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Non-degeneracy of the pairing (snake relation)


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or equivalently


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Ref:

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

\label{eq1}2(1)
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}\ }))(2)
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(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)
There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name. Cannot find a definition or applicable library operation named 𝐝 with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. FriCAS will attempt to step through and interpret the code. There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name.
Cannot find a definition or applicable library operation named 𝐝 with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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,[i,j])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(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: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Check that the twisted snake relation holds

fricas
test
  (  I Ω   )  /
  (  I X   )  /
  (   U I  )  =  I
There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation.
Cannot find a definition or applicable library operation named I with argument type(s) LinearOperator(OrderedVariableList([1,2]),Expression(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Dimension

Since the "snake" is twisted, dimension is as expected.


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fricas
d:=
    Ω /
    U

\label{eq3}�� \over U(3)
Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))

This "twisted dimension " depends on U!


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fricas
d':=
     Ω /
     X /
     U

\label{eq4}�� \over{U \  X}(4)
Type: LinearOperator(OrderedVariableList([1,2]),Expression(Integer))




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