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Octonion Algebra is Frobenius in Just One Way

Edit detail for Octonion Algebra is Frobenius in Just One Way revision 7 of 8

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Editor: Bill Page Time: 2011/05/06 10:17:07 GMT-7
Note: use new LOP

changed:
-)library CARTEN MONAL PROP LIN CALEY
-\end{axiom}
)library CARTEN MONAL PROP LOP CALEY
\end{axiom}

changed:
-𝐋 := LinearOperator(dim, OVAR [], ℚ)
-𝐞:ℒ 𝐋      := basisVectors()
-𝐝:ℒ 𝐋      := basisForms()
𝐋 := LinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ)
𝐞:ℒ 𝐋      := basisOut()
𝐝:ℒ 𝐋      := basisIn()

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
http://en.wikipedia.org/wiki/Octonion

We need the Axiom LinearOperator? library.

fricas

(1) -> )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

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

Type: Void

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

Type: Void

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-- 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.

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dim:=8

$\label{eq1}8$

(1)

Type: PositiveInteger?

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macro ℒ == List

Type: Void

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macro ℂ == CaleyDickson

Type: Void

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macro ℚ == Expression Integer

Type: Void

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𝐋 := LinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ)

$\label{eq2}\hbox{\axiomType{LinearOperator}\ } \left({{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[ 0, \: 1, \: 2, \: 3, \: 4, \: 5, \: 6, \: 7 \right]}\right)}, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}}\right)$

(2)

Type: Type

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𝐞:ℒ 𝐋      := basisOut()

$\label{eq3}\left[{|_{0}}, \:{|_{1}}, \:{|_{2}}, \:{|_{3}}, \:{|_{4}}, \:{|_{5}}, \:{|_{6}}, \:{|_{7}}\right]$

(3)

Type: List(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))

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𝐝:ℒ 𝐋      := basisIn()

$\label{eq4}\left[{|_{\ }^{0}}, \:{|_{\ }^{1}}, \:{|_{\ }^{2}}, \:{|_{\ }^{3}}, \:{|_{\ }^{4}}, \:{|_{\ }^{5}}, \:{|_{\ }^{6}}, \:{|_{\ }^{7}}\right]$

(4)

Type: List(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))

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

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

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X:𝐋:=[2,1]; -- twist

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),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

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

$\label{eq5}1$

(5)

Type: PositiveInteger?

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

$\label{eq6}1$

(6)

Type: PositiveInteger?

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

$\label{eq7}q_{2}$

(7)

Type: Symbol

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--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

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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,q[2]))

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

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

(9)

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

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

Type: Void

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

fricas

Compiling function S with type CaleyDickson(CaleyDickson(
      CaleyDickson(Expression(Integer),i,1),j,1),k,q[2]) -> 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)

