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Octonion Algebra Is Frobenius In Just One Way

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

Ref:

We need the Axiom LinearOperator library.

fricas
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP 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 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

fricas
dim:=8

\label{eq1}8(1)
Type: PositiveInteger?
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
ℒ := ClosedLinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ)

\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 0, 1, 2, 3, 4, 5, 6, 7 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
fricas
ⅇ:List ℒ      := basisOut()

\label{eq3}\left[{|_{\  0}}, \:{|_{\  1}}, \:{|_{\  2}}, \:{|_{\  3}}, \:{|_{\  4}}, \:{|_{\  5}}, \:{|_{\  6}}, \:{|_{\  7}}\right](3)
Type: List(ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
fricas
ⅆ:List ℒ      := basisIn()

\label{eq4}\left[{|^{\  0}}, \:{|^{\  1}}, \:{|^{\  2}}, \:{|^{\  3}}, \:{|^{\  4}}, \:{|^{\  5}}, \:{|^{\  6}}, \:{|^{\  7}}\right](4)
Type: List(ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
fricas
I:ℒ:=[1];   -- identity for composition
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
X:ℒ:=[2,1]; -- twist
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
V:ℒ:=ev(1) -- evaluation

\label{eq5}{|^{\  0 \  0}}+{|^{\  1 \  1}}+{|^{\  2 \  2}}+{|^{\  3 \  3}}+{|^{\  4 \  4}}+{|^{\  5 \  5}}+{|^{\  6 \  6}}+{|^{\  7 \  7}}(5)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
Λ:ℒ:=co(1) -- co-evaluation

\label{eq6}{|_{\  0 \  0}}+{|_{\  1 \  1}}+{|_{\  2 \  2}}+{|_{\  3 \  3}}+{|_{\  4 \  4}}+{|_{\  5 \  5}}+{|_{\  6 \  6}}+{|_{\  7 \  7}}(6)
Type: ClosedLinearOperator(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

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

\label{eq7}1(7)
Type: PositiveInteger?
fricas
--q1:=sb('q,[1])
q1:=1  -- not co-quaternion

\label{eq8}1(8)
Type: PositiveInteger?
fricas
q2:=sb('q,[2])

\label{eq9}q_{2}(9)
Type: Symbol
fricas
--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

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

\label{eq10}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}\right](10)
Type: List(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,q[2]))
fricas
-- Multiplication table:
M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]

\label{eq11}\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}}
(11)
Type: Matrix(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,q[2]))
fricas
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> 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(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{eq12}\begin{array}{@{}l}
\displaystyle
{|_{\  0}^{\  0 \  0}}+{|_{\  1}^{\  0 \  1}}+{|_{\  2}^{\  0 \  2}}+{|_{\  3}^{\  0 \  3}}+{{{q_{2}}\over{\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}}+ 
\
\
\displaystyle
{|_{\  5}^{\  1 \  4}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  1 \  5}}}-{|_{\  7}^{\  1 \  6}}+{|_{\  6}^{\  1 \  7}}+{|_{\  2}^{\  2 \  0}}-{|_{\  3}^{\  2 \  1}}- 
\
\
\displaystyle
{|_{\  0}^{\  2 \  2}}+{|_{\  1}^{\  2 \  3}}+{|_{\  6}^{\  2 \  4}}+{|_{\  7}^{\  2 \  5}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  2 \  6}}}-{|_{\  5}^{\  2 \  7}}+ 
\
\
\displaystyle
{|_{\  3}^{\  3 \  0}}+{|_{\  2}^{\  3 \  1}}-{|_{\  1}^{\  3 \  2}}-{|_{\  0}^{\  3 \  3}}+{|_{\  7}^{\  3 \  4}}-{|_{\  6}^{\  3 \  5}}+ 
\
\
\displaystyle
{|_{\  5}^{\  3 \  6}}-{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  3 \  7}}}+{{{q_{2}}\over{\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
{{{q_{2}}\over{\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}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  6 \  2}}}- 
\
\
\displaystyle
{|_{\  5}^{\  6 \  3}}-{{q_{2}}\ {|_{\  2}^{\  6 \  4}}}+{{\overline{q_{2}}}\ {|_{\  3}^{\  6 \  5}}}-{{q_{2}}\ {|_{\  0}^{\  6 \  6}}}-{{\overline{q_{2}}}\ {|_{\  1}^{\  6 \  7}}}+ 
\
\
\displaystyle
{|_{\  7}^{\  7 \  0}}-{|_{\  6}^{\  7 \  1}}+{|_{\  5}^{\  7 \  2}}+{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  7 \  3}}}-{{q_{2}}\ {|_{\  3}^{\  7 \  4}}}- 
\
\
\displaystyle
{{\overline{q_{2}}}\ {|_{\  2}^{\  7 \  5}}}+{{\overline{q_{2}}}\ {|_{\  1}^{\  7 \  6}}}-{{q_{2}}\ {|_{\  0}^{\  7 \  7}}}
(12)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
arity Y

\label{eq13}{{+}^{2}}\over +(13)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
fricas
matrix [[(ⅇ.i*ⅇ.j)/Y for i in 1..dim] for j in 1..dim]

\label{eq14}\left[ 
\begin{array}{cccccccc}
{|_{\  0}}&{|_{\  1}}&{|_{\  2}}&{|_{\  3}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}&{|_{\  5}}&{|_{\  6}}&{|_{\  7}}
\
{|_{\  1}}& -{|_{\  0}}& -{|_{\  3}}&{|_{\  2}}& -{|_{\  5}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}&{|_{\  7}}& -{|_{\  6}}
\
{|_{\  2}}&{|_{\  3}}& -{|_{\  0}}& -{|_{\  1}}& -{|_{\  6}}& -{|_{\  7}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}&{|_{\  5}}
\
{|_{\  3}}& -{|_{\  2}}&{|_{\  1}}& -{|_{\  0}}& -{|_{\  7}}&{|_{\  6}}& -{|_{\  5}}&{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}
\
{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}&{|_{\  5}}&{|_{\  6}}&{|_{\  7}}& -{{q_{2}}\ {|_{\  0}}}& -{{q_{2}}\ {|_{\  1}}}& -{{q_{2}}\ {|_{\  2}}}& -{{q_{2}}\ {|_{\  3}}}
\
{|_{\  5}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}&{|_{\  7}}& -{|_{\  6}}&{{q_{2}}\ {|_{\  1}}}& -{{q_{2}}\ {|_{\  0}}}&{{\overline{q_{2}}}\ {|_{\  3}}}& -{{\overline{q_{2}}}\ {|_{\  2}}}
\
{|_{\  6}}& -{|_{\  7}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}&{|_{\  5}}&{{q_{2}}\ {|_{\  2}}}& -{{\overline{q_{2}}}\ {|_{\  3}}}& -{{q_{2}}\ {|_{\  0}}}&{{\overline{q_{2}}}\ {|_{\  1}}}
\
{|_{\  7}}&{|_{\  6}}& -{|_{\  5}}& -{{{q_{2}}\over{\overline{q_{2}}}}\ {|_{\  4}}}&{{q_{2}}\ {|_{\  3}}}&{{\overline{q_{2}}}\ {|_{\  2}}}& -{{\overline{q_{2}}}\ {|_{\  1}}}& -{{q_{2}}\ {|_{\  0}}}
(14)
Type: Matrix(ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(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)

\label{eq15}\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}}}
(15)
Type: ClosedLinearOperator(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:


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\label{eq16}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (16)
(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: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

The Cartan-Killing Trace

fricas
Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V

\label{eq17}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  0 \  0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  1 \  1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  2 \  2}}}+ 
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  3 \  3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  4 \  4}}}+ 
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  5 \  5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  6 \  6}}}+ 
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  7 \  7}}}
(17)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

\label{eq18}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  0 \  0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  1 \  1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  2 \  2}}}+ 
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  3 \  3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  4 \  4}}}+ 
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  5 \  5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  6 \  6}}}+ 
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  7 \  7}}}
(18)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
test(Ù=Ú)

\label{eq19} \mbox{\rm true} (19)
Type: Boolean

forms a non-degenerate associative scalar product for Y

fricas
Ũ := Ù

\label{eq20}\begin{array}{@{}l}
\displaystyle
{{{{7 \ {\overline{q_{2}}}}+{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  0 \  0}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  1 \  1}}}+{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  2 \  2}}}+ 
\
\
\displaystyle
{{{-{7 \ {\overline{q_{2}}}}-{q_{2}}}\over{\overline{q_{2}}}}\ {|^{\  3 \  3}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  4 \  4}}}+ 
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  5 \  5}}}+{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  6 \  6}}}+ 
\
\
\displaystyle
{{{-{7 \ {q_{2}}\ {\overline{q_{2}}}}-{{q_{2}}^{2}}}\over{\overline{q_{2}}}}\ {|^{\  7 \  7}}}
(20)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

\label{eq21} \mbox{\rm false} (21)
Type: Boolean
fricas
determinant [[retract((ⅇ.i * ⅇ.j)/Ũ) for j in 1..dim] for i in 1..dim]

\label{eq22}{\left(
\begin{array}{@{}l}
\displaystyle
-{{5764801}\ {{q_{2}}^{4}}\ {{\overline{q_{2}}}^{8}}}-{{65883
44}\ {{q_{2}}^{5}}\ {{\overline{q_{2}}}^{7}}}- 
\
\
\displaystyle
{{3294172}\ {{q_{2}}^{6}}\ {{\overline{q_{2}}}^{6}}}-{{941192}\ {{q_{2}}^{7}}\ {{\overline{q_{2}}}^{5}}}-{{168070}\ {{q_{2}}^{8}}\ {{\overline{q_{2}}}^{4}}}- 
\
\
\displaystyle
{{19208}\ {{q_{2}}^{9}}\ {{\overline{q_{2}}}^{3}}}-{{1372}\ {{q_{2}}^{10}}\ {{\overline{q_{2}}}^{2}}}-{{56}\ {{q_{2}}^{11}}\ {\overline{q_{2}}}}-{{q_{2}}^{12}}
(22)
Type: Expression(Integer)

General Solution

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);
Type: Matrix(Expression(Integer))
fricas
u := transpose matrix [concat map(variables,ravel U)::List Symbol];
Type: Matrix(Polynomial(Integer))
fricas
J::OutputForm * u::OutputForm = 0

\label{eq23}(23)
Type: Equation(OutputForm?)
fricas
nrows(J),ncols(J)

\label{eq24}\left[{512}, \:{64}\right](24)
Type: Tuple(PositiveInteger?)

