Sedenion Algebra is Frobenius in just one way!

Linear operators over a 16-dimensional vector space representing Sedenion 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
• http://en.wikipedia.org/wiki/Sedenion

We need the Axiom LinearOperator library.

)library CARTEN MONAL PROP LOP CALEY

   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
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/LOP.NRLIB/LOP
   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])

-- subscript
macro sb == subscript

ℒ is the domain of 16-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

dim:=16

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macro ℂ == CaleyDickson

macro ℚ == Expression Integer

ℒ := LinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7,'8,'9,'10,'11,'12,'13,'14,'15], ℚ)

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ⅇ:List ℒ      := basisOut()

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ⅆ:List ℒ      := basisIn()

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I:ℒ:=[1];   -- identity for composition

X:ℒ:=[2,1]; -- twist

Now generate structure constants for Sedenion 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, split-octonions and seneions can be specified by Caley-Dickson parameters

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

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

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--q2:=sb('q,[2])
q2:=1  -- not split-octonion

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--q3:=sb('q,[3])
q3:=1  -- not split-sedennion

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QQ := ℂ(ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2),'l,q3);

Basis: Each B.i is a sedennion number

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

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-- Multiplication table:
M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]

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

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

Compiling function S with type CaleyDickson(CaleyDickson(
      CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,1),l,1)
       -> 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);

arity Y

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matrix [[(ⅇ.i*ⅇ.j)/Y for i in 1..dim] for j in 1..dim]

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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);

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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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;

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)::List Symbol);

--u := transpose matrix [concat map(variables,ravel U)::List Symbol];
--J::OutputForm * u::OutputForm = 0
nrows(J),ncols(J)

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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);

ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )

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This defines a family of Frobenius algebras:

zero? eval(ω,ℰ)

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The pairing is necessarily diagonal!

Ų:ℒ := eval(U,ℰ)

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matrix [[(ⅇ.i ⅇ.j)/Ų for i in 1..dim] for j in 1..dim]

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The scalar product must be non-degenerate:

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

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factor Ů

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Definition 3

Co-pairing

Solve the Snake Relation as a system of linear equations.

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

mU:=transpose inverse map(retract,Um);

Ω:=Σ(Σ(mU(i,j)*(ⅇ.i*ⅇ.j), i,1..dim), j,1..dim)

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matrix [[Ω/(ⅆ.i*ⅆ.j) for i in 1..dim] for j in 1..dim]

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Check "dimension" and the snake relations.

d:ℒ:=
       Ω    /
       X    /
       Ų

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test
    (    I Ω     )  /
    (     Ų I    )  =  I

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test
    (     Ω I    )  /
    (    I Ų     )  =  I

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Definition 4

Co-algebra

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

Too slow:

  \begin{axiom}
  W:=(Y,I)/Ų;
  λ:=(Ω,I,Ω)/(I,W,I)
  \end{axiom}

λ:= (I,Ω) / (Y,I)

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test( (Ω,I) / (I,Y) = λ )

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Frobenius Condition

Like Octonion algebra Sedenion algebra also fails the Frobenius Condition!

Too slow to complete here:

  \begin{axiom}

  Χ := Y / λ ;

  Χr := (λ,I)/(I,Y)
  test(Χr = Χ )

  Χl := (I,λ)/(Y,I);
  --test( Χl = Χ )
  test( Χr = Χl )

  \end{axiom}

Perhaps this is not too surprising since like Octonion Seden algebra is non-associative (in fact also non-alternative). Nevertheless Sedenions are "Frobenius" in a more general sense just because there is a non-degenerate associative pairing.

i = Unit of the algebra

i:=ⅇ.1

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test
         i     /
         λ     =    Ω

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Handle

H:ℒ :=
         λ     /
         X     /
         Y

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Definition 5

  i 
  U
  

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

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Y=U
ι  

test
        Y    /
        ι       = Ų

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For example:

ex1:=[q[3]=1,p[1]=1]

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Ų0:ℒ  :=eval(Ų,ex1)

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Ω0:ℒ  :=eval(Ω,ex1)$ℒ

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λ0:ℒ  :=eval(λ,ex1)$ℒ

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H0:ℒ :=eval(H,ex1)$ℒ

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