$\label{eq10}\begin{array}{@{}l} \displaystyle {|_{0}^{0 \ 0}}+{|_{1}^{0 \ 1}}+{|_{2}^{0 \ 2}}+{|_{3}^{0 \ 3}}+{{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{0 \ 4}}}+{|_{5}^{0 \ 5}}+ \ \ \displaystyle {|_{6}^{0 \ 6}}+{|_{7}^{0 \ 7}}+{|_{1}^{1 \ 0}}-{|_{0}^{1 \ 1}}+{|_{3}^{1 \ 2}}-{|_{2}^{1 \ 3}}+{|_{5}^{1 \ 4}}- \ \ \displaystyle {{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{1 \ 5}}}-{|_{7}^{1 \ 6}}+{|_{6}^{1 \ 7}}+{|_{2}^{2 \ 0}}-{|_{3}^{2 \ 1}}-{|_{0}^{2 \ 2}}+ \ \ \displaystyle {|_{1}^{2 \ 3}}+{|_{6}^{2 \ 4}}+{|_{7}^{2 \ 5}}-{{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{2 \ 6}}}-{|_{5}^{2 \ 7}}+{|_{3}^{3 \ 0}}+ \ \ \displaystyle {|_{2}^{3 \ 1}}-{|_{1}^{3 \ 2}}-{|_{0}^{3 \ 3}}+{|_{7}^{3 \ 4}}-{|_{6}^{3 \ 5}}+{|_{5}^{3 \ 6}}- \ \ \displaystyle {{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{3 \ 7}}}+{{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{4 \ 0}}}-{|_{5}^{4 \ 1}}-{|_{6}^{4 \ 2}}-{|_{7}^{4 \ 3}}- \ \ \displaystyle {{q_{2}}\ {|_{0}^{4 \ 4}}}+{{q_{2}}\ {|_{1}^{4 \ 5}}}+{{q_{2}}\ {|_{2}^{4 \ 6}}}+{{q_{2}}\ {|_{3}^{4 \ 7}}}+{|_{5}^{5 \ 0}}+ \ \ \displaystyle {{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{5 \ 1}}}-{|_{7}^{5 \ 2}}+{|_{6}^{5 \ 3}}-{{q_{2}}\ {|_{1}^{5 \ 4}}}-{{q_{2}}\ {|_{0}^{5 \ 5}}}- \ \ \displaystyle {{\overline{q_{2}}}\ {|_{3}^{5 \ 6}}}+{{\overline{q_{2}}}\ {|_{2}^{5 \ 7}}}+{|_{6}^{6 \ 0}}+{|_{7}^{6 \ 1}}+{{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{6 \ 2}}}-{|_{5}^{6 \ 3}}- \ \ \displaystyle {{q_{2}}\ {|_{2}^{6 \ 4}}}+{{\overline{q_{2}}}\ {|_{3}^{6 \ 5}}}-{{q_{2}}\ {|_{0}^{6 \ 6}}}-{{\overline{q_{2}}}\ {|_{1}^{6 \ 7}}}+{|_{7}^{7 \ 0}}-{|_{6}^{7 \ 1}}+ \ \ \displaystyle {|_{5}^{7 \ 2}}+{{\frac{q_{2}}{\overline{q_{2}}}}\ {|_{4}^{7 \ 3}}}-{{q_{2}}\ {|_{3}^{7 \ 4}}}-{{\overline{q_{2}}}\ {|_{2}^{7 \ 5}}}+{{\overline{q_{2}}}\ {|_{1}^{7 \ 6}}}- \ \ \displaystyle {{q_{2}}\ {|_{0}^{7 \ 7}}}$

(10)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

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arity Y

$\label{eq11}\frac{2}{1}$

(11)

Type: Prop(LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))

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

   >> System error:
   The value
  (0 . 0)
is not of type
  UNSIGNED-BYTE

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)

