Edit detail for Octonion Algebra is Frobenius in Just One Way revision 3 of 8

changed:
-Octonion Algebra is Frobenius in just one way!
-
-  8-dimensional vector space representing Octonion Algebra
-
-\begin{axiom}
-)set output tex off
-)set output algebra on
-\end{axiom}
Octonion Algebra Is Frobenius In Just One Way

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

Ref:

- http://arxiv.org/abs/1103.5113

  $S_3$-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.
\begin{axiom}
)library CARTEN MONAL PROP LIN CALEY
\end{axiom}

Use the following macros for convenient notation
\begin{axiom}
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
-- list
macro Ξ(f,i,n)==[f for i in n]
-- subscript
macro sb == subscript
\end{axiom}

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

changed:
-R ==> EXPR INT
-T ==> CartesianTensor(1,dim,EXPR INT)
-X:List T := [unravel [(i=j => 1;0) for j in 1..dim] for i in 1..dim]
-X(1),X(2)
-\end{axiom}
-
-Generate structure constants for Octonion Algebra (the Caley-Dickson way)
-\begin{axiom}
-Q ==> Quaternion R
-O ==> DirectProduct(2,Q)
-B0:=map(x+->octon(x.1,x.2,x.3,x.4,x.5,x.6,x.7,x.8),1$SQMATRIX(dim,R)::List List R)
-pair(x:Q,y:Q):O==directProduct vector [x,y]
-caleyOne:=pair(1,0)
-B:=map(x+->pair(quatern(x.1,x.2,x.3,x.4),quatern(x.5,x.6,x.7,x.8)),1$SQMATRIX(dim,R)::List List R)
-\end{axiom}
-\begin{axiom}
-caleyMul(x:O,y:O):O == pair((x.1)*(y.1) - conjugate(y.2)*(x.2), (y.2)*(x.1) + (x.2)*conjugate(y.1))
-caleyMul(caleyOne,caleyOne)
-M0:=matrix [[B0.i*B0.j for j in 1..dim] for i in 1..dim]  
-M:=matrix [[caleyMul(B.i,B.j) for j in 1..dim] for i in 1..dim]
-\end{axiom}
-\begin{axiom}
-caleyConj(x:O):O == pair(conjugate(x.1), -x.2)
-caleyInv(x:O):O == inv(caleyMul(caleyConj x,x).1) * caleyConj(x)
-S0(y)==map(x+->(x*inv(y)=1 or x*inv(y)=-1 => x*inv(y);0),M0)
-S0(B0.1)
-S(y)==map(x+->(caleyMul(x,caleyInv y)=caleyOne => 1;caleyMul(x,caleyInv y)=-caleyOne => -1;0),M)
-S(B.1)
-Yg0:T:=unravel concat concat(map(S0,B0)::List List List R);
-Yg:T:=unravel concat concat(map(S,B)::List List List R)
-test(Yg0=Yg)
-\end{axiom}
macro ℒ == List
macro ℂ == CaleyDickson
macro ℚ == Expression Integer
𝐋 := LinearOperator(dim, OVAR [], ℚ)
𝐞:ℒ 𝐋      := basisVectors()
𝐝:ℒ 𝐋      := basisForms()
o:𝐋:=1;     -- identity for product
I:𝐋:=[1];   -- identity for composition
X:𝐋:=[2,1]; -- twist
\end{axiom}

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
\begin{axiom}
--q0:=sb('q,[0])
q0:=1  -- not split-complex
--q1:=sb('q,[1])
q1:=1  -- not co-quaternion
q2:=sb('q,[2])
--q2:=1  -- split-octonion
QQ := ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2);
\end{axiom}