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);
Type: List(Vector(Expression(Integer)))
fricas
ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )
>> Error detected within library code: reducing over an empty list needs the 3 argument form

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) ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),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)
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) ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),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 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 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +
>> Error detected within library code: failed

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 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 + 0 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +

\label{eq25}\left[ 
\begin{array}{cccccccc}
{�� \ {|_{\  0 \  0}}}&{�� \ {|_{\  1 \  0}}}&{�� \ {|_{\  2 \  0}}}&{�� \ {|_{\  3 \  0}}}&{�� \ {|_{\  4 \  0}}}&{�� \ {|_{\  5 \  0}}}&{�� \ {|_{\  6 \  0}}}&{�� \ {|_{\  7 \  0}}}
\
{�� \ {|_{\  0 \  1}}}&{�� \ {|_{\  1 \  1}}}&{�� \ {|_{\  2 \  1}}}&{�� \ {|_{\  3 \  1}}}&{�� \ {|_{\  4 \  1}}}&{�� \ {|_{\  5 \  1}}}&{�� \ {|_{\  6 \  1}}}&{�� \ {|_{\  7 \  1}}}
\
{�� \ {|_{\  0 \  2}}}&{�� \ {|_{\  1 \  2}}}&{�� \ {|_{\  2 \  2}}}&{�� \ {|_{\  3 \  2}}}&{�� \ {|_{\  4 \  2}}}&{�� \ {|_{\  5 \  2}}}&{�� \ {|_{\  6 \  2}}}&{�� \ {|_{\  7 \  2}}}
\
{�� \ {|_{\  0 \  3}}}&{�� \ {|_{\  1 \  3}}}&{�� \ {|_{\  2 \  3}}}&{�� \ {|_{\  3 \  3}}}&{�� \ {|_{\  4 \  3}}}&{�� \ {|_{\  5 \  3}}}&{�� \ {|_{\  6 \  3}}}&{�� \ {|_{\  7 \  3}}}
\
{�� \ {|_{\  0 \  4}}}&{�� \ {|_{\  1 \  4}}}&{�� \ {|_{\  2 \  4}}}&{�� \ {|_{\  3 \  4}}}&{�� \ {|_{\  4 \  4}}}&{�� \ {|_{\  5 \  4}}}&{�� \ {|_{\  6 \  4}}}&{�� \ {|_{\  7 \  4}}}
\
{�� \ {|_{\  0 \  5}}}&{�� \ {|_{\  1 \  5}}}&{�� \ {|_{\  2 \  5}}}&{�� \ {|_{\  3 \  5}}}&{�� \ {|_{\  4 \  5}}}&{�� \ {|_{\  5 \  5}}}&{�� \ {|_{\  6 \  5}}}&{�� \ {|_{\  7 \  5}}}
\
{�� \ {|_{\  0 \  6}}}&{�� \ {|_{\  1 \  6}}}&{�� \ {|_{\  2 \  6}}}&{�� \ {|_{\  3 \  6}}}&{�� \ {|_{\  4 \  6}}}&{�� \ {|_{\  5 \  6}}}&{�� \ {|_{\  6 \  6}}}&{�� \ {|_{\  7 \  6}}}
\
{�� \ {|_{\  0 \  7}}}&{�� \ {|_{\  1 \  7}}}&{�� \ {|_{\  2 \  7}}}&{�� \ {|_{\  3 \  7}}}&{�� \ {|_{\  4 \  7}}}&{�� \ {|_{\  5 \  7}}}&{�� \ {|_{\  6 \  7}}}&{�� \ {|_{\  7 \  7}}}
(25)
Type: Matrix(ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer)))
fricas
mU:=transpose inverse map(retract,Um)
>> Error detected within library code: failed

Check "dimension" and the snake relations.

fricas
d:ℒ:=
       Ω    /
       X    /
       Ų
2 0 + - -- 0 2 + arity warning: ---- 2 0 + - -- 0 2 + 2 + 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +

\label{eq26}\begin{array}{@{}l}
\displaystyle
{�� \  �� \ {|_{\  0 \  0}^{\  0 \  0}}}+{�� \  �� \ {|_{\  1 \  0}^{\  0 \  1}}}+{�� \  �� \ {|_{\  2 \  0}^{\  0 \  2}}}+{�� \  �� \ {|_{\  3 \  0}^{\  0 \  3}}}+{�� \  �� \ {|_{\  4 \  0}^{\  0 \  4}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  5 \  0}^{\  0 \  5}}}+{�� \  �� \ {|_{\  6 \  0}^{\  0 \  6}}}+{�� \  �� \ {|_{\  7 \  0}^{\  0 \  7}}}+{�� \  �� \ {|_{\  0 \  1}^{\  1 \  0}}}+{�� \  �� \ {|_{\  1 \  1}^{\  1 \  1}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  2 \  1}^{\  1 \  2}}}+{�� \  �� \ {|_{\  3 \  1}^{\  1 \  3}}}+{�� \  �� \ {|_{\  4 \  1}^{\  1 \  4}}}+{�� \  �� \ {|_{\  5 \  1}^{\  1 \  5}}}+{�� \  �� \ {|_{\  6 \  1}^{\  1 \  6}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  7 \  1}^{\  1 \  7}}}+{�� \  �� \ {|_{\  0 \  2}^{\  2 \  0}}}+{�� \  �� \ {|_{\  1 \  2}^{\  2 \  1}}}+{�� \  �� \ {|_{\  2 \  2}^{\  2 \  2}}}+{�� \  �� \ {|_{\  3 \  2}^{\  2 \  3}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  4 \  2}^{\  2 \  4}}}+{�� \  �� \ {|_{\  5 \  2}^{\  2 \  5}}}+{�� \  �� \ {|_{\  6 \  2}^{\  2 \  6}}}+{�� \  �� \ {|_{\  7 \  2}^{\  2 \  7}}}+{�� \  �� \ {|_{\  0 \  3}^{\  3 \  0}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  1 \  3}^{\  3 \  1}}}+{�� \  �� \ {|_{\  2 \  3}^{\  3 \  2}}}+{�� \  �� \ {|_{\  3 \  3}^{\  3 \  3}}}+{�� \  �� \ {|_{\  4 \  3}^{\  3 \  4}}}+{�� \  �� \ {|_{\  5 \  3}^{\  3 \  5}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  6 \  3}^{\  3 \  6}}}+{�� \  �� \ {|_{\  7 \  3}^{\  3 \  7}}}+{�� \  �� \ {|_{\  0 \  4}^{\  4 \  0}}}+{�� \  �� \ {|_{\  1 \  4}^{\  4 \  1}}}+{�� \  �� \ {|_{\  2 \  4}^{\  4 \  2}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  3 \  4}^{\  4 \  3}}}+{�� \  �� \ {|_{\  4 \  4}^{\  4 \  4}}}+{�� \  �� \ {|_{\  5 \  4}^{\  4 \  5}}}+{�� \  �� \ {|_{\  6 \  4}^{\  4 \  6}}}+{�� \  �� \ {|_{\  7 \  4}^{\  4 \  7}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  0 \  5}^{\  5 \  0}}}+{�� \  �� \ {|_{\  1 \  5}^{\  5 \  1}}}+{�� \  �� \ {|_{\  2 \  5}^{\  5 \  2}}}+{�� \  �� \ {|_{\  3 \  5}^{\  5 \  3}}}+{�� \  �� \ {|_{\  4 \  5}^{\  5 \  4}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  5 \  5}^{\  5 \  5}}}+{�� \  �� \ {|_{\  6 \  5}^{\  5 \  6}}}+{�� \  �� \ {|_{\  7 \  5}^{\  5 \  7}}}+{�� \  �� \ {|_{\  0 \  6}^{\  6 \  0}}}+{�� \  �� \ {|_{\  1 \  6}^{\  6 \  1}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  2 \  6}^{\  6 \  2}}}+{�� \  �� \ {|_{\  3 \  6}^{\  6 \  3}}}+{�� \  �� \ {|_{\  4 \  6}^{\  6 \  4}}}+{�� \  �� \ {|_{\  5 \  6}^{\  6 \  5}}}+{�� \  �� \ {|_{\  6 \  6}^{\  6 \  6}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  7 \  6}^{\  6 \  7}}}+{�� \  �� \ {|_{\  0 \  7}^{\  7 \  0}}}+{�� \  �� \ {|_{\  1 \  7}^{\  7 \  1}}}+{�� \  �� \ {|_{\  2 \  7}^{\  7 \  2}}}+{�� \  �� \ {|_{\  3 \  7}^{\  7 \  3}}}+ 
\
\
\displaystyle
{�� \  �� \ {|_{\  4 \  7}^{\  7 \  4}}}+{�� \  �� \ {|_{\  5 \  7}^{\  7 \  5}}}+{�� \  �� \ {|_{\  6 \  7}^{\  7 \  6}}}+{�� \  �� \ {|_{\  7 \  7}^{\  7 \  7}}}
(26)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
test
    (    I Ω     )  /
    (     Ų I    )  =  I

\label{eq27} \mbox{\rm false} (27)
Type: Boolean
fricas
test
    (     Ω I    )  /
    (    I Ų     )  =  I
There are no library operations named Ω Use HyperDoc Browse or issue )what op Ω to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω with argument type(s) ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
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.