$\label{eq12}\begin{array}{@{}l} \displaystyle {{u^{1, \: 1}}\ {|_{\ }^{0 \ 0}}}+{{u^{1, \: 2}}\ {|_{\ }^{0 \ 1}}}+{{u^{1, \: 3}}\ {|_{\ }^{0 \ 2}}}+{{u^{1, \: 4}}\ {|_{\ }^{0 \ 3}}}+ \ \ \displaystyle {{u^{1, \: 5}}\ {|_{\ }^{0 \ 4}}}+{{u^{1, \: 6}}\ {|_{\ }^{0 \ 5}}}+{{u^{1, \: 7}}\ {|_{\ }^{0 \ 6}}}+{{u^{1, \: 8}}\ {|_{\ }^{0 \ 7}}}+ \ \ \displaystyle {{u^{2, \: 1}}\ {|_{\ }^{1 \ 0}}}+{{u^{2, \: 2}}\ {|_{\ }^{1 \ 1}}}+{{u^{2, \: 3}}\ {|_{\ }^{1 \ 2}}}+{{u^{2, \: 4}}\ {|_{\ }^{1 \ 3}}}+ \ \ \displaystyle {{u^{2, \: 5}}\ {|_{\ }^{1 \ 4}}}+{{u^{2, \: 6}}\ {|_{\ }^{1 \ 5}}}+{{u^{2, \: 7}}\ {|_{\ }^{1 \ 6}}}+{{u^{2, \: 8}}\ {|_{\ }^{1 \ 7}}}+ \ \ \displaystyle {{u^{3, \: 1}}\ {|_{\ }^{2 \ 0}}}+{{u^{3, \: 2}}\ {|_{\ }^{2 \ 1}}}+{{u^{3, \: 3}}\ {|_{\ }^{2 \ 2}}}+{{u^{3, \: 4}}\ {|_{\ }^{2 \ 3}}}+ \ \ \displaystyle {{u^{3, \: 5}}\ {|_{\ }^{2 \ 4}}}+{{u^{3, \: 6}}\ {|_{\ }^{2 \ 5}}}+{{u^{3, \: 7}}\ {|_{\ }^{2 \ 6}}}+{{u^{3, \: 8}}\ {|_{\ }^{2 \ 7}}}+ \ \ \displaystyle {{u^{4, \: 1}}\ {|_{\ }^{3 \ 0}}}+{{u^{4, \: 2}}\ {|_{\ }^{3 \ 1}}}+{{u^{4, \: 3}}\ {|_{\ }^{3 \ 2}}}+{{u^{4, \: 4}}\ {|_{\ }^{3 \ 3}}}+ \ \ \displaystyle {{u^{4, \: 5}}\ {|_{\ }^{3 \ 4}}}+{{u^{4, \: 6}}\ {|_{\ }^{3 \ 5}}}+{{u^{4, \: 7}}\ {|_{\ }^{3 \ 6}}}+{{u^{4, \: 8}}\ {|_{\ }^{3 \ 7}}}+ \ \ \displaystyle {{u^{5, \: 1}}\ {|_{\ }^{4 \ 0}}}+{{u^{5, \: 2}}\ {|_{\ }^{4 \ 1}}}+{{u^{5, \: 3}}\ {|_{\ }^{4 \ 2}}}+{{u^{5, \: 4}}\ {|_{\ }^{4 \ 3}}}+ \ \ \displaystyle {{u^{5, \: 5}}\ {|_{\ }^{4 \ 4}}}+{{u^{5, \: 6}}\ {|_{\ }^{4 \ 5}}}+{{u^{5, \: 7}}\ {|_{\ }^{4 \ 6}}}+{{u^{5, \: 8}}\ {|_{\ }^{4 \ 7}}}+ \ \ \displaystyle {{u^{6, \: 1}}\ {|_{\ }^{5 \ 0}}}+{{u^{6, \: 2}}\ {|_{\ }^{5 \ 1}}}+{{u^{6, \: 3}}\ {|_{\ }^{5 \ 2}}}+{{u^{6, \: 4}}\ {|_{\ }^{5 \ 3}}}+ \ \ \displaystyle {{u^{6, \: 5}}\ {|_{\ }^{5 \ 4}}}+{{u^{6, \: 6}}\ {|_{\ }^{5 \ 5}}}+{{u^{6, \: 7}}\ {|_{\ }^{5 \ 6}}}+{{u^{6, \: 8}}\ {|_{\ }^{5 \ 7}}}+ \ \ \displaystyle {{u^{7, \: 1}}\ {|_{\ }^{6 \ 0}}}+{{u^{7, \: 2}}\ {|_{\ }^{6 \ 1}}}+{{u^{7, \: 3}}\ {|_{\ }^{6 \ 2}}}+{{u^{7, \: 4}}\ {|_{\ }^{6 \ 3}}}+ \ \ \displaystyle {{u^{7, \: 5}}\ {|_{\ }^{6 \ 4}}}+{{u^{7, \: 6}}\ {|_{\ }^{6 \ 5}}}+{{u^{7, \: 7}}\ {|_{\ }^{6 \ 6}}}+{{u^{7, \: 8}}\ {|_{\ }^{6 \ 7}}}+ \ \ \displaystyle {{u^{8, \: 1}}\ {|_{\ }^{7 \ 0}}}+{{u^{8, \: 2}}\ {|_{\ }^{7 \ 1}}}+{{u^{8, \: 3}}\ {|_{\ }^{7 \ 2}}}+{{u^{8, \: 4}}\ {|_{\ }^{7 \ 3}}}+ \ \ \displaystyle {{u^{8, \: 5}}\ {|_{\ }^{7 \ 4}}}+{{u^{8, \: 6}}\ {|_{\ }^{7 \ 5}}}+{{u^{8, \: 7}}\ {|_{\ }^{7 \ 6}}}+{{u^{8, \: 8}}\ {|_{\ }^{7 \ 7}}}$

(12)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),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:

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

$\label{eq13} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(13)

(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;

   >> System error:
   The value
  (0 . 0)
is not of type
  UNSIGNED-BYTE

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.

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)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))

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

Type: Matrix(Polynomial(Integer))

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J::OutputForm * u::OutputForm = 0

$\label{eq14}\begin{array}{@{}l} \displaystyle {{\left[ \begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0$

(14)

Type: Equation(OutputForm?)

