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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
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Octonion Algebra is Frobenius in Just One Way

Edit detail for Octonion Algebra is Frobenius in Just One Way revision 4 of 8

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Editor: Bill Page Time: 2011/04/26 23:35:34 GMT-7
Note: update

removed:
-o:𝐋:=1;     -- identity for product

removed:
-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:
-In general the pairing is not symmetric!
The pairing is necessarily diagonal!

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

changed:
-    o   Ω    /
-    o   X    /
-    o   Ų    o
       Ω    /
       X    /
       Ų

changed:
-         o    Ω Ω  I    /
-         o   I Y I I    /
-         o   I  X  I    /
-         o   I I  Ų     o
-
-\begin{axiom}
         (    Ω Ω  I    ) /
         (   I Y I I    ) /
         (   I  X  I    ) /
         (   I I  Ų     )

\begin{axiom}

changed:
-     o    i     /
-     o    λ     =    Ω
-
-\end{axiom}
         i     /
         λ     =    Ω

\end{axiom}

changed:
-     o    λ    o /
-     o    Y    o
-
-\end{axiom}
         λ     /
         Y

\end{axiom}

changed:
-    o    i I    /
-    o     Ų    o
    (    i I    ) /
    (     Ų     )

changed:
-   o     Y     /
-   o     ι     o  = Ų
-
-\end{axiom}
        Y    /
        ι       = Ų

\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

$\label{eq1}8$

(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)

$\label{eq2}\hbox{\axiomType{LinearOperator}\ } (8, \hbox{\axiomType{OrderedVariableList}\ } ([ ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisVectors()

$\label{eq3}\left[{|_{1}}, \:{|_{2}}, \:{|_{3}}, \:{|_{4}}, \:{|_{5}}, \:{|_{6}}, \:{|_{7}}, \:{|_{8}}\right]$

(3)

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

axiom

𝐝:ℒ 𝐋      := basisForms()

$\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}, \:{|_{\ }^{3}}, \:{|_{\ }^{4}}, \:{|_{\ }^{5}}, \:{|_{\ }^{6}}, \:{|_{\ }^{7}}, \:{|_{\ }^{8}}\right]$

(4)

Type: List(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

$\label{eq5}1$

(5)

Type: PositiveInteger?

axiom

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

$\label{eq6}1$

(6)

Type: PositiveInteger?

axiom

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

$\label{eq7}q_{2}$

(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)

$\label{eq8}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}\right]$

(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)

$\label{eq9}\left[ \begin{array}{cccccccc} 1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k} \ i & - 1 & -{ij}& j &{- ik}& k &{{ij}k}& -{jk} \ j &{ij}& - 1 & - i & -{jk}&{-{ij}k}& k &{ik} \ {ij}& - j & i & - 1 &{-{ij}k}&{jk}&{- ik}& k \ k &{ik}&{jk}&{{ij}k}& -{q_{2}}&{-{q_{2}}i}&{-{q_{2}}j}&{{-{q_{2}}i}j} \ {ik}& - k &{{ij}k}& -{jk}&{{q_{2}}i}& -{q_{2}}&{{{q_{2}}i}j}&{-{q_{2}}j} \ {jk}&{-{ij}k}& - k &{ik}&{{q_{2}}j}&{{-{q_{2}}i}j}& -{q_{2}}&{{q_{2}}i} \ {{ij}k}&{jk}&{- ik}& - k &{{{q_{2}}i}j}&{{q_{2}}j}&{-{q_{2}}i}& -{q_{2}}$

(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)