Basis: Each B.i is a octonion number
\begin{axiom}
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)
-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real real(x/y),M)
-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ;
-- 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
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)
\end{axiom}

changed:
-U:T := unravel(concat
-  [[script(u,[[],[j,i]])
-    for i in 1..dim]
-      for j in 1..dim]
-        )
-\end{axiom}
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
\end{axiom}


changed:
-  In other words, if the (3,0)-tensor::
-
-    i  j  k   i  j  k   i  j  k
-     \ | /     \/  /     \  \/
-      \|/   =   \ /   -   \ /
-       0         0         0
  In other words, if the (3,0)-tensor:
$$
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\rput(1.6948438,0.205){=}
\end{pspicture} 
}
$$

changed:
-\begin{axiom}
-ω := reindex(reindex(U,[2,1])*reindex(Yg,[1,3,2]),[3,2,1])-U*Yg;
-\end{axiom}
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.

\begin{axiom}
ω:𝐋 :=(Y*I)/U  - (I*Y)/U;
\end{axiom}

Note: The only purpose of the o symbols on the left above is to serve as a constant left-side margin as required by Axiom. The symbols on the right describe the relation between row.


changed:
-  is called *pre-Frobenius*.
  is called a [Frobenius Algebra].

added:


changed:
-J := jacobian(ravel ω,concat(map(variables,ravel U))::List Symbol);
-uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
-J::OutputForm * uu::OutputForm = 0;
-nrows(J)
-ncols(J)
-\end{axiom}
J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);
u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
J::OutputForm * u::OutputForm = 0
nrows(J),ncols(J)
\end{axiom}


changed:
-\begin{axiom}
-NJ:=nullSpace(J)
-SS:=map((x,y)+->x=y,concat map(variables,ravel U),
-  entries reduce(+,[p[i]*NJ.i for i in 1..#NJ]))
-Ug:T := unravel(map(x+->subst(x,SS),ravel U))
-\end{axiom}
-
-This defines a family of pre-Frobenius algebras:
-\begin{axiom}
-test(unravel(map(x+->subst(x,SS),ravel ω))$T=0*ω)
-\end{axiom}

\begin{axiom}
Ñ:=nullSpace(J);
ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )
\end{axiom}

This defines a family of Frobenius algebras:
\begin{axiom}
zero? eval(ω,ℰ)
\end{axiom}

In general the pairing is not symmetric!
\begin{axiom}
Ų:𝐋 := eval(U,ℰ)
matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim)
\end{axiom}

This is the most general form of the "dot product" of two quaternions
\begin{axiom}
(a*b)/Ų
(a*a)/Ų
\end{axiom}

changed:
-Ud:DMP([p[i] for i in 1..#NJ],INT) := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim]
-factor Ud
-\end{axiom}
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim)
factor Ů
\end{axiom}

changed:
-\begin{axiom}
-Ωg:T:=unravel concat(transpose(1/Ud*adjoint([[Ug[i,j] for j in 1..dim] for i in 1..dim]).adjMat)::List List FRAC POLY INT)
-\end{axiom}
-<center><pre>
-dimension
-Ω
-U
-</pre></center>
-\begin{axiom}
-contract(contract(Ωg,1,Ug,1),1,2)
-\end{axiom}

Solve the [Snake Relation] as a system of linear equations.
\begin{axiom}
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/Ų, i,1..dim), j,1..dim)
mU:=transpose inverse map(retract,Um)
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)
\end{axiom}

Check "dimension" and the snake relations.
\begin{axiom}

d:𝐋:=
    o   Ω    /
    o   X    /
    o   Ų    o

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

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

\end{axiom}

changed:
-  Co-multiplication
-\begin{axiom}
-λg:=reindex(contract(contract(Ug*Yg,1,Ωg,1),1,Ωg,1),[2,3,1]);
--- just for display
-reindex(λg,[3,1,2])
-\end{axiom}
-<center><pre>
-i  
-λ=Ω
-</pre></center>
-\begin{axiom}
-test(λg*X(1)=Ωg)
-\end{axiom}
  Co-algebra

This expression is expensive to compute::

