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The Algebra of Complex Numbers Is Frobenius In Many Ways

Linear operators over a 2-dimensional vector space representing the algebra of complex numbers

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
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
fricas
-- subscript and superscripts
macro sb == subscript
Type: Void
fricas
macro sp == superscript
Type: Void

𝐋 is the domain of 2-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

fricas
dim:=2

\label{eq1}2(1)
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := ClosedLinearOperator(OVAR ['1,'i], 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 Complex Algebra

The basis consists of the real and imaginary units. We use complex 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 can be specified by Caley-Dickson parameter (i2)

fricas
--i2:=sp('i,[2])
i2:= -1  -- complex

\label{eq2}- 1(2)
Type: Integer
fricas
QQ := ℂ(ℚ,'i,-i2);
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 complex number

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

Units

fricas
e:=𝐞.1; i:=𝐞.2;
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.

Multiplication of arbitrary ccomplex numbers a and b

fricas
a:=Σ(sb('a,[i])*𝐞.i, 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.

Multiplication is Associative

fricas
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )
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(Y)
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 = \{ 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{eq3}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (3)
(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{eq4}��(4)
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)::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 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 pre-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.

Frobenius Form (co-unit)

fricas
d:=ε1*𝐝.1+εi*𝐝.2
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.

Express scalar product in terms of Frobenius form

fricas
𝔓:=solve(𝔇,Ξ(sb('p,[i]), i,1..#Ñ)).1
There are 2 exposed and 2 unexposed library operations named # having 1 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) Variable(Ñ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

In general the pairing is not symmetric!

fricas
u1:=matrix Ξ(Ξ(retract((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.

The scalar product must be non-degenerate:

fricas
Ů:=determinant u1
There are 3 exposed and 1 unexposed library operations named determinant having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op determinant 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 determinant with argument type(s) Variable(u1)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Cartan-Killing is a special case

fricas
ck:=solve(equate(eval(Ũ,𝔓)=Ų),[ε1,εi]).1
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(Ù) Variable(𝔓)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Frobenius scalar product of complex numbers a and b

fricas
a:=sb('a,[1])*e+sb('a,[2])*i

\label{eq5}{{a_{2}}\  i}+{{a_{1}}\  e}(5)
Type: Polynomial(Integer)
fricas
b:=sb('b,[1])*e+sb('b,[2])*i

\label{eq6}{{b_{2}}\  i}+{{b_{1}}\  e}(6)
Type: Polynomial(Integer)
fricas
(a,a)/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(Polynomial(Integer)) Variable(Ų)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

fricas
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
𝐋 is not a valid type.

The common demoninator is 1/\sqrt{\mathring{U}}

fricas
squareFreePart((factor denom Ů)::ℚ) / squareFreePart((factor numer Ů)::ℚ)
There are 3 exposed and 3 unexposed library operations named denom having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op denom 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 denom with argument type(s) Variable(Ů)
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
O:𝐋:=
       Ω    /
       Ų
𝐋 is not a valid type.

Cartan-Killing co-scalar

fricas
eval(Ω,ck)
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(ck)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Definition 4

Co-algebra

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

fricas
W:=
  (Y I) /
    Ų
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.

Cartan-Killing co-multiplication

fricas
eval(λ,ck)
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(λ) Variable(ck)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
test
     (    I ΩX     )  /
     (     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(ΩX)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Co-associativity

fricas
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ 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(λ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
test
         e     /
         λ     =    ΩX

\label{eq7} \mbox{\rm false} (7)
Type: Boolean

Frobenius Condition (fork)

fricas
H :=
         Y    /
         λ

\label{eq8}\frac{Y}{��}(8)
Type: Fraction(Polynomial(Integer))
fricas
test
     (   λ I   )  /
     (  I Y    )  =  H
There are no library operations named λ Use HyperDoc Browse or issue )what op λ to learn if there is any operation containing " λ " in its name.
Cannot find a definition or applicable library operation named λ 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.

The Cartan-Killing form makes H of the Frobenius condition idempotent

fricas
test( eval(H,ck)=eval(H/H,ck) )
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) Fraction(Polynomial(Integer)) Variable(ck)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

And it is unique.

fricas
h1:=map(numer,ravel(H-H/H)::List FRAC POLY INT);
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.

Handle

fricas
Φ :=
         λ     /
         Y

\label{eq9}\frac{��}{Y}(9)
Type: Fraction(Polynomial(Integer))

The Cartan-Killing form makes Φ of the identity

fricas
test( eval(Φ,ck)=I )
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) Fraction(Polynomial(Integer)) Variable(ck)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

and it can be the identity in only this one way.

fricas
solve(equate(Φ=I),[ε1,εi])
There are no library operations named equate Use HyperDoc Browse or issue )what op equate to learn if there is any operation containing " equate " in its name.
Cannot find a definition or applicable library operation named equate with argument type(s) Equation(Fraction(Polynomial(Integer)))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

If handle is identity then fork is idempotent but the converse is not true

fricas
Φ1:=map(numer,ravel(Φ-I)::List FRAC POLY INT);
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.

Figure 12

fricas
φφ:=          _
  ( Ω  Ω  ) / _
  ( X I I ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y );
There are no library operations named Ω Use HyperDoc Browse or issue )what op Ω to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω 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.

For Cartan-Killing this is just the co-scalar

fricas
test(eval(φφ,ck)=eval(Ω,ck))
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(φφ) Variable(ck)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Bi-algebra conditions

fricas
ΦΦ:=          _
  (  λ λ  ) / _
  ( I I X ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y ) ;
There are no library operations named λ Use HyperDoc Browse or issue )what op λ to learn if there is any operation containing " λ " in its name.
Cannot find a definition or applicable library operation named λ 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.

The Cartan Killing form is a bi-algebra

fricas
bi1:=map(numer,ravel(ΦΦ-H)::List FRAC POLY INT);
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.




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