Edit detail for Octonion Algebra is Frobenius in Just One Way revision 4 of 8

Octonion Algebra Is Frobenius In Just One Way

Linear operators over a 8-dimensional vector space representing octonnion algebra

Ref:

• http://arxiv.org/abs/1103.5113

  -permuted Frobenius Algebras

  Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)

• http://mat.uab.es/~kock/TQFT.html

  Frobenius algebras and 2D topological quantum field theories

  Joachim Kock

• http://en.wikipedia.org/wiki/Frobenius_algebra

We need the Axiom LinearOperator? library.

(1) -> )library CARTEN MONAL PROP LIN CALEY

   )library cannot find the file LIN.
   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
   CaleyDickson is now explicitly exposed in frame initial 
   CaleyDickson will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])

-- list
macro Ξ(f,i,n)==[f for i in n]

-- subscript
macro sb == subscript

𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

dim:=8

(1)

macro ℒ == List

macro ℂ == CaleyDickson

macro ℚ == Expression Integer

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.

Now generate structure constants for Octonion Algebra

The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Split-complex, co-quaternions and split-octonions can be specified by Caley-Dickson parameters

--q0:=sb('q,[0])
q0:=1  -- not split-complex

(2)

--q1:=sb('q,[1])
q1:=1  -- not co-quaternion

(3)

q2:=sb('q,[2])

(4)

--q2:=1  -- split-octonion
QQ := ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2);

Basis: Each B.i is a octonion number

B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)

(5)

-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

(6)

-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real real(x/y),M)

-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ;

Compiling function S with type CaleyDickson(CaleyDickson(
      CaleyDickson(Expression(Integer),i,1),j,1),k,q[2]) -> Matrix(
      Expression(Integer))

-- structure constants form a tensor operator
Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,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.

A scalar product is denoted by the (2,0)-tensor

U:=Σ(Σ(script('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.

Definition 1

We say that the scalar product is associative if the tensor equation holds:

    Y   =   Y
     U     U

In other words, if the (3,0)-tensor:

(7)

Using the LinearOperator? domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

ω:𝐋 :=(Y*I)/U  - (I*Y)/U;

   𝐋 is not a valid type.

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

We may consider the problem where multiplication Y is given, and look for all associative scalar products

This problem can be solved using linear algebra.

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial 
J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);

   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) 
                                 Variable(ω)

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

The matrix J transforms the coefficients of the tensor into coefficients of the tensor . We are looking for the general linear family of tensors such that J transforms into for any such .

If the null space of the J matrix is not empty we can use the basis to find all non-trivial solutions for U:

Ñ:=nullSpace(J);

   There are 3 exposed and 3 unexposed library operations named 
      nullSpace having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op nullSpace
      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 
      nullSpace with argument type(s) 
                                 Variable(J)

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

This defines a family of Frobenius algebras:

zero? eval(ω,ℰ)

   There are 10 exposed and 6 unexposed library operations named eval 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op eval
      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 eval 
      with argument type(s) 
                                 Variable(ω)
                                 Variable(ℰ)

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

The pairing is necessarily diagonal!

Ų:𝐋 := eval(U,ℰ)

   𝐋 is not a valid type.

The scalar product must be non-degenerate:

Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,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.

Definition 3

Co-pairing

Solve the [Snake Relation]? as a system of linear equations.

Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/Ų, 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 "dimension" and the snake relations.

d:𝐋:=
       Ω    /
       X    /
       Ų

   𝐋 is not a valid type.

Definition 4

Co-algebra

This expression is expensive to compute:

    λ:𝐋 :=
         (    Ω Ω  I    ) /
         (   I Y I I    ) /
         (   I  X  I    ) /
         (   I I  Ų     )

λ:𝐋 :=
     (    I Ω     )  /
     (     Y I    )

   𝐋 is not a valid type.

Frobenius Condition

It takes to long to computer here but it turns out that Octonion algebra fails the Frobenius Condition::

\begin{axiom}

Χ := Y / λ

test ( λ I ) / ( I Y ) = Χ

test ( I λ ) / ( Y I ) = Χ

\end{axiom}

Perhaps this is not surprising since Octonion algebra is not associative. Nevertheless it is "Frobenius" in a more general sense because there is a non-degenerate associative pairing and co-pairing.

i = Unit of the algebra

i:=𝐞.1

   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.

Handle

H:𝐋 :=
         λ     /
         Y

   𝐋 is not a valid type.

Definition 5

  i 
  U
  

ι:𝐋:=
    (    i I    ) /
    (     Ų     )

   𝐋 is not a valid type.

Y=U
ι  

test
        Y    /
        ι       = Ų

(8)

For example:

ex1:=[q[2]=1,p[1]=1]

(9)

Ų0:𝐋  :=eval(Ų,ex1)

   𝐋 is not a valid type.