    λ:𝐋 :=
         o    Ω Ω  I    /
         o   I Y I I    /
         o   I  X  I    /
         o   I I  Ų     o

\begin{axiom}

λ:𝐋 :=
     (    I Ω     )  /
     (     Y I    )

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

\end{axiom}

Frobenius Condition

  It takes to long to computer here but it turns out that Octonion
algebra fails the Frobenius Condition::

  !\begin{axiom}

  Χ :=
         Y    /
         λ

  test
     (   λ I   )  /
     (  I Y    )  =  Χ

  test
     (   I λ   )  /
     (    Y I  )  =  Χ

  \end{axiom}

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

i = Unit of the algebra
\begin{axiom}
i:=𝐞.1
test
     o    i     /
     o    λ     =    Ω

\end{axiom}

Handle
\begin{axiom}

H:𝐋 :=
     o    λ    o /
     o    Y    o

\end{axiom}

changed:
-ιg:=X(1)*Ug

ι:𝐋:=
    o    i I    /
    o     Ų    o


changed:
-test(ιg * Yg = Ug)
-\end{axiom}
test
   o     Y     /
   o     ι     o  = Ų

\end{axiom}

changed:
-Ug0:T:=unravel eval(ravel Ug,[p[1]=1])
-Ωg0:T:=unravel eval(ravel Ωg,[p[1]=1])
-λg0:T:=unravel eval(ravel λg,[p[1]=1]);
-reindex(λg0,[3,1,2])
-\end{axiom}
ex1:=[q[2]=1,p[1]=1]
Ų0:𝐋  :=eval(Ų,ex1)
Ω0:𝐋  :=eval(Ω,ex1)$𝐋
λ0:𝐋  :=eval(λ,ex1)$𝐋
H0:𝐋 :=eval(H,ex1)$𝐋
\end{axiom}

Octonion Algebra Is Frobenius In Just One Way

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

axiom

)library CARTEN MONAL PROP LIN CALEY

   CartesianTensor is now explicitly exposed in frame initial 
   CartesianTensor will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN
   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/PROP.NRLIB/PROP
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN
   CaleyDickson is now explicitly exposed in frame initial 
   CaleyDickson will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

axiom

-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])

Type: Void

axiom

-- list
macro Ξ(f,i,n)==[f for i in n]

Type: Void

axiom

-- 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.

axiom

dim:=8

(1)

Type: PositiveInteger?

axiom

macro ℒ == List

Type: Void

axiom

macro ℂ == CaleyDickson

Type: Void

axiom

macro ℚ == Expression Integer

Type: Void

axiom

𝐋 := LinearOperator(dim, OVAR [], ℚ)

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisVectors()

(3)

Type: List(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

axiom

𝐝:ℒ 𝐋      := basisForms()

(4)

Type: List(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

axiom

o:𝐋:=1;     -- identity for product

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

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

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

X:𝐋:=[2,1]; -- twist

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

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

axiom

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

(5)

Type: PositiveInteger?

axiom

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

(6)

Type: PositiveInteger?

axiom

q2:=sb('q,[2])

(7)

Type: Symbol

axiom

--q2:=1  -- split-octonion
QQ := ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2);

Type: Type

Basis: Each B.i is a octonion number

axiom

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

(8)

Type: List(CaleyDickson?(CaleyDickson?(CaleyDickson?(Expression(Integer),i,1),j,1),k,*01q(2)))

axiom

-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

(9)

Type: Matrix(CaleyDickson?(CaleyDickson?(CaleyDickson?(Expression(Integer),i,1),j,1),k,*01q(2)))

axiom

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

Type: Void

axiom

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

axiom

Compiling function S with type CaleyDickson(CaleyDickson(
      CaleyDickson(Expression(Integer),i,1),j,1),k,*01q(2)) -> Matrix(
      Expression(Integer))

Type: List(List(List(Expression(Integer))))

axiom

-- 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)

(10)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

arity Y

(11)

Type: Prop(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

axiom

matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

(12)

Type: Matrix(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

A scalar product is denoted by the (2,0)-tensor

axiom

U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

(13)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

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:

(14)

(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.

axiom

ω:𝐋 :=(Y*I)/U  - (I*Y)/U;

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

Note: The only purpose of the o symbols on the left above is to serve as a constant left-side margin as required by Axiom. The symbols on the right describe the relation between row.