fricas
W:=(Y,I)/Ų
3 + 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +

\label{eq28}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\  0 \  0}^{\  0 \  0 \  0}}}+{�� \ {|_{\  0 \  1}^{\  0 \  0 \  1}}}+{�� \ {|_{\  0 \  2}^{\  0 \  0 \  2}}}+{�� \ {|_{\  0 \  3}^{\  0 \  0 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  4}^{\  0 \  0 \  4}}}+{�� \ {|_{\  0 \  5}^{\  0 \  0 \  5}}}+{�� \ {|_{\  0 \  6}^{\  0 \  0 \  6}}}+{�� \ {|_{\  0 \  7}^{\  0 \  0 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  0}^{\  0 \  1 \  0}}}+{�� \ {|_{\  1 \  1}^{\  0 \  1 \  1}}}+{�� \ {|_{\  1 \  2}^{\  0 \  1 \  2}}}+{�� \ {|_{\  1 \  3}^{\  0 \  1 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  4}^{\  0 \  1 \  4}}}+{�� \ {|_{\  1 \  5}^{\  0 \  1 \  5}}}+{�� \ {|_{\  1 \  6}^{\  0 \  1 \  6}}}+{�� \ {|_{\  1 \  7}^{\  0 \  1 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  0}^{\  0 \  2 \  0}}}+{�� \ {|_{\  2 \  1}^{\  0 \  2 \  1}}}+{�� \ {|_{\  2 \  2}^{\  0 \  2 \  2}}}+{�� \ {|_{\  2 \  3}^{\  0 \  2 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  4}^{\  0 \  2 \  4}}}+{�� \ {|_{\  2 \  5}^{\  0 \  2 \  5}}}+{�� \ {|_{\  2 \  6}^{\  0 \  2 \  6}}}+{�� \ {|_{\  2 \  7}^{\  0 \  2 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  0}^{\  0 \  3 \  0}}}+{�� \ {|_{\  3 \  1}^{\  0 \  3 \  1}}}+{�� \ {|_{\  3 \  2}^{\  0 \  3 \  2}}}+{�� \ {|_{\  3 \  3}^{\  0 \  3 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  4}^{\  0 \  3 \  4}}}+{�� \ {|_{\  3 \  5}^{\  0 \  3 \  5}}}+{�� \ {|_{\  3 \  6}^{\  0 \  3 \  6}}}+{�� \ {|_{\  3 \  7}^{\  0 \  3 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  0 \  4 \  0}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  0 \  4 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  0 \  4 \  2}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  0 \  4 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  0 \  4 \  4}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  0 \  4 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  0 \  4 \  6}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  0 \  4 \  7}}}+{�� \ {|_{\  5 \  0}^{\  0 \  5 \  0}}}+{�� \ {|_{\  5 \  1}^{\  0 \  5 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  2}^{\  0 \  5 \  2}}}+{�� \ {|_{\  5 \  3}^{\  0 \  5 \  3}}}+{�� \ {|_{\  5 \  4}^{\  0 \  5 \  4}}}+{�� \ {|_{\  5 \  5}^{\  0 \  5 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  0 \  5 \  6}}}+{�� \ {|_{\  5 \  7}^{\  0 \  5 \  7}}}+{�� \ {|_{\  6 \  0}^{\  0 \  6 \  0}}}+{�� \ {|_{\  6 \  1}^{\  0 \  6 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  2}^{\  0 \  6 \  2}}}+{�� \ {|_{\  6 \  3}^{\  0 \  6 \  3}}}+{�� \ {|_{\  6 \  4}^{\  0 \  6 \  4}}}+{�� \ {|_{\  6 \  5}^{\  0 \  6 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  6}^{\  0 \  6 \  6}}}+{�� \ {|_{\  6 \  7}^{\  0 \  6 \  7}}}+{�� \ {|_{\  7 \  0}^{\  0 \  7 \  0}}}+{�� \ {|_{\  7 \  1}^{\  0 \  7 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  2}^{\  0 \  7 \  2}}}+{�� \ {|_{\  7 \  3}^{\  0 \  7 \  3}}}+{�� \ {|_{\  7 \  4}^{\  0 \  7 \  4}}}+{�� \ {|_{\  7 \  5}^{\  0 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  6}^{\  0 \  7 \  6}}}+{�� \ {|_{\  7 \  7}^{\  0 \  7 \  7}}}+{�� \ {|_{\  1 \  0}^{\  1 \  0 \  0}}}+{�� \ {|_{\  1 \  1}^{\  1 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  2}^{\  1 \  0 \  2}}}+{�� \ {|_{\  1 \  3}^{\  1 \  0 \  3}}}+{�� \ {|_{\  1 \  4}^{\  1 \  0 \  4}}}+{�� \ {|_{\  1 \  5}^{\  1 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  6}^{\  1 \  0 \  6}}}+{�� \ {|_{\  1 \  7}^{\  1 \  0 \  7}}}-{�� \ {|_{\  0 \  0}^{\  1 \  1 \  0}}}-{�� \ {|_{\  0 \  1}^{\  1 \  1 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  0 \  2}^{\  1 \  1 \  2}}}-{�� \ {|_{\  0 \  3}^{\  1 \  1 \  3}}}-{�� \ {|_{\  0 \  4}^{\  1 \  1 \  4}}}-{�� \ {|_{\  0 \  5}^{\  1 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  0 \  6}^{\  1 \  1 \  6}}}-{�� \ {|_{\  0 \  7}^{\  1 \  1 \  7}}}+{�� \ {|_{\  3 \  0}^{\  1 \  2 \  0}}}+{�� \ {|_{\  3 \  1}^{\  1 \  2 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  2}^{\  1 \  2 \  2}}}+{�� \ {|_{\  3 \  3}^{\  1 \  2 \  3}}}+{�� \ {|_{\  3 \  4}^{\  1 \  2 \  4}}}+{�� \ {|_{\  3 \  5}^{\  1 \  2 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  6}^{\  1 \  2 \  6}}}+{�� \ {|_{\  3 \  7}^{\  1 \  2 \  7}}}-{�� \ {|_{\  2 \  0}^{\  1 \  3 \  0}}}-{�� \ {|_{\  2 \  1}^{\  1 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  2 \  2}^{\  1 \  3 \  2}}}-{�� \ {|_{\  2 \  3}^{\  1 \  3 \  3}}}-{�� \ {|_{\  2 \  4}^{\  1 \  3 \  4}}}-{�� \ {|_{\  2 \  5}^{\  1 \  3 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  2 \  6}^{\  1 \  3 \  6}}}-{�� \ {|_{\  2 \  7}^{\  1 \  3 \  7}}}+{�� \ {|_{\  5 \  0}^{\  1 \  4 \  0}}}+{�� \ {|_{\  5 \  1}^{\  1 \  4 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  2}^{\  1 \  4 \  2}}}+{�� \ {|_{\  5 \  3}^{\  1 \  4 \  3}}}+{�� \ {|_{\  5 \  4}^{\  1 \  4 \  4}}}+{�� \ {|_{\  5 \  5}^{\  1 \  4 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  1 \  4 \  6}}}+{�� \ {|_{\  5 \  7}^{\  1 \  4 \  7}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  1 \  5 \  0}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  1 \  5 \  1}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  1 \  5 \  2}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  1 \  5 \  3}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  1 \  5 \  4}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  1 \  5 \  5}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  1 \  5 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  1 \  5 \  7}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  0}^{\  1 \  6 \  0}}}-{�� \ {|_{\  7 \  1}^{\  1 \  6 \  1}}}-{�� \ {|_{\  7 \  2}^{\  1 \  6 \  2}}}-{�� \ {|_{\  7 \  3}^{\  1 \  6 \  3}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  4}^{\  1 \  6 \  4}}}-{�� \ {|_{\  7 \  5}^{\  1 \  6 \  5}}}-{�� \ {|_{\  7 \  6}^{\  1 \  6 \  6}}}-{�� \ {|_{\  7 \  7}^{\  1 \  6 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  0}^{\  1 \  7 \  0}}}+{�� \ {|_{\  6 \  1}^{\  1 \  7 \  1}}}+{�� \ {|_{\  6 \  2}^{\  1 \  7 \  2}}}+{�� \ {|_{\  6 \  3}^{\  1 \  7 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  4}^{\  1 \  7 \  4}}}+{�� \ {|_{\  6 \  5}^{\  1 \  7 \  5}}}+{�� \ {|_{\  6 \  6}^{\  1 \  7 \  6}}}+{�� \ {|_{\  6 \  7}^{\  1 \  7 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  0}^{\  2 \  0 \  0}}}+{�� \ {|_{\  2 \  1}^{\  2 \  0 \  1}}}+{�� \ {|_{\  2 \  2}^{\  2 \  0 \  2}}}+{�� \ {|_{\  2 \  3}^{\  2 \  0 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  4}^{\  2 \  0 \  4}}}+{�� \ {|_{\  2 \  5}^{\  2 \  0 \  5}}}+{�� \ {|_{\  2 \  6}^{\  2 \  0 \  6}}}+{�� \ {|_{\  2 \  7}^{\  2 \  0 \  7}}}- 
\
\
\displaystyle
{�� \ {|_{\  3 \  0}^{\  2 \  1 \  0}}}-{�� \ {|_{\  3 \  1}^{\  2 \  1 \  1}}}-{�� \ {|_{\  3 \  2}^{\  2 \  1 \  2}}}-{�� \ {|_{\  3 \  3}^{\  2 \  1 \  3}}}- 
\
\
\displaystyle
{�� \ {|_{\  3 \  4}^{\  2 \  1 \  4}}}-{�� \ {|_{\  3 \  5}^{\  2 \  1 \  5}}}-{�� \ {|_{\  3 \  6}^{\  2 \  1 \  6}}}-{�� \ {|_{\  3 \  7}^{\  2 \  1 \  7}}}- 
\
\
\displaystyle
{�� \ {|_{\  0 \  0}^{\  2 \  2 \  0}}}-{�� \ {|_{\  0 \  1}^{\  2 \  2 \  1}}}-{�� \ {|_{\  0 \  2}^{\  2 \  2 \  2}}}-{�� \ {|_{\  0 \  3}^{\  2 \  2 \  3}}}- 
\
\
\displaystyle
{�� \ {|_{\  0 \  4}^{\  2 \  2 \  4}}}-{�� \ {|_{\  0 \  5}^{\  2 \  2 \  5}}}-{�� \ {|_{\  0 \  6}^{\  2 \  2 \  6}}}-{�� \ {|_{\  0 \  7}^{\  2 \  2 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  0}^{\  2 \  3 \  0}}}+{�� \ {|_{\  1 \  1}^{\  2 \  3 \  1}}}+{�� \ {|_{\  1 \  2}^{\  2 \  3 \  2}}}+{�� \ {|_{\  1 \  3}^{\  2 \  3 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  4}^{\  2 \  3 \  4}}}+{�� \ {|_{\  1 \  5}^{\  2 \  3 \  5}}}+{�� \ {|_{\  1 \  6}^{\  2 \  3 \  6}}}+{�� \ {|_{\  1 \  7}^{\  2 \  3 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  0}^{\  2 \  4 \  0}}}+{�� \ {|_{\  6 \  1}^{\  2 \  4 \  1}}}+{�� \ {|_{\  6 \  2}^{\  2 \  4 \  2}}}+{�� \ {|_{\  6 \  3}^{\  2 \  4 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  4}^{\  2 \  4 \  4}}}+{�� \ {|_{\  6 \  5}^{\  2 \  4 \  5}}}+{�� \ {|_{\  6 \  6}^{\  2 \  4 \  6}}}+{�� \ {|_{\  6 \  7}^{\  2 \  4 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  0}^{\  2 \  5 \  0}}}+{�� \ {|_{\  7 \  1}^{\  2 \  5 \  1}}}+{�� \ {|_{\  7 \  2}^{\  2 \  5 \  2}}}+{�� \ {|_{\  7 \  3}^{\  2 \  5 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  4}^{\  2 \  5 \  4}}}+{�� \ {|_{\  7 \  5}^{\  2 \  5 \  5}}}+{�� \ {|_{\  7 \  6}^{\  2 \  5 \  6}}}+{�� \ {|_{\  7 \  7}^{\  2 \  5 \  7}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  2 \  6 \  0}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  2 \  6 \  1}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  2 \  6 \  2}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  2 \  6 \  3}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  2 \  6 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  2 \  6 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  2 \  6 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  2 \  6 \  7}}}-{�� \ {|_{\  5 \  0}^{\  2 \  7 \  0}}}-{�� \ {|_{\  5 \  1}^{\  2 \  7 