$\label{eq10}\begin{array}{@{}l} \displaystyle {|_{1}^{1 \ 1}}+{|_{2}^{1 \ 2}}+{|_{3}^{1 \ 3}}+{|_{4}^{1 \ 4}}+{|_{5}^{1 \ 5}}+{|_{6}^{1 \ 6}}+{|_{7}^{1 \ 7}}+{|_{8}^{1 \ 8}}+ \ \ \displaystyle {|_{2}^{2 \ 1}}-{|_{1}^{2 \ 2}}+{|_{4}^{2 \ 3}}-{|_{3}^{2 \ 4}}+{|_{6}^{2 \ 5}}-{|_{5}^{2 \ 6}}-{|_{8}^{2 \ 7}}+{|_{7}^{2 \ 8}}+ \ \ \displaystyle {|_{3}^{3 \ 1}}-{|_{4}^{3 \ 2}}-{|_{1}^{3 \ 3}}+{|_{2}^{3 \ 4}}+{|_{7}^{3 \ 5}}+{|_{8}^{3 \ 6}}-{|_{5}^{3 \ 7}}-{|_{6}^{3 \ 8}}+ \ \ \displaystyle {|_{4}^{4 \ 1}}+{|_{3}^{4 \ 2}}-{|_{2}^{4 \ 3}}-{|_{1}^{4 \ 4}}+{|_{8}^{4 \ 5}}-{|_{7}^{4 \ 6}}+{|_{6}^{4 \ 7}}-{|_{5}^{4 \ 8}}+ \ \ \displaystyle {|_{5}^{5 \ 1}}-{|_{6}^{5 \ 2}}-{|_{7}^{5 \ 3}}-{|_{8}^{5 \ 4}}-{{q_{2}}\ {|_{1}^{5 \ 5}}}+{{q_{2}}\ {|_{2}^{5 \ 6}}}+{{q_{2}}\ {|_{3}^{5 \ 7}}}+ \ \ \displaystyle {{q_{2}}\ {|_{4}^{5 \ 8}}}+{|_{6}^{6 \ 1}}+{|_{5}^{6 \ 2}}-{|_{8}^{6 \ 3}}+{|_{7}^{6 \ 4}}-{{q_{2}}\ {|_{2}^{6 \ 5}}}-{{q_{2}}\ {|_{1}^{6 \ 6}}}- \ \ \displaystyle {{q_{2}}\ {|_{4}^{6 \ 7}}}+{{q_{2}}\ {|_{3}^{6 \ 8}}}+{|_{7}^{7 \ 1}}+{|_{8}^{7 \ 2}}+{|_{5}^{7 \ 3}}-{|_{6}^{7 \ 4}}-{{q_{2}}\ {|_{3}^{7 \ 5}}}+ \ \ \displaystyle {{q_{2}}\ {|_{4}^{7 \ 6}}}-{{q_{2}}\ {|_{1}^{7 \ 7}}}-{{q_{2}}\ {|_{2}^{7 \ 8}}}+{|_{8}^{8 \ 1}}-{|_{7}^{8 \ 2}}+{|_{6}^{8 \ 3}}+{|_{5}^{8 \ 4}}- \ \ \displaystyle {{q_{2}}\ {|_{4}^{8 \ 5}}}-{{q_{2}}\ {|_{3}^{8 \ 6}}}+{{q_{2}}\ {|_{2}^{8 \ 7}}}-{{q_{2}}\ {|_{1}^{8 \ 8}}}$

(10)

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

axiom

arity Y

$\label{eq11}2 \over 1$

(11)

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

axiom

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

$\label{eq12}\left[ \begin{array}{cccccccc} {|_{1}}&{|_{2}}&{|_{3}}&{|_{4}}&{|_{5}}&{|_{6}}&{|_{7}}&{|_{8}} \ {|_{2}}& -{|_{1}}& -{|_{4}}&{|_{3}}& -{|_{6}}&{|_{5}}&{|_{8}}& -{|_{7}} \ {|_{3}}&{|_{4}}& -{|_{1}}& -{|_{2}}& -{|_{7}}& -{|_{8}}&{|_{5}}&{|_{6}} \ {|_{4}}& -{|_{3}}&{|_{2}}& -{|_{1}}& -{|_{8}}&{|_{7}}& -{|_{6}}&{|_{5}} \ {|_{5}}&{|_{6}}&{|_{7}}&{|_{8}}& -{{q_{2}}\ {|_{1}}}& -{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{4}}} \ {|_{6}}& -{|_{5}}&{|_{8}}& -{|_{7}}&{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{1}}}&{{q_{2}}\ {|_{4}}}& -{{q_{2}}\ {|_{3}}} \ {|_{7}}& -{|_{8}}& -{|_{5}}&{|_{6}}&{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{4}}}& -{{q_{2}}\ {|_{1}}}&{{q_{2}}\ {|_{2}}} \ {|_{8}}&{|_{7}}& -{|_{6}}& -{|_{5}}&{{q_{2}}\ {|_{4}}}&{{q_{2}}\ {|_{3}}}& -{{q_{2}}\ {|_{2}}}& -{{q_{2}}\ {|_{1}}}$

(12)