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nrows(J),ncols(J)

$\label{eq15}\left[ 1, \:{64}\right]$

(15)

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:

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Ñ:=nullSpace(J);

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

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

$\label{eq16}\begin{array}{@{}l} \displaystyle \left[{{u^{1, \: 1}}={p_{64}}}, \:{{u^{1, \: 2}}={p_{63}}}, \:{{u^{1, \: 3}}={p_{62}}}, \:{{u^{1, \: 4}}={p_{61}}}, \: \right. \ \ \displaystyle \left.{{u^{1, \: 5}}={p_{60}}}, \:{{u^{1, \: 6}}={p_{59}}}, \:{{u^{1, \: 7}}={p_{58}}}, \:{{u^{1, \: 8}}={p_{57}}}, \: \right. \ \ \displaystyle \left.{{u^{2, \: 1}}={p_{56}}}, \:{{u^{2, \: 2}}={p_{55}}}, \:{{u^{2, \: 3}}={p_{54}}}, \:{{u^{2, \: 4}}={p_{53}}}, \: \right. \ \ \displaystyle \left.{{u^{2, \: 5}}={p_{52}}}, \:{{u^{2, \: 6}}={p_{51}}}, \:{{u^{2, \: 7}}={p_{50}}}, \:{{u^{2, \: 8}}={p_{49}}}, \: \right. \ \ \displaystyle \left.{{u^{3, \: 1}}={p_{48}}}, \:{{u^{3, \: 2}}={p_{47}}}, \:{{u^{3, \: 3}}={p_{46}}}, \:{{u^{3, \: 4}}={p_{45}}}, \: \right. \ \ \displaystyle \left.{{u^{3, \: 5}}={p_{44}}}, \:{{u^{3, \: 6}}={p_{43}}}, \:{{u^{3, \: 7}}={p_{42}}}, \:{{u^{3, \: 8}}={p_{41}}}, \: \right. \ \ \displaystyle \left.{{u^{4, \: 1}}={p_{40}}}, \:{{u^{4, \: 2}}={p_{39}}}, \:{{u^{4, \: 3}}={p_{38}}}, \:{{u^{4, \: 4}}={p_{37}}}, \: \right. \ \ \displaystyle \left.{{u^{4, \: 5}}={p_{36}}}, \:{{u^{4, \: 6}}={p_{35}}}, \:{{u^{4, \: 7}}={p_{34}}}, \:{{u^{4, \: 8}}={p_{33}}}, \: \right. \ \ \displaystyle \left.{{u^{5, \: 1}}={p_{32}}}, \:{{u^{5, \: 2}}={p_{31}}}, \:{{u^{5, \: 3}}={p_{30}}}, \:{{u^{5, \: 4}}={p_{29}}}, \: \right. \ \ \displaystyle \left.{{u^{5, \: 5}}={p_{28}}}, \:{{u^{5, \: 6}}={p_{27}}}, \:{{u^{5, \: 7}}={p_{26}}}, \:{{u^{5, \: 8}}={p_{25}}}, \: \right. \ \ \displaystyle \left.{{u^{6, \: 1}}={p_{24}}}, \:{{u^{6, \: 2}}={p_{23}}}, \:{{u^{6, \: 3}}={p_{22}}}, \:{{u^{6, \: 4}}={p_{21}}}, \: \right. \ \ \displaystyle \left.{{u^{6, \: 5}}={p_{20}}}, \:{{u^{6, \: 6}}={p_{19}}}, \:{{u^{6, \: 7}}={p_{18}}}, \:{{u^{6, \: 8}}={p_{17}}}, \: \right. \ \ \displaystyle \left.{{u^{7, \: 1}}={p_{16}}}, \:{{u^{7, \: 2}}={p_{15}}}, \:{{u^{7, \: 3}}={p_{14}}}, \:{{u^{7, \: 4}}={p_{13}}}, \: \right. \ \ \displaystyle \left.{{u^{7, \: 5}}={p_{12}}}, \:{{u^{7, \: 6}}={p_{11}}}, \:{{u^{7, \: 7}}={p_{10}}}, \:{{u^{7, \: 8}}={p_{9}}}, \:{{u^{8, \: 1}}={p_{8}}}, \: \right. \ \ \displaystyle \left.{{u^{8, \: 2}}={p_{7}}}, \:{{u^{8, \: 3}}={p_{6}}}, \:{{u^{8, \: 4}}={p_{5}}}, \:{{u^{8, \: 5}}={p_{4}}}, \:{{u^{8, \: 6}}={p_{3}}}, \: \right. \ \ \displaystyle \left.{{u^{8, \: 7}}={p_{2}}}, \:{{u^{8, \: 8}}={p_{1}}}\right]$