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  2}^{\  2 \  7 \  2}}}-{�� \ {|_{\  5 \  3}^{\  2 \  7 \  3}}}-{�� \ {|_{\  5 \  4}^{\  2 \  7 \  4}}}-{�� \ {|_{\  5 \  5}^{\  2 \  7 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  2 \  7 \  6}}}-{�� \ {|_{\  5 \  7}^{\  2 \  7 \  7}}}+{�� \ {|_{\  3 \  0}^{\  3 \  0 \  0}}}+{�� \ {|_{\  3 \  1}^{\  3 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  2}^{\  3 \  0 \  2}}}+{�� \ {|_{\  3 \  3}^{\  3 \  0 \  3}}}+{�� \ {|_{\  3 \  4}^{\  3 \  0 \  4}}}+{�� \ {|_{\  3 \  5}^{\  3 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  6}^{\  3 \  0 \  6}}}+{�� \ {|_{\  3 \  7}^{\  3 \  0 \  7}}}+{�� \ {|_{\  2 \  0}^{\  3 \  1 \  0}}}+{�� \ {|_{\  2 \  1}^{\  3 \  1 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  2}^{\  3 \  1 \  2}}}+{�� \ {|_{\  2 \  3}^{\  3 \  1 \  3}}}+{�� \ {|_{\  2 \  4}^{\  3 \  1 \  4}}}+{�� \ {|_{\  2 \  5}^{\  3 \  1 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  6}^{\  3 \  1 \  6}}}+{�� \ {|_{\  2 \  7}^{\  3 \  1 \  7}}}-{�� \ {|_{\  1 \  0}^{\  3 \  2 \  0}}}-{�� \ {|_{\  1 \  1}^{\  3 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  1 \  2}^{\  3 \  2 \  2}}}-{�� \ {|_{\  1 \  3}^{\  3 \  2 \  3}}}-{�� \ {|_{\  1 \  4}^{\  3 \  2 \  4}}}-{�� \ {|_{\  1 \  5}^{\  3 \  2 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  1 \  6}^{\  3 \  2 \  6}}}-{�� \ {|_{\  1 \  7}^{\  3 \  2 \  7}}}-{�� \ {|_{\  0 \  0}^{\  3 \  3 \  0}}}-{�� \ {|_{\  0 \  1}^{\  3 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  0 \  2}^{\  3 \  3 \  2}}}-{�� \ {|_{\  0 \  3}^{\  3 \  3 \  3}}}-{�� \ {|_{\  0 \  4}^{\  3 \  3 \  4}}}-{�� \ {|_{\  0 \  5}^{\  3 \  3 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  0 \  6}^{\  3 \  3 \  6}}}-{�� \ {|_{\  0 \  7}^{\  3 \  3 \  7}}}+{�� \ {|_{\  7 \  0}^{\  3 \  4 \  0}}}+{�� \ {|_{\  7 \  1}^{\  3 \  4 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  2}^{\  3 \  4 \  2}}}+{�� \ {|_{\  7 \  3}^{\  3 \  4 \  3}}}+{�� \ {|_{\  7 \  4}^{\  3 \  4 \  4}}}+{�� \ {|_{\  7 \  5}^{\  3 \  4 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  6}^{\  3 \  4 \  6}}}+{�� \ {|_{\  7 \  7}^{\  3 \  4 \  7}}}-{�� \ {|_{\  6 \  0}^{\  3 \  5 \  0}}}-{�� \ {|_{\  6 \  1}^{\  3 \  5 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  2}^{\  3 \  5 \  2}}}-{�� \ {|_{\  6 \  3}^{\  3 \  5 \  3}}}-{�� \ {|_{\  6 \  4}^{\  3 \  5 \  4}}}-{�� \ {|_{\  6 \  5}^{\  3 \  5 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  6}^{\  3 \  5 \  6}}}-{�� \ {|_{\  6 \  7}^{\  3 \  5 \  7}}}+{�� \ {|_{\  5 \  0}^{\  3 \  6 \  0}}}+{�� \ {|_{\  5 \  1}^{\  3 \  6 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  2}^{\  3 \  6 \  2}}}+{�� \ {|_{\  5 \  3}^{\  3 \  6 \  3}}}+{�� \ {|_{\  5 \  4}^{\  3 \  6 \  4}}}+{�� \ {|_{\  5 \  5}^{\  3 \  6 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  3 \  6 \  6}}}+{�� \ {|_{\  5 \  7}^{\  3 \  6 \  7}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  3 \  7 \  0}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  3 \  7 \  1}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  3 \  7 \  2}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  3 \  7 \  3}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  3 \  7 \  4}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  3 \  7 \  5}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  3 \  7 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  3 \  7 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  4 \  0 \  0}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  4 \  0 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  4 \  0 \  2}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  4 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  0 \  4}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  4 \  0 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  4 \  0 \  6}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  4 \  0 \  7}}}-{�� \ {|_{\  5 \  0}^{\  4 \  1 \  0}}}-{�� \ {|_{\  5 \  1}^{\  4 \  1 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  2}^{\  4 \  1 \  2}}}-{�� \ {|_{\  5 \  3}^{\  4 \  1 \  3}}}-{�� \ {|_{\  5 \  4}^{\  4 \  1 \  4}}}-{�� \ {|_{\  5 \  5}^{\  4 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  4 \  1 \  6}}}-{�� \ {|_{\  5 \  7}^{\  4 \  1 \  7}}}-{�� \ {|_{\  6 \  0}^{\  4 \  2 \  0}}}-{�� \ {|_{\  6 \  1}^{\  4 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  2}^{\  4 \  2 \  2}}}-{�� \ {|_{\  6 \  3}^{\  4 \  2 \  3}}}-{�� \ {|_{\  6 \  4}^{\  4 \  2 \  4}}}-{�� \ {|_{\  6 \  5}^{\  4 \  2 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  6}^{\  4 \  2 \  6}}}-{�� \ {|_{\  6 \  7}^{\  4 \  2 \  7}}}-{�� \ {|_{\  7 \  0}^{\  4 \  3 \  0}}}-{�� \ {|_{\  7 \  1}^{\  4 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  2}^{\  4 \  3 \  2}}}-{�� \ {|_{\  7 \  3}^{\  4 \  3 \  3}}}-{�� \ {|_{\  7 \  4}^{\  4 \  3 \  4}}}-{�� \ {|_{\  7 \  5}^{\  4 \  3 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  6}^{\  4 \  3 \  6}}}-{�� \ {|_{\  7 \  7}^{\  4 \  3 \  7}}}-{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  4 \  4 \  0}}}-{{q_{2}}\  �� \ {|_{\  0 \  1}^{\  4 \  4 \  1}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  2}^{\  4 \  4 \  2}}}-{{q_{2}}\  �� \ {|_{\  0 \  3}^{\  4 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  0 \  4}^{\  4 \  4 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  5}^{\  4 \  4 \  5}}}-{{q_{2}}\  �� \ {|_{\  0 \  6}^{\  4 \  4 \  6}}}-{{q_{2}}\  �� \ {|_{\  0 \  7}^{\  4 \  4 \  7}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  0}^{\  4 \  5 \  0}}}+{{q_{2}}\  �� \ {|_{\  1 \  1}^{\  4 \  5 \  1}}}+{{q_{2}}\  �� \ {|_{\  1 \  2}^{\  4 \  5 \  2}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  3}^{\  4 \  5 \  3}}}+{{q_{2}}\  �� \ {|_{\  1 \  4}^{\  4 \  5 \  4}}}+{{q_{2}}\  �� \ {|_{\  1 \  5}^{\  4 \  5 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  6}^{\  4 \  5 \  6}}}+{{q_{2}}\  �� \ {|_{\  1 \  7}^{\  4 \  5 \  7}}}+{{q_{2}}\  �� \ {|_{\  2 \  0}^{\  4 \  6 \  0}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  1}^{\  4 \  6 \  1}}}+{{q_{2}}\  �� \ {|_{\  2 \  2}^{\  4 \  6 \  2}}}+{{q_{2}}\  �� \ {|_{\  2 \  3}^{\  4 \  6 \  3}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  4}^{\  4 \  6 \  4}}}+{{q_{2}}\  �� \ {|_{\  2 \  5}^{\  4 \  6 \  5}}}+{{q_{2}}\  �� \ {|_{\  2 \  6}^{\  4 \  6 \  6}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  7}^{\  4 \  6 \  7}}}+{{q_{2}}\  �� \ {|_{\  3 \  0}^{\  4 \  7 \  0}}}+{{q_{2}}\  �� \ {|_{\  3 \  1}^{\  4 \  7 \  1}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  2}^{\  4 \  7 \  2}}}+{{q_{2}}\  �� \ {|_{\  3 \  3}^{\  4 \  7 \  3}}}+{{q_{2}}\  �� \ {|_{\  3 \  4}^{\  4 \  7 \  4}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  5}^{\  4 \  7 \  5}}}+{{q_{2}}\  �� \ {|_{\  3 \  6}^{\  4 \  7 \  6}}}+{{q_{2}}\  �� \ {|_{\  3 \  7}^{\  4 \  7 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  0}^{\  5 \  0 \  0}}}+{�� \ {|_{\  5 \  1}^{\  5 \  0 \  1}}}+{�� \ {|_{\  5 \  2}^{\  5 \  0 \  2}}}+{�� \ {|_{\  5 \  3}^{\  5 \  0 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  4}^{\  5 \  0 \  4}}}+{�� \ {|_{\  5 \  5}^{\  5 \  0 \  5}}}+{�� \ {|_{\  5 \  6}^{\  5 \  0 \  6}}}+{�� \ {|_{\  5 \  7}^{\  5 \  0 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  5 \  1 \  0}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  5 \  1 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  5 \  1 \  2}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  5 \  1 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  5 \  1 \  4}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  5 \  1 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  5 \  1 \  6}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  5 \  1 \  7}}}-{�� \ {|_{\  7 \  0}^{\  5 \  2 \  0}}}-{�� \ {|_{\  7 \  1}^{\  5 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  2}^{\  5 \  2 \  2}}}-{�� \ {|_{\  7 \  3}^{\  5 \  2 \  3}}}-{�� \ {|_{\  7 \  4}^{\  5 \  2 \  4}}}-{�� \ {|_{\  7 \  5}^{\  5 \  2 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  6}^{\  5 \  2 \  6}}}-{�� \ {|_{\  7 \  7}^{\  5 \  2 \  7}}}+{�� \ {|_{\  6 \  0}^{\  5 \  3 \  0}}}+{�� \ {|_{\  6 \  1}^{\  5 \  3 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  2}^{\  5 \  3 \  2}}}+{�� \ {|_{\  6 \  3}^{\  5 \  3 \  3}}}+{�� \ {|_{\  6 \  4}^{\  5 \  3 \  4}}}+{�� \ {|_{\  6 \  5}^{\  5 \  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  6}^{\  5 \  3 \  6}}}+{�� \ {|_{\  6 \  7}^{\  5 \  3 \  7}}}-{{q_{2}}\  �� \ {|_{\  1 \  0}^{\  5 \  4 \  0}}}-{{q_{2}}\  �� \ {|_{\  1 \  1}^{\  5 \  4 \  1}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  2}^{\  5 \  4 \  2}}}-{{q_{2}}\  �� \ {|_{\  1 \  3}^{\  5 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  1 \  4}^{\  5 \  4 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  5}^{\  5 \  4 \  5}}}-{{q_{2}}\  �� \ {|_{\  1 \  6}^{\  5 \  4 \  6}}}-{{q_{2}}\  �� \ {|_{\  1 \  7}^{\  5 \  4 \  7}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  5 \  5 \  0}}}-{{q_{2}}\  �� \ {|_{\  0 \  1}^{\  5 \  5 \  1}}}-{{q_{2}}\  �� \ {|_{\  0 \  2}^{\  5 \  5 \  2}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  3}^{\  5 