d':=
Ω /
X /
U
Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
Internal Error
The function contract with signature $(Integer)$(Integer)\$(Integer) is
missing from domain CartesianTensor12(Expression (Integer))