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Sedenion Algebra is Frobenius in just one way!

Linear operators over a 16-dimensional vector space representing Sedenion Algebra

Ref:

We need the Axiom LinearOperator library.

fricas
)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

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

fricas
dim:=16

\label{eq1}16(1)
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
ℒ := LinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7,'8,'9,'10,'11,'12,'13,'14,'15], ℚ)

\label{eq2}\hbox{\axiomType{LinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
fricas
ⅇ:List ℒ      := basisOut()
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

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

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

\label{eq3}1(3)
fricas
--q1:=sb('q,[1])
q1:=1  -- not co-quaternion

\label{eq4}1(4)
fricas
--q2:=sb('q,[2])
q2:=1  -- not split-octonion

\label{eq5}1(5)
fricas
--q3:=sb('q,[3])
q3:=1  -- not split-sedennion

\label{eq6}1(6)
fricas
QQ := ℂ(ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2),'l,q3);
Type: Type

Basis: Each B.i is a sedennion number

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

\label{eq7}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}, \: l , \:{il}, \:{jl}, \:{{ij}l}, \:{kl}, \:{{ik}l}, \:{{jk}l}, \:{{{ij}k}l}\right](7)
Type: List(CaleyDickson(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,1),l,1))
fricas
-- Multiplication table:
M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]

\label{eq8}\left[ 
\begin{array}{cccccccccccccccc}
1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k}& l &{il}&{jl}&{{ij}l}&{kl}&{{ik}l}&{{jk}l}&{{{ij}k}l}
\
i & - 1 & -{ij}& j &{- ik}& k &{{ij}k}& -{jk}&{- il}& l &{{ij}l}&{- jl}&{{ik}l}& -{kl}&{{-{ij}k}l}&{{jk}l}
\
j &{ij}& - 1 & - i & -{jk}&{-{ij}k}& k &{ik}&{- jl}&{-{ij}l}& l &{il}&{{jk}l}&{{{ij}k}l}& -{kl}&{{- ik}l}
\
{ij}& - j & i & - 1 &{-{ij}k}&{jk}&{- ik}& k &{-{ij}l}&{jl}&{- il}& l &{{{ij}k}l}&{-{jk}l}&{{ik}l}& -{kl}
\
k &{ik}&{jk}&{{ij}k}& - 1 & - i & - j & -{ij}& -{kl}&{{- ik}l}&{-{jk}l}&{{-{ij}k}l}& l &{il}&{jl}&{{ij}l}
\
{ik}& - k &{{ij}k}& -{jk}& i & - 1 &{ij}& - j &{{- ik}l}&{kl}&{{-{ij}k}l}&{{jk}l}&{- il}& l &{-{ij}l}&{jl}
\
{jk}&{-{ij}k}& - k &{ik}& j & -{ij}& - 1 & i &{-{jk}l}&{{{ij}k}l}&{kl}&{{- ik}l}&{- jl}&{{ij}l}& l &{- il}
\
{{ij}k}&{jk}&{- ik}& - k &{ij}& j & - i & - 1 &{{-{ij}k}l}&{-{jk}l}&{{ik}l}&{kl}&{-{ij}l}&{- jl}&{il}& l 
\
l &{il}&{jl}&{{ij}l}&{kl}&{{ik}l}&{{jk}l}&{{{ij}k}l}& - 1 & - i & - j & -{ij}& - k &{- ik}& -{jk}&{-{ij}k}
\
{il}& - l &{{ij}l}&{- jl}&{{ik}l}& -{kl}&{{-{ij}k}l}&{{jk}l}& i & - 1 &{ij}& - j &{ik}& - k &{-{ij}k}&{jk}
\
{jl}&{-{ij}l}& - l &{il}&{{jk}l}&{{{ij}k}l}& -{kl}&{{- ik}l}& j & -{ij}& - 1 & i &{jk}&{{ij}k}& - k &{- ik}
\
{{ij}l}&{jl}&{- il}& - l &{{{ij}k}l}&{-{jk}l}&{{ik}l}& -{kl}&{ij}& j & - i & - 1 &{{ij}k}& -{jk}&{ik}& - k 
\
{kl}&{{- ik}l}&{-{jk}l}&{{-{ij}k}l}& - l &{il}&{jl}&{{ij}l}& k &{- ik}& -{jk}&{-{ij}k}& - 1 & i & j &{ij}
\
{{ik}l}&{kl}&{{-{ij}k}l}&{{jk}l}&{- il}& - l &{-{ij}l}&{jl}&{ik}& k &{-{ij}k}&{jk}& - i & - 1 & -{ij}& j 
\
{{jk}l}&{{{ij}k}l}&{kl}&{{- ik}l}&{- jl}&{{ij}l}& - l &{- il}&{jk}&{{ij}k}& k &{- ik}& - j &{ij}& - 1 & - i 
\
{{{ij}k}l}&{-{jk}l}&{{ik}l}&{kl}&{-{ij}l}&{- jl}&{il}& - l &{{ij}k}& -{jk}&{ik}& k & -{ij}& - j & i & - 1 
(8)
Type: Matrix(CaleyDickson(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,1),l,1))
fricas
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real real real(x/y),M)
Type: Void
fricas
-- The result is a nested list
ѕ :=map(S,B)::List List List ℚ;
fricas
Compiling function S with type CaleyDickson(CaleyDickson(
      CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,1),l,1)
       -> Matrix(Expression(Integer))
Type: List(List(List(Expression(Integer))))
fricas
-- 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 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 "$" to specify which version of the function you need. FriCAS will attempt to step through and interpret the code. Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