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.

axiom

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial 
J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);

Type: Matrix(Expression(Integer))

axiom

u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];

Type: Matrix(Polynomial(Integer))

axiom

J::OutputForm * u::OutputForm = 0

(15)

Type: Equation(OutputForm?)

axiom

nrows(J),ncols(J)

(16)

Type: Tuple(PositiveInteger?)

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:

axiom

Ñ:=nullSpace(J);

Type: List(Vector(Expression(Integer)))

axiom

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

(17)

Type: List(Equation(Expression(Integer)))

This defines a family of Frobenius algebras:

axiom

zero? eval(ω,ℰ)

(18)

Type: Boolean

In general the pairing is not symmetric!

axiom

Ų:𝐋 := eval(U,ℰ)

(19)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim)

(20)

Type: Matrix(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

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

axiom

(a*b)/Ų

                   1 2 0
                  (-)  -
                   1   0
   arity warning: ------
                     2
                     -
                     0

(21)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

(a*a)/Ų

                   1 2 0
                  (-)  -
                   1   0
   arity warning: ------
                     2
                     -
                     0

(22)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

The scalar product must be non-degenerate:

axiom

Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim)

(23)

Type: Expression(Integer)

axiom

factor Ů

(24)

Type: Factored(Expression(Integer))

Definition 3

Co-pairing

Solve the [Snake Relation]? as a system of linear equations.

axiom

Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/Ų, i,1..dim), j,1..dim)

(25)

Type: Matrix(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

axiom

mU:=transpose inverse map(retract,Um)

(26)

Type: Matrix(Expression(Integer))

axiom

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

(27)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)

(28)

Type: Matrix(LinearOperator?(8,OrderedVariableList?([]),Expression(Integer)))

Check "dimension" and the snake relations.

axiom

d:𝐋:=
    o   Ω    /
    o   X    /
    o   Ų    o

(29)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

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

(30)

Type: Boolean

axiom

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

(31)

Type: Boolean

Definition 4

Co-algebra

This expression is expensive to compute:

    λ:𝐋 :=
         o    Ω Ω  I    /
         o   I Y I I    /
         o   I  X  I    /
         o   I I  Ų     o

axiom

λ:𝐋 :=
     (    I Ω     )  /
     (     Y I    )

(32)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

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

(33)

Type: Boolean

Frobenius Condition

It takes to long to computer here but it turns out that Octonion algebra fails the Frobenius Condition::

\begin{axiom}

Χ := Y / λ

test ( λ I ) / ( I Y ) = Χ

test ( I λ ) / ( Y I ) = Χ

\end{axiom}

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

i = Unit of the algebra

axiom

i:=𝐞.1

(34)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

test
     o    i     /
     o    λ     =    Ω

(35)

Type: Boolean

Handle

axiom

H:𝐋 :=
     o    λ    o /
     o    Y    o

(36)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

Definition 5

Co-unit

  i 
  U
  

axiom

ι:𝐋:=
    o    i I    /
    o     Ų    o

(37)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

Y=U
ι  

axiom

test
   o     Y     /
   o     ι     o  = Ų

(38)

Type: Boolean

For example:

axiom

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

(39)

Type: List(Equation(Polynomial(Integer)))

axiom

Ų0:𝐋  :=eval(Ų,ex1)

(40)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

Ω0:𝐋  :=eval(Ω,ex1)$𝐋

(41)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

λ0:𝐋  :=eval(λ,ex1)$𝐋

(42)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))

axiom

H0:𝐋 :=eval(H,ex1)$𝐋

(43)

Type: LinearOperator?(8,OrderedVariableList?([]),Expression(Integer))