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  0 \  4}^{\  5 \  5 \  4}}}-{{q_{2}}\  �� \ {|_{\  0 \  5}^{\  5 \  5 \  5}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  6}^{\  5 \  5 \  6}}}-{{q_{2}}\  �� \ {|_{\  0 \  7}^{\  5 \  5 \  7}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3 \  0}^{\  5 \  6 \  0}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  1}^{\  5 \  6 \  1}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3 \  2}^{\  5 \  6 \  2}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3 \  3}^{\  5 \  6 \  3}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  4}^{\  5 \  6 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3 \  5}^{\  5 \  6 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3 \  6}^{\  5 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  7}^{\  5 \  6 \  7}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  0}^{\  5 \  7 \  0}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  1}^{\  5 \  7 \  1}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2 \  2}^{\  5 \  7 \  2}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  3}^{\  5 \  7 \  3}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  4}^{\  5 \  7 \  4}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2 \  5}^{\  5 \  7 \  5}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  6}^{\  5 \  7 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  7}^{\  5 \  7 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  0}^{\  6 \  0 \  0}}}+{�� \ {|_{\  6 \  1}^{\  6 \  0 \  1}}}+{�� \ {|_{\  6 \  2}^{\  6 \  0 \  2}}}+{�� \ {|_{\  6 \  3}^{\  6 \  0 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  4}^{\  6 \  0 \  4}}}+{�� \ {|_{\  6 \  5}^{\  6 \  0 \  5}}}+{�� \ {|_{\  6 \  6}^{\  6 \  0 \  6}}}+{�� \ {|_{\  6 \  7}^{\  6 \  0 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  0}^{\  6 \  1 \  0}}}+{�� \ {|_{\  7 \  1}^{\  6 \  1 \  1}}}+{�� \ {|_{\  7 \  2}^{\  6 \  1 \  2}}}+{�� \ {|_{\  7 \  3}^{\  6 \  1 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  4}^{\  6 \  1 \  4}}}+{�� \ {|_{\  7 \  5}^{\  6 \  1 \  5}}}+{�� \ {|_{\  7 \  6}^{\  6 \  1 \  6}}}+{�� \ {|_{\  7 \  7}^{\  6 \  1 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  6 \  2 \  0}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  6 \  2 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  6 \  2 \  2}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  6 \  2 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  6 \  2 \  4}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  6 \  2 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  6 \  2 \  6}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  6 \  2 \  7}}}-{�� \ {|_{\  5 \  0}^{\  6 \  3 \  0}}}-{�� \ {|_{\  5 \  1}^{\  6 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  2}^{\  6 \  3 \  2}}}-{�� \ {|_{\  5 \  3}^{\  6 \  3 \  3}}}-{�� \ {|_{\  5 \  4}^{\  6 \  3 \  4}}}-{�� \ {|_{\  5 \  5}^{\  6 \  3 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  6 \  3 \  6}}}-{�� \ {|_{\  5 \  7}^{\  6 \  3 \  7}}}-{{q_{2}}\  �� \ {|_{\  2 \  0}^{\  6 \  4 \  0}}}-{{q_{2}}\  �� \ {|_{\  2 \  1}^{\  6 \  4 \  1}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  2}^{\  6 \  4 \  2}}}-{{q_{2}}\  �� \ {|_{\  2 \  3}^{\  6 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  2 \  4}^{\  6 \  4 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  5}^{\  6 \  4 \  5}}}-{{q_{2}}\  �� \ {|_{\  2 \  6}^{\  6 \  4 \  6}}}-{{q_{2}}\  �� \ {|_{\  2 \  7}^{\  6 \  4 \  7}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  0}^{\  6 \  5 \  0}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3 \  1}^{\  6 \  5 \  1}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3 \  2}^{\  6 \  5 \  2}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  3}^{\  6 \  5 \  3}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3 \  4}^{\  6 \  5 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3 \  5}^{\  6 \  5 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  6}^{\  6 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3 \  7}^{\  6 \  5 \  7}}}-{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  6 \  6 \  0}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  1}^{\  6 \  6 \  1}}}-{{q_{2}}\  �� \ {|_{\  0 \  2}^{\  6 \  6 \  2}}}-{{q_{2}}\  �� \ {|_{\  0 \  3}^{\  6 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  4}^{\  6 \  6 \  4}}}-{{q_{2}}\  �� \ {|_{\  0 \  5}^{\  6 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  0 \  6}^{\  6 \  6 \  6}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  7}^{\  6 \  6 \  7}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  0}^{\  6 \  7 \  0}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  1}^{\  6 \  7 \  1}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1 \  2}^{\  6 \  7 \  2}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  3}^{\  6 \  7 \  3}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  4}^{\  6 \  7 \  4}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1 \  5}^{\  6 \  7 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  6}^{\  6 \  7 \  6}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  7}^{\  6 \  7 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  0}^{\  7 \  0 \  0}}}+{�� \ {|_{\  7 \  1}^{\  7 \  0 \  1}}}+{�� \ {|_{\  7 \  2}^{\  7 \  0 \  2}}}+{�� \ {|_{\  7 \  3}^{\  7 \  0 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  4}^{\  7 \  0 \  4}}}+{�� \ {|_{\  7 \  5}^{\  7 \  0 \  5}}}+{�� \ {|_{\  7 \  6}^{\  7 \  0 \  6}}}+{�� \ {|_{\  7 \  7}^{\  7 \  0 \  7}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  0}^{\  7 \  1 \  0}}}-{�� \ {|_{\  6 \  1}^{\  7 \  1 \  1}}}-{�� \ {|_{\  6 \  2}^{\  7 \  1 \  2}}}-{�� \ {|_{\  6 \  3}^{\  7 \  1 \  3}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  4}^{\  7 \  1 \  4}}}-{�� \ {|_{\  6 \  5}^{\  7 \  1 \  5}}}-{�� \ {|_{\  6 \  6}^{\  7 \  1 \  6}}}-{�� \ {|_{\  6 \  7}^{\  7 \  1 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  0}^{\  7 \  2 \  0}}}+{�� \ {|_{\  5 \  1}^{\  7 \  2 \  1}}}+{�� \ {|_{\  5 \  2}^{\  7 \  2 \  2}}}+{�� \ {|_{\  5 \  3}^{\  7 \  2 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  4}^{\  7 \  2 \  4}}}+{�� \ {|_{\  5 \  5}^{\  7 \  2 \  5}}}+{�� \ {|_{\  5 \  6}^{\  7 \  2 \  6}}}+{�� \ {|_{\  5 \  7}^{\  7 \  2 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  0}^{\  7 \  3 \  0}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  1}^{\  7 \  3 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  2}^{\  7 \  3 \  2}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  3}^{\  7 \  3 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  7 \  3 \  4}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  5}^{\  7 \  3 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  6}^{\  7 \  3 \  6}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  7}^{\  7 \  3 \  7}}}-{{q_{2}}\  �� \ {|_{\  3 \  0}^{\  7 \  4 \  0}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  1}^{\  7 \  4 \  1}}}-{{q_{2}}\  �� \ {|_{\  3 \  2}^{\  7 \  4 \  2}}}-{{q_{2}}\  �� \ {|_{\  3 \  3}^{\  7 \  4 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  4}^{\  7 \  4 \  4}}}-{{q_{2}}\  �� \ {|_{\  3 \  5}^{\  7 \  4 \  5}}}-{{q_{2}}\  �� \ {|_{\  3 \  6}^{\  7 \  4 \  6}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  7}^{\  7 \  4 \  7}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  0}^{\  7 \  5 \  0}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  1}^{\  7 \  5 \  1}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2 \  2}^{\  7 \  5 \  2}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  3}^{\  7 \  5 \  3}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  4}^{\  7 \  5 \  4}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2 \  5}^{\  7 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  6}^{\  7 \  5 \  6}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  7}^{\  7 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1 \  0}^{\  7 \  6 \  0}}}+{�� \ {\overline{q_{2}}}\ {|_{\  1 \  1}^{\  7 \  6 \  1}}}+{�� \ {\overline{q_{2}}}\ {|_{\  1 \  2}^{\  7 \  6 \  2}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1 \  3}^{\  7 \  6 \  3}}}+{�� \ {\overline{q_{2}}}\ {|_{\  1 \  4}^{\  7 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  1 \  5}^{\  7 \  6 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1 \  6}^{\  7 \  6 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  1 \  7}^{\  7 \  6 \  7}}}-{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  7 \  7 \  0}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  1}^{\  7 \  7 \  1}}}-{{q_{2}}\  �� \ {|_{\  0 \  2}^{\  7 \  7 \  2}}}-{{q_{2}}\  �� \ {|_{\  0 \  3}^{\  7 \  7 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  4}^{\  7 \  7 \  4}}}-{{q_{2}}\  �� \ {|_{\  0 \  5}^{\  7 \  7 \  5}}}-{{q_{2}}\  �� \ {|_{\  0 \  6}^{\  7 \  7 \  6}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  7}^{\  7 \  7 \  7}}}
(28)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
λ:=(Ω,I,Ω)/(I,W,I)
4 + + - -- + 4 + arity warning: ---- 5 0 + - -- 0 4 +
>> System error: Heap exhausted (no more space for allocation). There are still 807796736 bytes available; the request was for 1073741840 bytes.
PROCEED WITH CAUTION.

