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Octonion Algebra Is Frobenius In Just One Way

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

Ref:

We need the Axiom LinearOperator library.

fricas
(1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY
>> System error: The value 15684 is not of type LIST

Use the following macros for convenient notation

fricas
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
fricas
-- subscript
macro sb == subscript
Type: Void

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

fricas
dim:=8

\label{eq1}8(1)
Type: PositiveInteger?
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
ℒ := ClosedLinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ)
There are no library operations named ClosedLinearOperator Use HyperDoc Browse or issue )what op ClosedLinearOperator to learn if there is any operation containing " ClosedLinearOperator " in its name.
Cannot find a definition or applicable library operation named ClosedLinearOperator with argument type(s) 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

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

\label{eq2}1(2)
Type: PositiveInteger?
fricas
--q1:=sb('q,[1])
q1:=1  -- not co-quaternion

\label{eq3}1(3)
Type: PositiveInteger?
fricas
q2:=sb('q,[2])

\label{eq4}q_{2}(4)
Type: Symbol
fricas
--q2:=1  -- split-octonion
QQ := ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2);
There are no library operations named CaleyDickson Use HyperDoc Browse or issue )what op CaleyDickson to learn if there is any operation containing " CaleyDickson " in its name.
Cannot find a definition or applicable library operation named CaleyDickson with argument type(s) Type Variable(i) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Basis: Each B.i is a octonion number

fricas
B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::List List ℚ)
QQ is not a valid type.

A scalar product is denoted by the (2,0)-tensor U = \{ u_{ij} \}

fricas
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:


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\label{eq5}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (5)
(three-point function) is zero.

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.

fricas
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]?.

The Cartan-Killing Trace

fricas
Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V
There are no exposed library operations named Y but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op Y to learn more about the available operations.
Cannot find a definition or applicable library operation named Y with argument type(s) Variable(Λ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

forms a non-degenerate associative scalar product for Y

fricas
Ũ := Ù

\label{eq6}��(6)
Type: Variable(Ù)
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
There are no exposed library operations named Y but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op Y to learn more about the available operations.
Cannot find a definition or applicable library operation named Y with argument type(s) Variable(I)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

General Solution

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

This problem can be solved using linear algebra.

fricas
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables,ravel U)::List 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 U into coefficients of the tensor \Phi. We are looking for the general linear family of tensors U=U(Y,p_i) such that J transforms U into \Phi=0 for any such U.

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

fricas
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:

fricas
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!

fricas
Ų:ℒ := eval(U,ℰ)
ℒ is not a valid type.

The scalar product must be non-degenerate:

fricas
Ů:=determinant [[retract((ⅇ.i * ⅇ.j)/Ų) for j in 1..dim] for i in 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.

fricas
Um:=matrix [[(ⅇ.i*ⅇ.j)/Ų for i in 1..dim] for j in 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.

fricas
d:ℒ:=
       Ω    /
       X    /
       Ų
ℒ is not a valid type.

Definition 4

Co-algebra

Compute the "three-point" function and use it to define co-multiplication.

fricas
W:=(Y,I)/Ų
There are 11 exposed and 15 unexposed library operations named / having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op / 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 / with argument type(s) Tuple(OrderedVariableList([Y,I])) Variable(Ų)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
test
     (    I Ω     )  /
     (     Y I    )  =  λ
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.

Frobenius Condition

Octonion algebra fails the Frobenius Condition!

fricas
H :=
       Y    /
       λ

\label{eq7}\frac{Y}{��}(7)
Type: Fraction(Polynomial(Integer))
fricas
Hr := (λ,I)/(I,Y)
There are 11 exposed and 15 unexposed library operations named / having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op / 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 / with argument type(s) Tuple(OrderedVariableList([λ,I])) Tuple(OrderedVariableList([I,Y]))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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

i = Unit of the algebra

fricas
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 and handle element

fricas
Φ:ℒ :=
         λ     /
         X     /
         Y
ℒ is not a valid type.

Definition 5

Co-unit
  i 
  U
  

fricas
ι:ℒ:=
    (    i I    ) /
    (     Ų     )
ℒ is not a valid type.

Y=U
ι  
fricas
test
        Y    /
        ι       = Ų

\label{eq8} \mbox{\rm false} (8)
Type: Boolean

For example:

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

\label{eq9}\left[{{q_{2}}= 1}, \:{{p_{1}}= 1}\right](9)
Type: List(Equation(Polynomial(Integer)))
fricas
Ų0:ℒ  :=eval(Ų,ex1)
ℒ is not a valid type.




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