(16)

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

This defines a family of Frobenius algebras:

fricas

zero? eval(ω,ℰ)

$\label{eq17} \mbox{\rm false}$

(17)

Type: Boolean

The pairing is necessarily diagonal!

fricas

Ų:𝐋 := eval(U,ℰ)

$\label{eq18}\begin{array}{@{}l} \displaystyle {{p_{64}}\ {|_{\ }^{0 \ 0}}}+{{p_{63}}\ {|_{\ }^{0 \ 1}}}+{{p_{62}}\ {|_{\ }^{0 \ 2}}}+{{p_{61}}\ {|_{\ }^{0 \ 3}}}+{{p_{6 0}}\ {|_{\ }^{0 \ 4}}}+ \ \ \displaystyle {{p_{59}}\ {|_{\ }^{0 \ 5}}}+{{p_{58}}\ {|_{\ }^{0 \ 6}}}+{{p_{57}}\ {|_{\ }^{0 \ 7}}}+{{p_{56}}\ {|_{\ }^{1 \ 0}}}+{{p_{5 5}}\ {|_{\ }^{1 \ 1}}}+ \ \ \displaystyle {{p_{54}}\ {|_{\ }^{1 \ 2}}}+{{p_{53}}\ {|_{\ }^{1 \ 3}}}+{{p_{52}}\ {|_{\ }^{1 \ 4}}}+{{p_{51}}\ {|_{\ }^{1 \ 5}}}+{{p_{5 0}}\ {|_{\ }^{1 \ 6}}}+ \ \ \displaystyle {{p_{49}}\ {|_{\ }^{1 \ 7}}}+{{p_{48}}\ {|_{\ }^{2 \ 0}}}+{{p_{47}}\ {|_{\ }^{2 \ 1}}}+{{p_{46}}\ {|_{\ }^{2 \ 2}}}+{{p_{4 5}}\ {|_{\ }^{2 \ 3}}}+ \ \ \displaystyle {{p_{44}}\ {|_{\ }^{2 \ 4}}}+{{p_{43}}\ {|_{\ }^{2 \ 5}}}+{{p_{42}}\ {|_{\ }^{2 \ 6}}}+{{p_{41}}\ {|_{\ }^{2 \ 7}}}+{{p_{4 0}}\ {|_{\ }^{3 \ 0}}}+ \ \ \displaystyle {{p_{39}}\ {|_{\ }^{3 \ 1}}}+{{p_{38}}\ {|_{\ }^{3 \ 2}}}+{{p_{37}}\ {|_{\ }^{3 \ 3}}}+{{p_{36}}\ {|_{\ }^{3 \ 4}}}+{{p_{3 5}}\ {|_{\ }^{3 \ 5}}}+ \ \ \displaystyle {{p_{34}}\ {|_{\ }^{3 \ 6}}}+{{p_{33}}\ {|_{\ }^{3 \ 7}}}+{{p_{32}}\ {|_{\ }^{4 \ 0}}}+{{p_{31}}\ {|_{\ }^{4 \ 1}}}+{{p_{3 0}}\ {|_{\ }^{4 \ 2}}}+ \ \ \displaystyle {{p_{29}}\ {|_{\ }^{4 \ 3}}}+{{p_{28}}\ {|_{\ }^{4 \ 4}}}+{{p_{27}}\ {|_{\ }^{4 \ 5}}}+{{p_{26}}\ {|_{\ }^{4 \ 6}}}+{{p_{2 5}}\ {|_{\ }^{4 \ 7}}}+ \ \ \displaystyle {{p_{24}}\ {|_{\ }^{5 \ 0}}}+{{p_{23}}\ {|_{\ }^{5 \ 1}}}+{{p_{22}}\ {|_{\ }^{5 \ 2}}}+{{p_{21}}\ {|_{\ }^{5 \ 3}}}+{{p_{2 0}}\ {|_{\ }^{5 \ 4}}}+ \ \ \displaystyle {{p_{19}}\ {|_{\ }^{5 \ 5}}}+{{p_{18}}\ {|_{\ }^{5 \ 6}}}+{{p_{17}}\ {|_{\ }^{5 \ 7}}}+{{p_{16}}\ {|_{\ }^{6 \ 0}}}+{{p_{1 5}}\ {|_{\ }^{6 \ 1}}}+ \ \ \displaystyle {{p_{14}}\ {|_{\ }^{6 \ 2}}}+{{p_{13}}\ {|_{\ }^{6 \ 3}}}+{{p_{12}}\ {|_{\ }^{6 \ 4}}}+{{p_{11}}\ {|_{\ }^{6 \ 5}}}+{{p_{1 0}}\ {|_{\ }^{6 \ 6}}}+ \ \ \displaystyle {{p_{9}}\ {|_{\ }^{6 \ 7}}}+{{p_{8}}\ {|_{\ }^{7 \ 0}}}+{{p_{7}}\ {|_{\ }^{7 \ 1}}}+{{p_{6}}\ {|_{\ }^{7 \ 2}}}+{{p_{5}}\ {|_{\ }^{7 \ 3}}}+{{p_{4}}\ {|_{\ }^{7 \ 4}}}+ \ \ \displaystyle {{p_{3}}\ {|_{\ }^{7 \ 5}}}+{{p_{2}}\ {|_{\ }^{7 \ 6}}}+{{p_{1}}\ {|_{\ }^{7 \ 7}}}$