fricas
test
     (    I Ω     )  /
     (     Y I    )  =  λ
2 + + - -- + 2 + arity warning: ---- 3 0 + - -- 0 2 +

\label{eq29} \mbox{\rm false} (29)
Type: Boolean
fricas
test
     (     Ω I    )  /
     (    I Y     )  =  λ
There are no library operations named Ω Use HyperDoc Browse or issue )what op Ω to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω with argument type(s) ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Frobenius Condition

Octonion algebra fails the Frobenius Condition!

fricas
H :=
       Y    /
       λ
2 + 0 -- - + 0 arity warning: ---- + 0 - - + 0

\label{eq30}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\  0}^{\  0 \  0}}}+{�� \ {|_{\  1}^{\  0 \  1}}}+{�� \ {|_{\  2}^{\  0 \  2}}}+{�� \ {|_{\  3}^{\  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  0 \  4}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5}^{\  0 \  5}}}+{�� \ {|_{\  6}^{\  0 \  6}}}+{�� \ {|_{\  7}^{\  0 \  7}}}+{�� \ {|_{\  1}^{\  1 \  0}}}-{�� \ {|_{\  0}^{\  1 \  1}}}+{�� \ {|_{\  3}^{\  1 \  2}}}- 
\
\
\displaystyle
{�� \ {|_{\  2}^{\  1 \  3}}}+{�� \ {|_{\  5}^{\  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  1 \  5}}}-{�� \ {|_{\  7}^{\  1 \  6}}}+{�� \ {|_{\  6}^{\  1 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2}^{\  2 \  0}}}-{�� \ {|_{\  3}^{\  2 \  1}}}-{�� \ {|_{\  0}^{\  2 \  2}}}+{�� \ {|_{\  1}^{\  2 \  3}}}+{�� \ {|_{\  6}^{\  2 \  4}}}+{�� \ {|_{\  7}^{\  2 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  2 \  6}}}-{�� \ {|_{\  5}^{\  2 \  7}}}+{�� \ {|_{\  3}^{\  3 \  0}}}+{�� \ {|_{\  2}^{\  3 \  1}}}-{�� \ {|_{\  1}^{\  3 \  2}}}- 
\
\
\displaystyle
{�� \ {|_{\  0}^{\  3 \  3}}}+{�� \ {|_{\  7}^{\  3 \  4}}}-{�� \ {|_{\  6}^{\  3 \  5}}}+{�� \ {|_{\  5}^{\  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  3 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  4 \  0}}}-{�� \ {|_{\  5}^{\  4 \  1}}}-{�� \ {|_{\  6}^{\  4 \  2}}}-{�� \ {|_{\  7}^{\  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  0}^{\  4 \  4}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1}^{\  4 \  5}}}+{{q_{2}}\  �� \ {|_{\  2}^{\  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  3}^{\  4 \  7}}}+{�� \ {|_{\  5}^{\  5 \  0}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  5 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  7}^{\  5 \  2}}}+{�� \ {|_{\  6}^{\  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  1}^{\  5 \  4}}}-{{q_{2}}\  �� \ {|_{\  0}^{\  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3}^{\  5 \  6}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2}^{\  5 \  7}}}+{�� \ {|_{\  6}^{\  6 \  0}}}+{�� \ {|_{\  7}^{\  6 \  1}}}+{{{{q_{2}}\  ��}\over{\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}}}- 
\
\
\displaystyle
{�� \ {|_{\  6}^{\  7 \  1}}}+{�� \ {|_{\  5}^{\  7 \  2}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  3}^{\  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2}^{\  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1}^{\  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  0}^{\  7 \  7}}}
(30)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
Hr := (λ,I)/(I,Y)
2 + + - -- + 2 + arity warning: ---- 3 0 + - -- 0 2 +

