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# Edit detail for Snake Relation revision 10 of 11

 1 2 3 4 5 6 7 8 9 10 11 Editor: test1 Time: 2013/04/23 19:43:07 GMT+0 Note:

changed:
-)library MONAL PROP LIN
)library MONAL PROP LOP

changed:
-𝐋 := LinearOperator(dim, OVAR [], ℚ)
-𝐞:ℒ 𝐋      := basisVectors()
-𝐝:ℒ 𝐋      := basisForms()
𝐋 := LinearOperator(OVAR ['1, '2], ℚ)
𝐞:ℒ 𝐋      := basisOut()
𝐝:ℒ 𝐋      := basisIn()


Non-degeneracy of the pairing

Ref:

We use the Axiom LinearOperator? library

fricas
)library MONAL PROP LOP
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 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)
 (2)
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()
 (3)
Type: List(LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer)))
fricas
𝐝:ℒ 𝐋      := basisIn()
 (4)
Type: List(LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer)))
fricas
I:𝐋:=[1]   -- identity for composition
 (5)
Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))
fricas
X:𝐋:=[2,1] -- twist
 (6)
Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))

## Pairing

A scalar product (pairing) is represented by

fricas
U:=Σ(Σ(sp('u,[i,j])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
 (7)
Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))

In general we do not require that it be symmetric.

## Co-pairing

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

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

Check that the snake relation holds

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

This "twisted" quantity does not.

fricas
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)$(Integer) is missing from domain CartesianTensor12(Expression (Integer)) ## Symmetric Pairing Repeat the calculation, assuming that U is symmetric. fricas sym:=groebner ravel(U-X/U) Function: contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer)) Internal Error The function contract with signature$(Integer)$(Integer)$(Integer) is
missing from domain CartesianTensor12(Expression (Integer))

fricas
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)$(Integer) is missing from domain CartesianTensor12(Expression (Integer)) Check that the snake relation holds fricas test ( I Ω ) / ( U I ) = I Function: contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer)) Internal Error The function contract with signature$(Integer)$(Integer)$(Integer) is
missing from domain CartesianTensor12(Expression (Integer))

These quantities no longer depends on !

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

Twist dimension or twist snake? --Bill Page, Sun, 08 May 2011 14:16:39 -0700 reply
TwistedSnakeRelation?