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

A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$

axiom

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

$\label{eq13}\begin{array}{@{}l} \displaystyle {{u^{1, \: 1}}\ {|_{\ }^{1 \ 1}}}+{{u^{1, \: 2}}\ {|_{\ }^{1 \ 2}}}+{{u^{1, \: 3}}\ {|_{\ }^{1 \ 3}}}+{{u^{1, \: 4}}\ {|_{\ }^{1 \ 4}}}+ \ \ \displaystyle {{u^{1, \: 5}}\ {|_{\ }^{1 \ 5}}}+{{u^{1, \: 6}}\ {|_{\ }^{1 \ 6}}}+{{u^{1, \: 7}}\ {|_{\ }^{1 \ 7}}}+{{u^{1, \: 8}}\ {|_{\ }^{1 \ 8}}}+ \ \ \displaystyle {{u^{2, \: 1}}\ {|_{\ }^{2 \ 1}}}+{{u^{2, \: 2}}\ {|_{\ }^{2 \ 2}}}+{{u^{2, \: 3}}\ {|_{\ }^{2 \ 3}}}+{{u^{2, \: 4}}\ {|_{\ }^{2 \ 4}}}+ \ \ \displaystyle {{u^{2, \: 5}}\ {|_{\ }^{2 \ 5}}}+{{u^{2, \: 6}}\ {|_{\ }^{2 \ 6}}}+{{u^{2, \: 7}}\ {|_{\ }^{2 \ 7}}}+{{u^{2, \: 8}}\ {|_{\ }^{2 \ 8}}}+ \ \ \displaystyle {{u^{3, \: 1}}\ {|_{\ }^{3 \ 1}}}+{{u^{3, \: 2}}\ {|_{\ }^{3 \ 2}}}+{{u^{3, \: 3}}\ {|_{\ }^{3 \ 3}}}+{{u^{3, \: 4}}\ {|_{\ }^{3 \ 4}}}+ \ \ \displaystyle {{u^{3, \: 5}}\ {|_{\ }^{3 \ 5}}}+{{u^{3, \: 6}}\ {|_{\ }^{3 \ 6}}}+{{u^{3, \: 7}}\ {|_{\ }^{3 \ 7}}}+{{u^{3, \: 8}}\ {|_{\ }^{3 \ 8}}}+ \ \ \displaystyle {{u^{4, \: 1}}\ {|_{\ }^{4 \ 1}}}+{{u^{4, \: 2}}\ {|_{\ }^{4 \ 2}}}+{{u^{4, \: 3}}\ {|_{\ }^{4 \ 3}}}+{{u^{4, \: 4}}\ {|_{\ }^{4 \ 4}}}+ \ \ \displaystyle {{u^{4, \: 5}}\ {|_{\ }^{4 \ 5}}}+{{u^{4, \: 6}}\ {|_{\ }^{4 \ 6}}}+{{u^{4, \: 7}}\ {|_{\ }^{4 \ 7}}}+{{u^{4, \: 8}}\ {|_{\ }^{4 \ 8}}}+ \ \ \displaystyle {{u^{5, \: 1}}\ {|_{\ }^{5 \ 1}}}+{{u^{5, \: 2}}\ {|_{\ }^{5 \ 2}}}+{{u^{5, \: 3}}\ {|_{\ }^{5 \ 3}}}+{{u^{5, \: 4}}\ {|_{\ }^{5 \ 4}}}+ \ \ \displaystyle {{u^{5, \: 5}}\ {|_{\ }^{5 \ 5}}}+{{u^{5, \: 6}}\ {|_{\ }^{5 \ 6}}}+{{u^{5, \: 7}}\ {|_{\ }^{5 \ 7}}}+{{u^{5, \: 8}}\ {|_{\ }^{5 \ 8}}}+ \ \ \displaystyle {{u^{6, \: 1}}\ {|_{\ }^{6 \ 1}}}+{{u^{6, \: 2}}\ {|_{\ }^{6 \ 2}}}+{{u^{6, \: 3}}\ {|_{\ }^{6 \ 3}}}+{{u^{6, \: 4}}\ {|_{\ }^{6 \ 4}}}+ \ \ \displaystyle {{u^{6, \: 5}}\ {|_{\ }^{6 \ 5}}}+{{u^{6, \: 6}}\ {|_{\ }^{6 \ 6}}}+{{u^{6, \: 7}}\ {|_{\ }^{6 \ 7}}}+{{u^{6, \: 8}}\ {|_{\ }^{6 \ 8}}}+ \ \ \displaystyle {{u^{7, \: 1}}\ {|_{\ }^{7 \ 1}}}+{{u^{7, \: 2}}\ {|_{\ }^{7 \ 2}}}+{{u^{7, \: 3}}\ {|_{\ }^{7 \ 3}}}+{{u^{7, \: 4}}\ {|_{\ }^{7 \ 4}}}+ \ \ \displaystyle {{u^{7, \: 5}}\ {|_{\ }^{7 \ 5}}}+{{u^{7, \: 6}}\ {|_{\ }^{7 \ 6}}}+{{u^{7, \: 7}}\ {|_{\ }^{7 \ 7}}}+{{u^{7, \: 8}}\ {|_{\ }^{7 \ 8}}}+ \ \ \displaystyle {{u^{8, \: 1}}\ {|_{\ }^{8 \ 1}}}+{{u^{8, \: 2}}\ {|_{\ }^{8 \ 2}}}+{{u^{8, \: 3}}\ {|_{\ }^{8 \ 3}}}+{{u^{8, \: 4}}\ {|_{\ }^{8 \ 4}}}+ \ \ \displaystyle {{u^{8, \: 5}}\ {|_{\ }^{8 \ 5}}}+{{u^{8, \: 6}}\ {|_{\ }^{8 \ 6}}}+{{u^{8, \: 7}}\ {|_{\ }^{8 \ 7}}}+{{u^{8, \: 8}}\ {|_{\ }^{8 \ 8}}}$