\label{eq31}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\  0 \  0}^{\  0 \  0 \  0}}}+{�� \ {|_{\  0 \  1}^{\  0 \  0 \  1}}}+{�� \ {|_{\  0 \  2}^{\  0 \  0 \  2}}}+{�� \ {|_{\  0 \  3}^{\  0 \  0 \  3}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  0 \  4}}}+{�� \ {|_{\  0 \  5}^{\  0 \  0 \  5}}}+{�� \ {|_{\  0 \  6}^{\  0 \  0 \  6}}}+{�� \ {|_{\  0 \  7}^{\  0 \  0 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  1}^{\  0 \  1 \  0}}}-{�� \ {|_{\  0 \  0}^{\  0 \  1 \  1}}}+{�� \ {|_{\  0 \  3}^{\  0 \  1 \  2}}}-{�� \ {|_{\  0 \  2}^{\  0 \  1 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  5}^{\  0 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  1 \  5}}}-{�� \ {|_{\  0 \  7}^{\  0 \  1 \  6}}}+{�� \ {|_{\  0 \  6}^{\  0 \  1 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  2}^{\  0 \  2 \  0}}}-{�� \ {|_{\  0 \  3}^{\  0 \  2 \  1}}}-{�� \ {|_{\  0 \  0}^{\  0 \  2 \  2}}}+{�� \ {|_{\  0 \  1}^{\  0 \  2 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  6}^{\  0 \  2 \  4}}}+{�� \ {|_{\  0 \  7}^{\  0 \  2 \  5}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  2 \  6}}}-{�� \ {|_{\  0 \  5}^{\  0 \  2 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  3}^{\  0 \  3 \  0}}}+{�� \ {|_{\  0 \  2}^{\  0 \  3 \  1}}}-{�� \ {|_{\  0 \  1}^{\  0 \  3 \  2}}}-{�� \ {|_{\  0 \  0}^{\  0 \  3 \  3}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  7}^{\  0 \  3 \  4}}}-{�� \ {|_{\  0 \  6}^{\  0 \  3 \  5}}}+{�� \ {|_{\  0 \  5}^{\  0 \  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  3 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  4 \  0}}}-{�� \ {|_{\  0 \  5}^{\  0 \  4 \  1}}}-{�� \ {|_{\  0 \  6}^{\  0 \  4 \  2}}}-{�� \ {|_{\  0 \  7}^{\  0 \  4 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  0 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  0 \  1}^{\  0 \  4 \  5}}}+{{q_{2}}\  �� \ {|_{\  0 \  2}^{\  0 \  4 \  6}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  3}^{\  0 \  4 \  7}}}+{�� \ {|_{\  0 \  5}^{\  0 \  5 \  0}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  5 \  1}}}-{�� \ {|_{\  0 \  7}^{\  0 \  5 \  2}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  6}^{\  0 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  0 \  1}^{\  0 \  5 \  4}}}-{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  0 \  5 \  5}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  0 \  3}^{\  0 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  0 \  2}^{\  0 \  5 \  7}}}+{�� \ {|_{\  0 \  6}^{\  0 \  6 \  0}}}+{�� \ {|_{\  0 \  7}^{\  0 \  6 \  1}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  6 \  2}}}-{�� \ {|_{\  0 \  5}^{\  0 \  6 \  3}}}-{{q_{2}}\  �� \ {|_{\  0 \  2}^{\  0 \  6 \  4}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  0 \  3}^{\  0 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  0 \  6 \  6}}}-{�� \ {\overline{q_{2}}}\ {|_{\  0 \  1}^{\  0 \  6 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  7}^{\  0 \  7 \  0}}}-{�� \ {|_{\  0 \  6}^{\  0 \  7 \  1}}}+{�� \ {|_{\  0 \  5}^{\  0 \  7 \  2}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  0 \  4}^{\  0 \  7 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  3}^{\  0 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  0 \  2}^{\  0 \  7 \  5}}}+{�� \ {\overline{q_{2}}}\ {|_{\  0 \  1}^{\  0 \  7 \  6}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  0 \  0}^{\  0 \  7 \  7}}}+{�� \ {|_{\  1 \  0}^{\  1 \  0 \  0}}}+{�� \ {|_{\  1 \  1}^{\  1 \  0 \  1}}}+{�� \ {|_{\  1 \  2}^{\  1 \  0 \  2}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  3}^{\  1 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  0 \  4}}}+{�� \ {|_{\  1 \  5}^{\  1 \  0 \  5}}}+{�� \ {|_{\  1 \  6}^{\  1 \  0 \  6}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  7}^{\  1 \  0 \  7}}}+{�� \ {|_{\  1 \  1}^{\  1 \  1 \  0}}}-{�� \ {|_{\  1 \  0}^{\  1 \  1 \  1}}}+{�� \ {|_{\  1 \  3}^{\  1 \  1 \  2}}}- 
\
\
\displaystyle
{�� \ {|_{\  1 \  2}^{\  1 \  1 \  3}}}+{�� \ {|_{\  1 \  5}^{\  1 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  1 \  5}}}-{�� \ {|_{\  1 \  7}^{\  1 \  1 \  6}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  6}^{\  1 \  1 \  7}}}+{�� \ {|_{\  1 \  2}^{\  1 \  2 \  0}}}-{�� \ {|_{\  1 \  3}^{\  1 \  2 \  1}}}-{�� \ {|_{\  1 \  0}^{\  1 \  2 \  2}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  1}^{\  1 \  2 \  3}}}+{�� \ {|_{\  1 \  6}^{\  1 \  2 \  4}}}+{�� \ {|_{\  1 \  7}^{\  1 \  2 \  5}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  2 \  6}}}- 
\
\
\displaystyle
{�� \ {|_{\  1 \  5}^{\  1 \  2 \  7}}}+{�� \ {|_{\  1 \  3}^{\  1 \  3 \  0}}}+{�� \ {|_{\  1 \  2}^{\  1 \  3 \  1}}}-{�� \ {|_{\  1 \  1}^{\  1 \  3 \  2}}}- 
\
\
\displaystyle
{�� \ {|_{\  1 \  0}^{\  1 \  3 \  3}}}+{�� \ {|_{\  1 \  7}^{\  1 \  3 \  4}}}-{�� \ {|_{\  1 \  6}^{\  1 \  3 \  5}}}+{�� \ {|_{\  1 \  5}^{\  1 \  3 \  6}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  3 \  7}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  4 \  0}}}-{�� \ {|_{\  1 \  5}^{\  1 \  4 \  1}}}-{�� \ {|_{\  1 \  6}^{\  1 \  4 \  2}}}- 
\
\
\displaystyle
{�� \ {|_{\  1 \  7}^{\  1 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  1 \  0}^{\  1 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  1 \  1}^{\  1 \  4 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  2}^{\  1 \  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  1 \  3}^{\  1 \  4 \  7}}}+{�� \ {|_{\  1 \  5}^{\  1 \  5 \  0}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  5 \  1}}}-{�� \ {|_{\  1 \  7}^{\  1 \  5 \  2}}}+{�� \ {|_{\  1 \  6}^{\  1 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  1 \  1}^{\  1 \  5 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  0}^{\  1 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  3}^{\  1 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  1 \  2}^{\  1 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  1 \  6}^{\  1 \  6 \  0}}}+{�� \ {|_{\  1 \  7}^{\  1 \  6 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  6 \  2}}}-{�� \ {|_{\  1 \  5}^{\  1 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1 \  2}^{\  1 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  1 \  3}^{\  1 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  1 \  0}^{\  1 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1 \  1}^{\  1 \  6 \  7}}}+{�� \ {|_{\  1 \  7}^{\  1 \  7 \  0}}}-{�� \ {|_{\  1 \  6}^{\  1 \  7 \  1}}}+{�� \ {|_{\  1 \  5}^{\  1 \  7 \  2}}}+ \
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  1 \  4}^{\  1 \  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  1 \  3}^{\  1 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  1 \  2}^{\  1 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1 \  1}^{\  1 \  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  1 \  0}^{\  1 \  7 \  7}}}+{�� \ {|_{\  2 \  0}^{\  2 \  0 \  0}}}+{�� \ {|_{\  2 \  1}^{\  2 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  2}^{\  2 \  0 \  2}}}+{�� \ {|_{\  2 \  3}^{\  2 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  0 \  4}}}+{�� \ {|_{\  2 \  5}^{\  2 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  6}^{\  2 \  0 \  6}}}+{�� \ {|_{\  2 \  7}^{\  2 \  0 \  7}}}+{�� \ {|_{\  2 \  1}^{\  2 \  1 \  0}}}-{�� \ {|_{\  2 \  0}^{\  2 \  1 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  3}^{\  2 \  1 \  2}}}-{�� \ {|_{\  2 \  2}^{\  2 \  1 \  3}}}+{�� \ {|_{\  2 \  5}^{\  2 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  2 \  7}^{\  2 \  1 \  6}}}+{�� \ {|_{\  2 \  6}^{\  2 \  1 \  7}}}+{�� \ {|_{\  2 \  2}^{\  2 \  2 \  0}}}-{�� \ {|_{\  2 \  3}^{\  2 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  2 \  0}^{\  2 \  2 \  2}}}+{�� \ {|_{\  2 \  1}^{\  2 \  2 \  3}}}+{�� \ {|_{\  2 \  6}^{\  2 \  2 \  4}}}+{�� \ {|_{\  2 \  7}^{\  2 \  2 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  2 \  6}}}-{�� \ {|_{\  2 \  5}^{\  2 \  2 \  7}}}+{�� \ {|_{\  2 \  3}^{\  2 \  3 \  0}}}+{�� \ {|_{\  2 \  2}^{\  2 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  2 \  1}^{\  2 \  3 \  2}}}-{�� \ {|_{\  2 \  0}^{\  2 \  3 \  3}}}+{�� \ {|_{\  2 \  7}^{\  2 \  3 \  4}}}-{�� \ {|_{\  2 \  6}^{\  2 \  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  5}^{\  2 \  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  3 \  7}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  4 \  0}}}-{�� \ {|_{\  2 \  5}^{\  2 \  4 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  2 \  6}^{\  2 \  4 \  2}}}-{�� \ {|_{\  2 \  7}^{\  2 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  2 \  0}^{\  2 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  2 \  1}^{\  2 \  4 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  2}^{\  2 \  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  2 \  3}^{\  2 \  4 \  7}}}+{�� \ {|_{\  2 \  5}^{\  2 \  5 \  0}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  5 \  1}}}-{�� \ {|_{\  2 \  7}^{\  2 \  5 \  2}}}+{�� \ {|_{\  2 \  6}^{\  2 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  2 \  1}^{\  2 \  5 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  0}^{\  2 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  3}^{\  2 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  2}^{\  2 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2 \  6}^{\  2 \  6 \  0}}}+{�� \ {|_{\  2 \  7}^{\  2 \  6 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  6 \  2}}}-{�� \ {|_{\  2 \  5}^{\  2 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  2 \  2}^{\  2 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2 \  3}^{\  2 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  2 \  0}^{\  2 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2 \  1}^{\  2 \  6 \  7}}}+{�� \ {|_{\  2 \  7}^{\  2 \  7 \  0}}}-{�� \ {|_{\  2 \  6}^{\  2 \  7 \  1}}}+{�� \ {|_{\  2 \  5}^{\  2 \  7 \  2}}}+ \
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  2 \  4}^{\  2 \  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  2 \  3}^{\  2 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  2 \  2}^{\  2 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2 \  1}^{\  2 \  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  2 \  0}^{\  2 \  7 \  7}}}+{�� \ {|_{\  3 \  0}^{\  3 \  0 \  0}}}+{�� \ {|_{\  3 \  1}^{\  3 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  2}^{\  3 \  0 \  2}}}+{�� \ {|_{\  3 \  3}^{\  3 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  0 \  4}}}+{�� \ {|_{\  3 \  5}^{\  3 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  6}^{\  3 \  0 \  6}}}+{�� \ {|_{\  3 \  7}^{\  3 \  0 \  7}}}+{�� \ {|_{\  3 \  1}^{\  3 \  1 \  0}}}-{�� \ {|_{\  3 \  0}^{\  3 \  1 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  3}^{\  3 \  1 \  2}}}-{�� \ {|_{\  3 \  2}^{\  3 \  1 \  3}}}+{�� \ {|_{\  3 \  5}^{\  3 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  3 \  7}^{\  3 \  1 \  6}}}+{�� \ {|_{\  3 \  6}^{\  3 \  1 \  7}}}+{�� \ {|_{\  3 \  2}^{\  3 \  2 \  0}}}-{�� \ {|_{\  3 \  3}^{\  3 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  3 \  0}^{\  3 \  2 \  2}}}+{�� \ {|_{\  3 \  1}^{\  3 \  2 \  3}}}+{�� \ {|_{\  3 \  6}^{\  3 \  2 \  4}}}+{�� \ {|_{\  3 \  7}^{\  3 \  2 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  2 \  6}}}-{�� \ {|_{\  3 \  5}^{\  3 \  2 \  7}}}+{�� \ {|_{\  3 \  3}^{\  3 \  3 \  0}}}+{�� \ {|_{\  3 \  2}^{\  3 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  3 \  1}^{\  3 \  3 \  2}}}-{�� \ {|_{\  3 \  0}^{\  3 \  3 \  3}}}+{�� \ {|_{\  3 \  7}^{\  3 \  3 \  4}}}-{�� \ {|_{\  3 \  6}^{\  3 \  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  5}^{\  3 \  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  3 \  7}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  4 \  0}}}-{�� \ {|_{\  3 \  5}^{\  3 \  4 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  3 \  6}^{\  3 \  4 \  2}}}-{�� \ {|_{\  3 \  7}^{\  3 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  3 \  0}^{\  3 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  3 \  1}^{\  3 \  4 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  2}^{\  3 \  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  3 \  3}^{\  3 \  4 \  7}}}+{�� \ {|_{\  3 \  5}^{\  3 \  5 \  0}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  5 \  1}}}-{�� \ {|_{\  3 \  7}^{\  3 \  5 \  2}}}+{�� \ {|_{\  3 \  6}^{\  3 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  3 \  1}^{\  3 \  5 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  0}^{\  3 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3 \  3}^{\  3 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3 \  2}^{\  3 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3 \  6}^{\  3 \  6 \  0}}}+{�� \ {|_{\  3 \  7}^{\  3 \  6 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  6 \  2}}}-{�� \ {|_{\  3 \  5}^{\  3 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  3 \  2}^{\  3 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3 \  3}^{\  3 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  3 \  0}^{\  3 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  1}^{\  3 \  6 \  7}}}+{�� \ {|_{\  3 \  7}^{\  3 \  7 \  0}}}-{�� \ {|_{\  3 \  6}^{\  3 \  7 \  1}}}+{�� \ {|_{\  3 \  5}^{\  3 \  7 \  2}}}+ \
