Edit detail for SandBox Quaternion Algebra is Frobenius in Many Ways revision 25 of 32

Quaternion Algebra Is Frobenius In Many Ways

Linear operators over a 4-dimensional vector space representing quaternion 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 ARITY CMONAL CPROP CLOP CALEY

   CartesianTensor is now explicitly exposed in frame initial 
   CartesianTensor will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN
   Arity is now explicitly exposed in frame initial 
   Arity will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY
   ClosedMonoidal is now explicitly exposed in frame initial 
   ClosedMonoidal will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL
   ClosedProp is now explicitly exposed in frame initial 
   ClosedProp will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP
   ClosedLinearOperator is now explicitly exposed in frame initial 
   ClosedLinearOperator will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP
   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 and superscripts
macro sb == subscript

macro sp == superscript

𝐋 is the domain of 4-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

dim:=4

(1)

macro ℒ == List

macro ℂ == CaleyDickson

macro ℚ == Expression Integer

𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k], ℚ)

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

   Cannot find a no-argument definition or library operation named 
      OrderedFinite .

Now generate structure constants for Quaternion 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 and co-quaternions can be specified by Caley-Dickson parameters (q0 = -1, q1 = -1)

q0:=sp('i,[2])

(2)

--q0:=σ::Symbol
--q0:=1  -- not split-complex
q1:=sp('j,[2])

(3)

--q1:=μ::Symbol
--q1:=1  -- quaternion
QQ := ℂ(ℂ(ℚ,'i,q0),'j,q1);

   >> System error:
   The function BOOT::PUT is undefined.

Basis: Each B.i is a quaternion number

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

   QQ is not a valid type.

Multiplication of arbitrary quaternions and

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

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:=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:

(4)

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]?.

The Cartan-Killing Trace

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

Ũ := Ù

(5)

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

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 2 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 2 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:

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.

In general the pairing is not symmetric!

Ų:𝐋 := eval(U,ℰ)

   𝐋 is not a valid type.

This is the most general form of the "dot product" of two quaternions

(a*b)/Ų

(6)

(a*a)/Ų

(7)

Cartan-Killing is a special case

ck:=solve(equate(Ũ=Ų),Ξ(sb('p,[i]), i,1..#Ñ)).1

   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(Symbol)

      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:

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.

Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)

   𝐋 is not a valid type.

Check "dimension" and the snake relations.

d:𝐋:=
       Ω    /
       Ų

   𝐋 is not a valid type.

Definition 4

Co-algebra

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

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.

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

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.

Frobenius Condition (fork)

H :=
         Y    /
         λ

(8)

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

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.

But it is not unique. E.g. other idempots

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

   There are 2 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

Φ:𝐋 :=
         λ     /
         Y

   𝐋 is not a valid type.

The Cartan-Killing form makes Φ of the identity

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) 
                                 Variable(Φ)
                                Variable(ck)

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

but it can be the identity in many ways. For example,

solve(equate(eval(Φ,[p[1]=1,p[2]=1,p[3]=1,q1=1])=I),[p[4]])

   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(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

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

   There are 2 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) 
                             Polynomial(Integer)

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

Scalars for figure 12 (only 1 non-zero for Cartan-Killing form)

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.

Bi-algebra conditions

ΦΦ:=          _
  ( λ  λ  ) / _
  ( X I I ) / _
  ( 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.

solve(equate(ΦΦ=H/X),[p[1],p[2],p[3],p[4]])

   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.

Definition 5

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.

Co-unit

i 
U

ι:𝐋:=
    (    i I   ) /
          Ų

   𝐋 is not a valid type.

Y=U
ι  

test
        Y     /
        ι     =  Ų

(9)

For example:

ex1:=[q0=1,q1=1,p[1]=1,p[2]=1,p[3]=1,p[4]=1]

(10)

Ų0:𝐋  :=eval(Ų,ex1)

   𝐋 is not a valid type.