(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:

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(4.82,0.92) \psbezier[linewidth=0.04](2.2,0.9)(2.2,0.1)(2.6,0.1)(2.6,0.9) \psline[linewidth=0.04cm](2.4,0.3)(2.4,-0.1) \psbezier[linewidth=0.04](2.4,-0.1)(2.4,-0.9)(3.0,-0.9)(3.0,-0.1) \psline[linewidth=0.04cm](3.0,-0.1)(3.0,0.9) \psbezier[linewidth=0.04](4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9) \psline[linewidth=0.04cm](4.6,0.3)(4.6,-0.1) \psbezier[linewidth=0.04](4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1) \psline[linewidth=0.04cm](4.0,-0.1)(4.0,0.9) \usefont{T1}{ptm}{m}{n} \rput(3.4948437,0.205){-} \psline[linewidth=0.04cm](0.6,-0.7)(0.6,0.9) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(1.2,-0.9)(1.2,-0.1) \psline[linewidth=0.04cm](0.0,-0.1)(0.0,0.9) \psline[linewidth=0.04cm](1.2,-0.1)(1.2,0.9) \usefont{T1}{ptm}{m}{n} \rput(1.6948438,0.205){=} \end{pspicture} }$

$\label{eq14} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(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))

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

$\label{eq15}$

(15)

Type: Equation(OutputForm?)

axiom

nrows(J),ncols(J)

$\label{eq16}\left[{512}, \:{64}\right]$

(16)

Type: Tuple(PositiveInteger?)

The matrix J transforms the coefficients of the tensor into coefficients of the tensor $\Phi$ . We are looking for the general linear family of tensors such that J transforms into $\Phi=0$ 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) )