(18)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

fricas

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

   >> System error:
   The value
  (0 . 0)
is not of type
  UNSIGNED-BYTE

The scalar product must be non-degenerate:

fricas

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

   >> System error:
   The value
  (0 . 0)
is not of type
  UNSIGNED-BYTE

Definition 3

Co-pairing

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

fricas

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

   >> System error:
   The value
  (0 . 0)
is not of type
  UNSIGNED-BYTE

Check "dimension" and the snake relations.

fricas

d:𝐋:=
       Ω    /
       X    /
       Ų

   >> System error:
   The value
  (1 . "failed")
is not of type
  UNSIGNED-BYTE

Definition 4

Co-algebra

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

fricas

W:=(Y,I)/Ų

   >> System error:
   The value
  (0 . 0)
is not of type
  UNSIGNED-BYTE

fricas

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

   >> System error:
   The value
  (1 . "failed")
is not of type
  UNSIGNED-BYTE

Frobenius Condition

Octonion algebra fails the Frobenius Condition!

fricas

Χ :=
       Y    /
       λ

   >> System error:
   The value
  (0 . 1)
is not of type
  UNSIGNED-BYTE

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

i = Unit of the algebra

fricas

i:=𝐞.1

$\label{eq19}|_{0}$

(19)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

fricas

test
         i     /
         λ     =    Ω

   >> System error:
   The value
  (0 . 1)
is not of type
  UNSIGNED-BYTE

Handle

fricas

H:𝐋 :=
         λ     /
         X     /
         Y

   >> System error:
   The value
  (1 . "failed")
is not of type
  UNSIGNED-BYTE

Definition 5

Co-unit

  i 
  U

fricas

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

   >> System error:
   The value
  (0 . 0)
is not of type
  UNSIGNED-BYTE

Y=U
ι

fricas

test
        Y    /
        ι       = Ų

   >> System error:
   The value
  (0 . 1)
is not of type
  UNSIGNED-BYTE

For example:

fricas

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

$\label{eq20}\left[{{q_{2}}= 1}, \:{{p_{1}}= 1}\right]$

(20)

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

fricas

Ų0:𝐋  :=eval(Ų,ex1)

