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Non-degeneracy of the pairing

Ref:


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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 convenient notation

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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)
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
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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()

\label{eq3}\left[{|_{1}}, \:{|_{2}}\right](3)
Type: List(LinearOperator(OrderedVariableList?([1,2]),Expression(Integer)))
fricas
𝐝:ℒ 𝐋      := basisIn()

\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}\right](4)
Type: List(LinearOperator(OrderedVariableList?([1,2]),Expression(Integer)))
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I:𝐋:=[1]   -- identity for composition

\label{eq5}{|_{1}^{1}}+{|_{2}^{2}}(5)
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))
fricas
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(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)

\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(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)

\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(OrderedVariableList?([1,2]),Expression(Integer))
fricas
Í:=(I*Ω)/(U*I);
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))
fricas
Ì:=(Ω*I)/(I*U);
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))
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equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g);
Type: Void
fricas
eq1:=equate(Í,I)
fricas
Compiling function equate with type (LinearOperator(
      OrderedVariableList([1,2]),Expression(Integer)), LinearOperator(
      OrderedVariableList([1,2]),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)))
fricas
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)))
fricas
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(sb('u,[i,j]), i,1..dim), j,1..dim));
Type: List(List(Equation(Expression(Integer))))
fricas
if #snake ~= 1 then error "no solution"
Type: Void
fricas
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(OrderedVariableList?([1,2]),Expression(Integer))
fricas
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(OrderedVariableList?([1,2]),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)

\label{eq13}\left[ 
\begin{array}{cc}
{u^{1, \: 1}}&{u^{2, \: 1}}
\
{u^{1, \: 2}}&{u^{2, \: 2}}
(13)
Type: Matrix(LinearOperator(OrderedVariableList?([1,2]),Expression(Integer)))
fricas
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))
fricas
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(OrderedVariableList?([1,2]),Expression(Integer))
fricas
-- compare
test(Ω=Ωm)

\label{eq16} \mbox{\rm true} (16)
Type: Boolean

Check that the snake relation holds

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

\label{eq17} \mbox{\rm true} (17)
Type: Boolean
fricas
test
    (   Ω I  )  /
    (  I U   )  =  I

\label{eq18} \mbox{\rm true} (18)
Type: Boolean

Dimension

This quantity depends on U!


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fricas
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(OrderedVariableList?([1,2]),Expression(Integer))

This "twisted" quantity does not.


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d':=
     Ω /
     X /
     U

\label{eq20}2(20)
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))

Symmetric Pairing

Repeat the calculation, assuming that U is symmetric.

fricas
sym:=groebner ravel(U-X/U)

\label{eq21}\left[{{u^{2, \: 1}}-{u^{1, \: 2}}}\right](21)
Type: List(Polynomial(Integer))
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vars:=map(x+->kernels(x).1,sym)::List Symbol

\label{eq22}\left[{u^{2, \: 1}}\right](22)
Type: List(Symbol)
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eqs:=solve(sym,vars).1

\label{eq23}\left[{{u^{2, \: 1}}={u^{1, \: 2}}}\right](23)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
U:=eval(U,eqs)

\label{eq24}{{u^{1, \: 1}}\ {|_{\ }^{1 \  1}}}+{{u^{1, \: 2}}\ {|_{\ }^{1 \  2}}}+{{u^{1, \: 2}}\ {|_{\ }^{2 \  1}}}+{{u^{2, \: 2}}\ {|_{\ }^{2 \  2}}}(24)
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))

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

\label{eq25}\left[ 
\begin{array}{cc}
{u^{1, \: 1}}&{u^{1, \: 2}}
\
{u^{1, \: 2}}&{u^{2, \: 2}}
(25)
Type: Matrix(LinearOperator(OrderedVariableList?([1,2]),Expression(Integer)))
fricas
mU:=transpose inverse map(retract,Um)

\label{eq26}\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}}}}
(26)
Type: Matrix(Expression(Integer))
fricas
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)

\label{eq27}\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}}}
(27)
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))

Check that the snake relation holds

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

\label{eq28} \mbox{\rm true} (28)
Type: Boolean
fricas
test
    (   Ω I  )  /
    (  I U   )  =  I

\label{eq29} \mbox{\rm true} (29)
Type: Boolean

These quantities no longer depends on U!

fricas
d:=
    Ω /
    U

\label{eq30}2(30)
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))

fricas
d':=
     Ω /
     X /
     U

\label{eq31}2(31)
Type: LinearOperator(OrderedVariableList?([1,2]),Expression(Integer))

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




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