$\label{eq17}\begin{array}{@{}l} \displaystyle \left[{{u^{1, \: 1}}= -{{p_{1}}\over{q_{2}}}}, \:{{u^{1, \: 2}}= 0}, \:{{u^{1, \: 3}}= 0}, \:{{u^{1, \: 4}}= 0}, \:{{u^{1, \: 5}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{1, \: 6}}= 0}, \:{{u^{1, \: 7}}= 0}, \:{{u^{1, \: 8}}= 0}, \:{{u^{2, \: 1}}= 0}, \:{{u^{2, \: 2}}={{p_{1}}\over{q_{2}}}}, \: \right. \ \ \displaystyle \left.{{u^{2, \: 3}}= 0}, \:{{u^{2, \: 4}}= 0}, \:{{u^{2, \: 5}}= 0}, \:{{u^{2, \: 6}}= 0}, \:{{u^{2, \: 7}}= 0}, \:{{u^{2, \: 8}}= 0}, \right. \ \ \displaystyle \left.\:{{u^{3, \: 1}}= 0}, \:{{u^{3, \: 2}}= 0}, \:{{u^{3, \: 3}}={{p_{1}}\over{q_{2}}}}, \:{{u^{3, \: 4}}= 0}, \:{{u^{3, \: 5}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{3, \: 6}}= 0}, \:{{u^{3, \: 7}}= 0}, \:{{u^{3, \: 8}}= 0}, \:{{u^{4, \: 1}}= 0}, \:{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \right. \ \ \displaystyle \left.\:{{u^{4, \: 4}}={{p_{1}}\over{q_{2}}}}, \:{{u^{4, \: 5}}= 0}, \:{{u^{4, \: 6}}= 0}, \:{{u^{4, \: 7}}= 0}, \:{{u^{4, \: 8}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{5, \: 1}}= 0}, \:{{u^{5, \: 2}}= 0}, \:{{u^{5, \: 3}}= 0}, \:{{u^{5, \: 4}}= 0}, \:{{u^{5, \: 5}}={p_{1}}}, \: \right. \ \ \displaystyle \left.{{u^{5, \: 6}}= 0}, \:{{u^{5, \: 7}}= 0}, \:{{u^{5, \: 8}}= 0}, \:{{u^{6, \: 1}}= 0}, \:{{u^{6, \: 2}}= 0}, \:{{u^{6, \: 3}}= 0}, \right. \ \ \displaystyle \left.\:{{u^{6, \: 4}}= 0}, \:{{u^{6, \: 5}}= 0}, \:{{u^{6, \: 6}}={p_{1}}}, \:{{u^{6, \: 7}}= 0}, \:{{u^{6, \: 8}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{7, \: 1}}= 0}, \:{{u^{7, \: 2}}= 0}, \:{{u^{7, \: 3}}= 0}, \:{{u^{7, \: 4}}= 0}, \:{{u^{7, \: 5}}= 0}, \:{{u^{7, \: 6}}= 0}, \right. \ \ \displaystyle \left.\:{{u^{7, \: 7}}={p_{1}}}, \:{{u^{7, \: 8}}= 0}, \:{{u^{8, \: 1}}= 0}, \:{{u^{8, \: 2}}= 0}, \:{{u^{8, \: 3}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{8, \: 4}}= 0}, \:{{u^{8, \: 5}}= 0}, \:{{u^{8, \: 6}}= 0}, \:{{u^{8, \: 7}}= 0}, \:{{u^{8, \: 8}}={p_{1}}}\right]$

(17)

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

This defines a family of Frobenius algebras:

axiom

zero? eval(ω,ℰ)

$\label{eq18} \mbox{\rm true}$

(18)

Type: Boolean

The pairing is necessarily diagonal!

axiom

Ų:𝐋 := eval(U,ℰ)

$\label{eq19}\begin{array}{@{}l} \displaystyle -{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1 \ 1}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{2 \ 2}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{3 \ 3}}}+{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{4 \ 4}}}+{{p_{1}}\ {|_{\ }^{5 \ 5}}}+ \ \ \displaystyle {{p_{1}}\ {|_{\ }^{6 \ 6}}}+{{p_{1}}\ {|_{\ }^{7 \ 7}}}+{{p_{1}}\ {|_{\ }^{8 \ 8}}}$

(19)

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

axiom

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

$\label{eq20}\left[ \begin{array}{cccccccc} -{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}$

(20)

Type: Matrix(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)

$\label{eq21}-{{{p_{1}}^8}\over{{q_{2}}^4}}$

(21)

Type: Expression(Integer)

axiom

factor Ů

$\label{eq22}-{{{p_{1}}^8}\over{{q_{2}}^4}}$

(22)

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)

$\label{eq23}\left[ \begin{array}{cccccccc} -{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 &{{p_{1}}\over{q_{2}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}$

(23)

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

axiom

mU:=transpose inverse map(retract,Um)

$\label{eq24}\left[ \begin{array}{cccccccc} -{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}$

(24)

Type: Matrix(Expression(Integer))

axiom

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

$\label{eq25}\begin{array}{@{}l} \displaystyle -{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}}}+{{1 \over{p_{1}}}\ {|_{5 \ 5}}}+{{1 \over{p_{1}}}\ {|_{6 \ 6}}}+ \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{7 \ 7}}}+{{1 \over{p_{1}}}\ {|_{8 \ 8}}}$

(25)

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

axiom

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

$\label{eq26}\left[ \begin{array}{cccccccc} -{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 &{{q_{2}}\over{p_{1}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}$

(26)

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

Check "dimension" and the snake relations.