$\label{eq21}\begin{array}{@{}l} \displaystyle {{p_{64}}\ {|_{\ }^{0 \ 0}}}+{{p_{63}}\ {|_{\ }^{0 \ 1}}}+{{p_{62}}\ {|_{\ }^{0 \ 2}}}+{{p_{61}}\ {|_{\ }^{0 \ 3}}}+{{p_{6 0}}\ {|_{\ }^{0 \ 4}}}+ \ \ \displaystyle {{p_{59}}\ {|_{\ }^{0 \ 5}}}+{{p_{58}}\ {|_{\ }^{0 \ 6}}}+{{p_{57}}\ {|_{\ }^{0 \ 7}}}+{{p_{56}}\ {|_{\ }^{1 \ 0}}}+{{p_{5 5}}\ {|_{\ }^{1 \ 1}}}+ \ \ \displaystyle {{p_{54}}\ {|_{\ }^{1 \ 2}}}+{{p_{53}}\ {|_{\ }^{1 \ 3}}}+{{p_{52}}\ {|_{\ }^{1 \ 4}}}+{{p_{51}}\ {|_{\ }^{1 \ 5}}}+{{p_{5 0}}\ {|_{\ }^{1 \ 6}}}+ \ \ \displaystyle {{p_{49}}\ {|_{\ }^{1 \ 7}}}+{{p_{48}}\ {|_{\ }^{2 \ 0}}}+{{p_{47}}\ {|_{\ }^{2 \ 1}}}+{{p_{46}}\ {|_{\ }^{2 \ 2}}}+{{p_{4 5}}\ {|_{\ }^{2 \ 3}}}+ \ \ \displaystyle {{p_{44}}\ {|_{\ }^{2 \ 4}}}+{{p_{43}}\ {|_{\ }^{2 \ 5}}}+{{p_{42}}\ {|_{\ }^{2 \ 6}}}+{{p_{41}}\ {|_{\ }^{2 \ 7}}}+{{p_{4 0}}\ {|_{\ }^{3 \ 0}}}+ \ \ \displaystyle {{p_{39}}\ {|_{\ }^{3 \ 1}}}+{{p_{38}}\ {|_{\ }^{3 \ 2}}}+{{p_{37}}\ {|_{\ }^{3 \ 3}}}+{{p_{36}}\ {|_{\ }^{3 \ 4}}}+{{p_{3 5}}\ {|_{\ }^{3 \ 5}}}+ \ \ \displaystyle {{p_{34}}\ {|_{\ }^{3 \ 6}}}+{{p_{33}}\ {|_{\ }^{3 \ 7}}}+{{p_{32}}\ {|_{\ }^{4 \ 0}}}+{{p_{31}}\ {|_{\ }^{4 \ 1}}}+{{p_{3 0}}\ {|_{\ }^{4 \ 2}}}+ \ \ \displaystyle {{p_{29}}\ {|_{\ }^{4 \ 3}}}+{{p_{28}}\ {|_{\ }^{4 \ 4}}}+{{p_{27}}\ {|_{\ }^{4 \ 5}}}+{{p_{26}}\ {|_{\ }^{4 \ 6}}}+{{p_{2 5}}\ {|_{\ }^{4 \ 7}}}+ \ \ \displaystyle {{p_{24}}\ {|_{\ }^{5 \ 0}}}+{{p_{23}}\ {|_{\ }^{5 \ 1}}}+{{p_{22}}\ {|_{\ }^{5 \ 2}}}+{{p_{21}}\ {|_{\ }^{5 \ 3}}}+{{p_{2 0}}\ {|_{\ }^{5 \ 4}}}+ \ \ \displaystyle {{p_{19}}\ {|_{\ }^{5 \ 5}}}+{{p_{18}}\ {|_{\ }^{5 \ 6}}}+{{p_{17}}\ {|_{\ }^{5 \ 7}}}+{{p_{16}}\ {|_{\ }^{6 \ 0}}}+{{p_{1 5}}\ {|_{\ }^{6 \ 1}}}+ \ \ \displaystyle {{p_{14}}\ {|_{\ }^{6 \ 2}}}+{{p_{13}}\ {|_{\ }^{6 \ 3}}}+{{p_{12}}\ {|_{\ }^{6 \ 4}}}+{{p_{11}}\ {|_{\ }^{6 \ 5}}}+{{p_{1 0}}\ {|_{\ }^{6 \ 6}}}+ \ \ \displaystyle {{p_{9}}\ {|_{\ }^{6 \ 7}}}+{{p_{8}}\ {|_{\ }^{7 \ 0}}}+{{p_{7}}\ {|_{\ }^{7 \ 1}}}+{{p_{6}}\ {|_{\ }^{7 \ 2}}}+{{p_{5}}\ {|_{\ }^{7 \ 3}}}+{{p_{4}}\ {|_{\ }^{7 \ 4}}}+ \ \ \displaystyle {{p_{3}}\ {|_{\ }^{7 \ 5}}}+{{p_{2}}\ {|_{\ }^{7 \ 6}}}+{|_{\ }^{7 \ 7}}$

(21)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

fricas

Ω0:𝐋  :=eval(Ω,ex1)$𝐋

$\label{eq22}��$

(22)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

fricas

λ0:𝐋  :=eval(λ,ex1)$𝐋

$\label{eq23}��$

(23)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

fricas

H0:𝐋 :=eval(H,ex1)$𝐋

$\label{eq24}H$

(24)

Type: LinearOperator?(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))