login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit
SandBox Quaternion Algebra is Frobenius in Many Ways
SandBoxPauliAlgebra  ← ↑  

Snake Relation
  
last edited 13 years ago by Bill Page

Edit detail for Snake Relation revision 3 of 11
1 2 3 4 5 6 7 8 9 10 11
Editor: Bill Page
Time: 2011/04/22 09:43:41 GMT-7
Note: dimension 
changed:
-We need the Axiom LinearOperator library.
We use the Axiom LinearOperator library

changed:
-
-Use the following macros for convenient notation
and convenient notation

removed:
--- summation

removed:
--- list

removed:
--- subscript

added:
macro sp == superscript

changed:
-𝐋 is the domain of 4-dimensional linear operators
Let 𝐋 be the domain of 2-dimensional linear operators

changed:
-U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
U:=Σ(Σ(sp('u,[i,j])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

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


changed:
-Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
-Iright:=(I*Ω)/(U*I);
-Ileft:=(Ω*I)/(I*U);
Ω:𝐋:=Σ(Σ(sb('u,[i,j])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
Í:=(I*Ω)/(U*I);
Ì:=(Ω*I)/(I*U);

changed:
-eq1:=equate(Iright,I);
-eq2:=equate(Ileft,I);
-snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));
eq1:=equate(Í,I)
eq2:=equate(Ì,I)
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(sb('u,[i,j]), i,1..dim), j,1..dim));

changed:
-The quantity "dimension" depends on $U$!
Dimension

  This quantity depends on $U$!

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.

A scalar product (pairing) is denoted 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.

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!

fricas
d:𝐋:=

       Ω      /
       U

   𝐋 is not a valid type.