axiom

d:𝐋:=
       Ω    /
       X    /
       Ų

$\label{eq27}8$

(27)

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

axiom

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

$\label{eq28} \mbox{\rm true}$

(28)

Type: Boolean

axiom

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

$\label{eq29} \mbox{\rm true}$

(29)

Type: Boolean

Definition 4

Co-algebra

This expression is expensive to compute:

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

axiom

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

$\label{eq30}\begin{array}{@{}l} \displaystyle -{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 1}^{1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 2}^{1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 3}^{1}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 4}^{1}}}+{{1 \over{p_{1}}}\ {|_{5 \ 5}^{1}}}+ \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{6 \ 6}^{1}}}+{{1 \over{p_{1}}}\ {|_{7 \ 7}^{1}}}+{{1 \over{p_{1}}}\ {|_{8 \ 8}^{1}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 2}^{2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 1}^{2}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{3 \ 4}^{2}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 3}^{2}}}-{{1 \over{p_{1}}}\ {|_{5 \ 6}^{2}}}+{{1 \over{p_{1}}}\ {|_{6 \ 5}^{2}}}+{{1 \over{p_{1}}}\ {|_{7 \ 8}^{2}}}- \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{8 \ 7}^{2}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 3}^{3}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 4}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 1}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 2}^{3}}}- \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{5 \ 7}^{3}}}-{{1 \over{p_{1}}}\ {|_{6 \ 8}^{3}}}+{{1 \over{p_{1}}}\ {|_{7 \ 5}^{3}}}+{{1 \over{p_{1}}}\ {|_{8 \ 6}^{3}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 4}^{4}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{2 \ 3}^{4}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 2}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 1}^{4}}}-{{1 \over{p_{1}}}\ {|_{5 \ 8}^{4}}}+{{1 \over{p_{1}}}\ {|_{6 \ 7}^{4}}}- \ \ \displaystyle {{1 \over{p_{1}}}\ {|_{7 \ 6}^{4}}}+{{1 \over{p_{1}}}\ {|_{8 \ 5}^{4}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 5}^{5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 6}^{5}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 7}^{5}}}+ \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{4 \ 8}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 1}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 2}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 3}^{5}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 4}^{5}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{1 \ 6}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 5}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 8}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 7}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 2}^{6}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{6 \ 1}^{6}}}+{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 4}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 3}^{6}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 7}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 8}^{7}}}- \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{3 \ 5}^{7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 6}^{7}}}+{{{q_{2}}\over{p_{1}}}\ {|_{5 \ 3}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 4}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 1}^{7}}}+ \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{8 \ 2}^{7}}}-{{{q_{2}}\over{p_{1}}}\ {|_{1 \ 8}^{8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{2 \ 7}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{3 \ 6}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{4 \ 5}^{8}}}+ \ \ \displaystyle {{{q_{2}}\over{p_{1}}}\ {|_{5 \ 4}^{8}}}+{{{q_{2}}\over{p_{1}}}\ {|_{6 \ 3}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{7 \ 2}^{8}}}-{{{q_{2}}\over{p_{1}}}\ {|_{8 \ 1}^{8}}}$

(30)

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

axiom

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

$\label{eq31} \mbox{\rm true}$

(31)

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

$\label{eq32}|_{1}$

(32)

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

axiom

test
         i     /
         λ     =    Ω

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean

Handle

axiom

H:𝐋 :=
         λ     /
         Y

$\label{eq34}\begin{array}{@{}l} \displaystyle -{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{1}^{1}}}-{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{2}^{2}}}-{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{3}^{3}}}-{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{4}^{4}}}-{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{5}^{5}}}- \ \ \displaystyle {{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{6}^{6}}}-{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{7}^{7}}}-{{{8 \ {q_{2}}}\over{p_{1}}}\ {|_{8}^{8}}}$

(34)

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

Definition 5

Co-unit

  i 
  U

axiom

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

$\label{eq35}-{{{p_{1}}\over{q_{2}}}\ {|_{\ }^{1}}}$

(35)

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

Y=U
ι

axiom

test
        Y    /
        ι       = Ų

$\label{eq36} \mbox{\rm true}$

(36)

Type: Boolean

For example:

axiom

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

$\label{eq37}\left[{{q_{2}}= 1}, \:{{p_{1}}= 1}\right]$

(37)

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

axiom

Ų0:𝐋  :=eval(Ų,ex1)

