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SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation
  
last edited 13 years ago by Bill Page

Edit detail for Snake Relation revision 7 of 11
1 2 3 4 5 6 7 8 9 10 11
Editor: Bill Page
Time: 2011/04/22 20:27:24 GMT-7
Note: diagrams 
changed:
-A scalar product (pairing) is denoted by
Pairing

  A scalar product (pairing) is represented by


added:
In general we do not require that it be symmetric.


added:
$$
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{
\begin{pspicture}(0,-0.92)(0.82,0.92)
\psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1)
\psbezier[linewidth=0.04](0.0,0.1)(0.0,0.9)(0.8,0.9)(0.8,0.1)
\psline[linewidth=0.04cm](0.0,0.1)(0.0,-0.1)
\psline[linewidth=0.04cm](0.8,0.1)(0.8,-0.1)
\end{pspicture} 
}
$$

changed:
-This one apparently does not.
This "twisted" quantity does not.
$$
\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-1.22)(0.82,1.22)
\psbezier[linewidth=0.04](0.0,-0.4)(0.0,-1.2)(0.8,-1.2)(0.8,-0.4)
\psbezier[linewidth=0.04](0.0,0.4)(0.0,1.2)(0.8,1.2)(0.8,0.4)
\psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,0.4)(0.0,-0.2)(0.8,0.2)(0.8,-0.4)
\psbezier[linewidth=0.04](0.8,0.4)(0.8,-0.2)(0.0,0.2)(0.0,-0.4)
\end{pspicture} 
}
$$

changed:
-Repeat the calculation, assuming that U is symmetric.
Symmetric Pairing

  Repeat the calculation, assuming that U is symmetric.

changed:
-Dimension
-
-  This quantity no longer depends on $U$!
These quantities no longer depends on $U$!

removed:
-

removed:
-

removed:
-This one apparently does not.

Non-degeneracy of the pairing

Ref:

\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.82)(5.22,0.82) \psbezier[linewidth=0.04](0.0,0.0)(0.0,-0.8)(0.6,-0.8)(0.6,0.0) \psbezier[linewidth=0.04](0.6,0.0)(0.6,0.8)(1.2,0.8)(1.2,0.0) \psline[linewidth=0.04cm](0.0,0.0)(0.0,0.6) \psline[linewidth=0.04cm](1.2,0.0)(1.2,-0.6) \psline[linewidth=0.04cm](2.6,0.6)(2.6,-0.6) \psbezier[linewidth=0.04](5.2,0.0)(5.2,-0.8)(4.6,-0.8)(4.6,0.0) \psbezier[linewidth=0.04](4.6,0.0)(4.6,0.8)(4.0,0.8)(4.0,0.0) \psline[linewidth=0.04cm](5.2,0.0)(5.2,0.6) \psline[linewidth=0.04cm](4.0,0.0)(4.0,-0.6) \usefont{T1}{ptm}{m}{n} \rput(1.8948437,0.105){=} \usefont{T1}{ptm}{m}{n} \rput(3.2948437,0.105){=} \end{pspicture} }  

We use the Axiom LinearOperator? library

fricas
(1) -> )library MONAL PROP LIN

   )library cannot find the file LIN.
   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

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
\label{eq1}2(1)
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := LinearOperator(dim, OVAR [], ℚ)

   There are no library operations named LinearOperator 
      Use HyperDoc Browse or issue
                           )what op LinearOperator
      to learn if there is any operation containing " LinearOperator " 
      in its name.

   Cannot find a definition or applicable library operation named 
      LinearOperator with argument type(s) 
                               PositiveInteger
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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 "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)

   𝐋 is not a valid type.

This is equivalent to a matrix inverse (transposed!)

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

   There are no library operations named 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
   Cannot find a definition or applicable library operation named 𝐞 
      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 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

   Cannot find a definition or applicable library operation named 𝐞 
      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.

Check that the snake relation holds

fricas
test
    (  I Ω   )  /
    (   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) 
                                 Variable(Ω)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Dimension

This quantity depends on U!

\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(0.82,0.92) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1) \psbezier[linewidth=0.04](0.0,0.1)(0.0,0.9)(0.8,0.9)(0.8,0.1) \psline[linewidth=0.04cm](0.0,0.1)(0.0,-0.1) \psline[linewidth=0.04cm](0.8,0.1)(0.8,-0.1) \end{pspicture} }  

fricas
d:=
    Ω /
    U
\label{eq2}\frac{��}{U}(2)
Type: Fraction(Polynomial(Integer))

This "twisted" quantity does not.

\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.22)(0.82,1.22) \psbezier[linewidth=0.04](0.0,-0.4)(0.0,-1.2)(0.8,-1.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.0,0.4)(0.0,1.2)(0.8,1.2)(0.8,0.4) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,0.4)(0.0,-0.2)(0.8,0.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.8,0.4)(0.8,-0.2)(0.0,0.2)(0.0,-0.4) \end{pspicture} }  
fricas
d':=
     Ω /
     X /
     U
\label{eq3}\frac{��}{U \ X}(3)
Type: Fraction(Polynomial(Integer))

Symmetric Pairing

Repeat the calculation, assuming that U is symmetric.

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

   There are 1 exposed and 0 unexposed library operations named ravel 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op ravel
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named ravel
      with argument type(s) 
                        Fraction(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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

   There are no library operations named 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
   Cannot find a definition or applicable library operation named 𝐞 
      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 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

   Cannot find a definition or applicable library operation named 𝐞 
      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.

Check that the snake relation holds

fricas
test
    (  I Ω   )  /
    (   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) 
                                 Variable(Ω)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

These quantities no longer depends on U!

fricas
d:=
    Ω /
    U
\label{eq4}\frac{��}{U}(4)
Type: Fraction(Polynomial(Integer))

fricas
d':=
     Ω /
     X /
     U
\label{eq5}\frac{��}{U \ X}(5)
Type: Fraction(Polynomial(Integer))