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{eq9}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (9)
(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;
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))

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 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 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(U)
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 12 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) LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer)) 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)

\label{eq10}U(10)
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
matrix [[(ⅇ.i ⅇ.j)/Ų for i in 1..dim] for j in 1..dim]
0 - 2 arity warning: ------ 0 1 2 - (-) 0 1 Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

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]
0 - 2 arity warning: ------ 0 1 2 - (-) 0 1 Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

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];
0 - 2 arity warning: ------ 0 1 2 - (-) 0 1 Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Check "dimension" and the snake relations.

fricas
d:ℒ:=
       Ω    /
       X    /
       Ų

\label{eq11}U \  X \  ��(11)
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
test
    (    I Ω     )  /
    (     Ų I    )  =  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(Ω)
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.

Too slow:

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

fricas
λ:= (I,Ω) / (Y,I)
There are 16 exposed and 12 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([I,Ω])) Tuple(OrderedVariableList([Y,I]))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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

fricas
i:=ⅇ.1
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Handle

fricas
H:ℒ :=
         λ     /
         X     /
         Y

\label{eq12}{X \  ��}\over Y(12)
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))

Definition 5

Co-unit
  i 
  U
  

fricas
ι:ℒ:=
    (    i I    ) /
    (     Ų     )
0 - 1 arity warning: ------ 0 1 1 - (-) 0 1 Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Y=U
ι  
fricas
test
        Y    /
        ι       = Ų

\label{eq13} \mbox{\rm false} (13)
Type: Boolean

For example:

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

\label{eq14}\left[{{q_{3}}= 1}, \:{{p_{1}}= 1}\right](14)
Type: List(Equation(Polynomial(Integer)))
fricas
Ų0:ℒ  :=eval(Ų,ex1)

\label{eq15}U(15)
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
Ω0:ℒ  :=eval(Ω,ex1)$ℒ

\label{eq16}��(16)
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
λ0:ℒ  :=eval(λ,ex1)$ℒ

\label{eq17}��(17)
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
H0:ℒ :=eval(H,ex1)$ℒ

\label{eq18}{X \  ��}\over Y(18)
Type: LinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))




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