$\label{eq38}-{|_{\ }^{1 \ 1}}+{|_{\ }^{2 \ 2}}+{|_{\ }^{3 \ 3}}+{|_{\ }^{4 \ 4}}+{|_{\ }^{5 \ 5}}+{|_{\ }^{6 \ 6}}+{|_{\ }^{7 \ 7}}+{|_{\ }^{8 \ 8}}$

(38)

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

axiom

Ω0:𝐋  :=eval(Ω,ex1)$𝐋

$\label{eq39}-{|_{1 \ 1}}+{|_{2 \ 2}}+{|_{3 \ 3}}+{|_{4 \ 4}}+{|_{5 \ 5}}+{|_{6 \ 6}}+{|_{7 \ 7}}+{|_{8 \ 8}}$

(39)

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

axiom

λ0:𝐋  :=eval(λ,ex1)$𝐋

$\label{eq40}\begin{array}{@{}l} \displaystyle -{|_{1 \ 1}^{1}}+{|_{2 \ 2}^{1}}+{|_{3 \ 3}^{1}}+{|_{4 \ 4}^{1}}+{|_{5 \ 5}^{1}}+{|_{6 \ 6}^{1}}+{|_{7 \ 7}^{1}}+ \ \ \displaystyle {|_{8 \ 8}^{1}}-{|_{1 \ 2}^{2}}-{|_{2 \ 1}^{2}}-{|_{3 \ 4}^{2}}+{|_{4 \ 3}^{2}}-{|_{5 \ 6}^{2}}+{|_{6 \ 5}^{2}}+{|_{7 \ 8}^{2}}- \ \ \displaystyle {|_{8 \ 7}^{2}}-{|_{1 \ 3}^{3}}+{|_{2 \ 4}^{3}}-{|_{3 \ 1}^{3}}-{|_{4 \ 2}^{3}}-{|_{5 \ 7}^{3}}-{|_{6 \ 8}^{3}}+{|_{7 \ 5}^{3}}+ \ \ \displaystyle {|_{8 \ 6}^{3}}-{|_{1 \ 4}^{4}}-{|_{2 \ 3}^{4}}+{|_{3 \ 2}^{4}}-{|_{4 \ 1}^{4}}-{|_{5 \ 8}^{4}}+{|_{6 \ 7}^{4}}-{|_{7 \ 6}^{4}}+ \ \ \displaystyle {|_{8 \ 5}^{4}}-{|_{1 \ 5}^{5}}+{|_{2 \ 6}^{5}}+{|_{3 \ 7}^{5}}+{|_{4 \ 8}^{5}}-{|_{5 \ 1}^{5}}-{|_{6 \ 2}^{5}}-{|_{7 \ 3}^{5}}- \ \ \displaystyle {|_{8 \ 4}^{5}}-{|_{1 \ 6}^{6}}-{|_{2 \ 5}^{6}}+{|_{3 \ 8}^{6}}-{|_{4 \ 7}^{6}}+{|_{5 \ 2}^{6}}-{|_{6 \ 1}^{6}}+{|_{7 \ 4}^{6}}- \ \ \displaystyle {|_{8 \ 3}^{6}}-{|_{1 \ 7}^{7}}-{|_{2 \ 8}^{7}}-{|_{3 \ 5}^{7}}+{|_{4 \ 6}^{7}}+{|_{5 \ 3}^{7}}-{|_{6 \ 4}^{7}}-{|_{7 \ 1}^{7}}+ \ \ \displaystyle {|_{8 \ 2}^{7}}-{|_{1 \ 8}^{8}}+{|_{2 \ 7}^{8}}-{|_{3 \ 6}^{8}}-{|_{4 \ 5}^{8}}+{|_{5 \ 4}^{8}}+{|_{6 \ 3}^{8}}-{|_{7 \ 2}^{8}}- \ \ \displaystyle {|_{8 \ 1}^{8}}$

(40)

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

axiom

H0:𝐋 :=eval(H,ex1)$𝐋

$\label{eq41}-{8 \ {|_{1}^{1}}}-{8 \ {|_{2}^{2}}}-{8 \ {|_{3}^{3}}}-{8 \ {|_{4}^{4}}}-{8 \ {|_{5}^{5}}}-{8 \ {|_{6}^{6}}}-{8 \ {|_{7}^{7}}}-{8 \ {|_{8}^{8}}}$

(41)

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