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  3 \  4}^{\  3 \  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  3 \  3}^{\  3 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  3 \  2}^{\  3 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  3 \  1}^{\  3 \  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  3 \  0}^{\  3 \  7 \  7}}}+{�� \ {|_{\  4 \  0}^{\  4 \  0 \  0}}}+{�� \ {|_{\  4 \  1}^{\  4 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  4 \  2}^{\  4 \  0 \  2}}}+{�� \ {|_{\  4 \  3}^{\  4 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  0 \  4}}}+{�� \ {|_{\  4 \  5}^{\  4 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  4 \  6}^{\  4 \  0 \  6}}}+{�� \ {|_{\  4 \  7}^{\  4 \  0 \  7}}}+{�� \ {|_{\  4 \  1}^{\  4 \  1 \  0}}}-{�� \ {|_{\  4 \  0}^{\  4 \  1 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  4 \  3}^{\  4 \  1 \  2}}}-{�� \ {|_{\  4 \  2}^{\  4 \  1 \  3}}}+{�� \ {|_{\  4 \  5}^{\  4 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  4 \  7}^{\  4 \  1 \  6}}}+{�� \ {|_{\  4 \  6}^{\  4 \  1 \  7}}}+{�� \ {|_{\  4 \  2}^{\  4 \  2 \  0}}}-{�� \ {|_{\  4 \  3}^{\  4 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  4 \  0}^{\  4 \  2 \  2}}}+{�� \ {|_{\  4 \  1}^{\  4 \  2 \  3}}}+{�� \ {|_{\  4 \  6}^{\  4 \  2 \  4}}}+{�� \ {|_{\  4 \  7}^{\  4 \  2 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  2 \  6}}}-{�� \ {|_{\  4 \  5}^{\  4 \  2 \  7}}}+{�� \ {|_{\  4 \  3}^{\  4 \  3 \  0}}}+{�� \ {|_{\  4 \  2}^{\  4 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  4 \  1}^{\  4 \  3 \  2}}}-{�� \ {|_{\  4 \  0}^{\  4 \  3 \  3}}}+{�� \ {|_{\  4 \  7}^{\  4 \  3 \  4}}}-{�� \ {|_{\  4 \  6}^{\  4 \  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  4 \  5}^{\  4 \  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  3 \  7}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  4 \  0}}}-{�� \ {|_{\  4 \  5}^{\  4 \  4 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  4 \  6}^{\  4 \  4 \  2}}}-{�� \ {|_{\  4 \  7}^{\  4 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  4 \  0}^{\  4 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  4 \  1}^{\  4 \  4 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  4 \  2}^{\  4 \  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  4 \  3}^{\  4 \  4 \  7}}}+{�� \ {|_{\  4 \  5}^{\  4 \  5 \  0}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  5 \  1}}}-{�� \ {|_{\  4 \  7}^{\  4 \  5 \  2}}}+{�� \ {|_{\  4 \  6}^{\  4 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  4 \  1}^{\  4 \  5 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  4 \  0}^{\  4 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  4 \  3}^{\  4 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  4 \  2}^{\  4 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  4 \  6}^{\  4 \  6 \  0}}}+{�� \ {|_{\  4 \  7}^{\  4 \  6 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  6 \  2}}}-{�� \ {|_{\  4 \  5}^{\  4 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  4 \  2}^{\  4 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  4 \  3}^{\  4 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  4 \  0}^{\  4 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  4 \  1}^{\  4 \  6 \  7}}}+{�� \ {|_{\  4 \  7}^{\  4 \  7 \  0}}}-{�� \ {|_{\  4 \  6}^{\  4 \  7 \  1}}}+{�� \ {|_{\  4 \  5}^{\  4 \  7 \  2}}}+ \
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4 \  4}^{\  4 \  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  4 \  3}^{\  4 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  4 \  2}^{\  4 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  4 \  1}^{\  4 \  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  4 \  0}^{\  4 \  7 \  7}}}+{�� \ {|_{\  5 \  0}^{\  5 \  0 \  0}}}+{�� \ {|_{\  5 \  1}^{\  5 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  2}^{\  5 \  0 \  2}}}+{�� \ {|_{\  5 \  3}^{\  5 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  0 \  4}}}+{�� \ {|_{\  5 \  5}^{\  5 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  5 \  0 \  6}}}+{�� \ {|_{\  5 \  7}^{\  5 \  0 \  7}}}+{�� \ {|_{\  5 \  1}^{\  5 \  1 \  0}}}-{�� \ {|_{\  5 \  0}^{\  5 \  1 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  3}^{\  5 \  1 \  2}}}-{�� \ {|_{\  5 \  2}^{\  5 \  1 \  3}}}+{�� \ {|_{\  5 \  5}^{\  5 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  7}^{\  5 \  1 \  6}}}+{�� \ {|_{\  5 \  6}^{\  5 \  1 \  7}}}+{�� \ {|_{\  5 \  2}^{\  5 \  2 \  0}}}-{�� \ {|_{\  5 \  3}^{\  5 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  0}^{\  5 \  2 \  2}}}+{�� \ {|_{\  5 \  1}^{\  5 \  2 \  3}}}+{�� \ {|_{\  5 \  6}^{\  5 \  2 \  4}}}+{�� \ {|_{\  5 \  7}^{\  5 \  2 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  2 \  6}}}-{�� \ {|_{\  5 \  5}^{\  5 \  2 \  7}}}+{�� \ {|_{\  5 \  3}^{\  5 \  3 \  0}}}+{�� \ {|_{\  5 \  2}^{\  5 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  1}^{\  5 \  3 \  2}}}-{�� \ {|_{\  5 \  0}^{\  5 \  3 \  3}}}+{�� \ {|_{\  5 \  7}^{\  5 \  3 \  4}}}-{�� \ {|_{\  5 \  6}^{\  5 \  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  5}^{\  5 \  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  3 \  7}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  4 \  0}}}-{�� \ {|_{\  5 \  5}^{\  5 \  4 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  5 \  4 \  2}}}-{�� \ {|_{\  5 \  7}^{\  5 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  5 \  0}^{\  5 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  5 \  1}^{\  5 \  4 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  5 \  2}^{\  5 \  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  5 \  3}^{\  5 \  4 \  7}}}+{�� \ {|_{\  5 \  5}^{\  5 \  5 \  0}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  5 \  1}}}-{�� \ {|_{\  5 \  7}^{\  5 \  5 \  2}}}+{�� \ {|_{\  5 \  6}^{\  5 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  5 \  1}^{\  5 \  5 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  5 \  0}^{\  5 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  5 \  3}^{\  5 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  5 \  2}^{\  5 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5 \  6}^{\  5 \  6 \  0}}}+{�� \ {|_{\  5 \  7}^{\  5 \  6 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  6 \  2}}}-{�� \ {|_{\  5 \  5}^{\  5 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  5 \  2}^{\  5 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  5 \  3}^{\  5 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  5 \  0}^{\  5 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  5 \  1}^{\  5 \  6 \  7}}}+{�� \ {|_{\  5 \  7}^{\  5 \  7 \  0}}}-{�� \ {|_{\  5 \  6}^{\  5 \  7 \  1}}}+{�� \ {|_{\  5 \  5}^{\  5 \  7 \  2}}}+ \
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  5 \  4}^{\  5 \  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  5 \  3}^{\  5 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  5 \  2}^{\  5 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  5 \  1}^{\  5 \  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  5 \  0}^{\  5 \  7 \  7}}}+{�� \ {|_{\  6 \  0}^{\  6 \  0 \  0}}}+{�� \ {|_{\  6 \  1}^{\  6 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  2}^{\  6 \  0 \  2}}}+{�� \ {|_{\  6 \  3}^{\  6 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  0 \  4}}}+{�� \ {|_{\  6 \  5}^{\  6 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  6}^{\  6 \  0 \  6}}}+{�� \ {|_{\  6 \  7}^{\  6 \  0 \  7}}}+{�� \ {|_{\  6 \  1}^{\  6 \  1 \  0}}}-{�� \ {|_{\  6 \  0}^{\  6 \  1 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  3}^{\  6 \  1 \  2}}}-{�� \ {|_{\  6 \  2}^{\  6 \  1 \  3}}}+{�� \ {|_{\  6 \  5}^{\  6 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  7}^{\  6 \  1 \  6}}}+{�� \ {|_{\  6 \  6}^{\  6 \  1 \  7}}}+{�� \ {|_{\  6 \  2}^{\  6 \  2 \  0}}}-{�� \ {|_{\  6 \  3}^{\  6 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  0}^{\  6 \  2 \  2}}}+{�� \ {|_{\  6 \  1}^{\  6 \  2 \  3}}}+{�� \ {|_{\  6 \  6}^{\  6 \  2 \  4}}}+{�� \ {|_{\  6 \  7}^{\  6 \  2 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  2 \  6}}}-{�� \ {|_{\  6 \  5}^{\  6 \  2 \  7}}}+{�� \ {|_{\  6 \  3}^{\  6 \  3 \  0}}}+{�� \ {|_{\  6 \  2}^{\  6 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  1}^{\  6 \  3 \  2}}}-{�� \ {|_{\  6 \  0}^{\  6 \  3 \  3}}}+{�� \ {|_{\  6 \  7}^{\  6 \  3 \  4}}}-{�� \ {|_{\  6 \  6}^{\  6 \  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  5}^{\  6 \  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  3 \  7}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  4 \  0}}}-{�� \ {|_{\  6 \  5}^{\  6 \  4 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  6 \  6}^{\  6 \  4 \  2}}}-{�� \ {|_{\  6 \  7}^{\  6 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  6 \  0}^{\  6 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  6 \  1}^{\  6 \  4 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  6 \  2}^{\  6 \  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  6 \  3}^{\  6 \  4 \  7}}}+{�� \ {|_{\  6 \  5}^{\  6 \  5 \  0}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  5 \  1}}}-{�� \ {|_{\  6 \  7}^{\  6 \  5 \  2}}}+{�� \ {|_{\  6 \  6}^{\  6 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  6 \  1}^{\  6 \  5 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  6 \  0}^{\  6 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  6 \  3}^{\  6 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  6 \  2}^{\  6 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6 \  6}^{\  6 \  6 \  0}}}+{�� \ {|_{\  6 \  7}^{\  6 \  6 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  6 \  2}}}-{�� \ {|_{\  6 \  5}^{\  6 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  6 \  2}^{\  6 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  6 \  3}^{\  6 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  6 \  0}^{\  6 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  6 \  1}^{\  6 \  6 \  7}}}+{�� \ {|_{\  6 \  7}^{\  6 \  7 \  0}}}-{�� \ {|_{\  6 \  6}^{\  6 \  7 \  1}}}+{�� \ {|_{\  6 \  5}^{\  6 \  7 \  2}}}+ \
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  6 \  4}^{\  6 \  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  6 \  3}^{\  6 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  6 \  2}^{\  6 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  6 \  1}^{\  6 \  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  6 \  0}^{\  6 \  7 \  7}}}+{�� \ {|_{\  7 \  0}^{\  7 \  0 \  0}}}+{�� \ {|_{\  7 \  1}^{\  7 \  0 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  2}^{\  7 \  0 \  2}}}+{�� \ {|_{\  7 \  3}^{\  7 \  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  0 \  4}}}+{�� \ {|_{\  7 \  5}^{\  7 \  0 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  6}^{\  7 \  0 \  6}}}+{�� \ {|_{\  7 \  7}^{\  7 \  0 \  7}}}+{�� \ {|_{\  7 \  1}^{\  7 \  1 \  0}}}-{�� \ {|_{\  7 \  0}^{\  7 \  1 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  3}^{\  7 \  1 \  2}}}-{�� \ {|_{\  7 \  2}^{\  7 \  1 \  3}}}+{�� \ {|_{\  7 \  5}^{\  7 \  1 \  4}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  1 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  7}^{\  7 \  1 \  6}}}+{�� \ {|_{\  7 \  6}^{\  7 \  1 \  7}}}+{�� \ {|_{\  7 \  2}^{\  7 \  2 \  0}}}-{�� \ {|_{\  7 \  3}^{\  7 \  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  0}^{\  7 \  2 \  2}}}+{�� \ {|_{\  7 \  1}^{\  7 \  2 \  3}}}+{�� \ {|_{\  7 \  6}^{\  7 \  2 \  4}}}+{�� \ {|_{\  7 \  7}^{\  7 \  2 \  5}}}- 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  2 \  6}}}-{�� \ {|_{\  7 \  5}^{\  7 \  2 \  7}}}+{�� \ {|_{\  7 \  3}^{\  7 \  3 \  0}}}+{�� \ {|_{\  7 \  2}^{\  7 \  3 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  1}^{\  7 \  3 \  2}}}-{�� \ {|_{\  7 \  0}^{\  7 \  3 \  3}}}+{�� \ {|_{\  7 \  7}^{\  7 \  3 \  4}}}-{�� \ {|_{\  7 \  6}^{\  7 \  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  5}^{\  7 \  3 \  6}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  3 \  7}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  4 \  0}}}-{�� \ {|_{\  7 \  5}^{\  7 \  4 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  7 \  6}^{\  7 \  4 \  2}}}-{�� \ {|_{\  7 \  7}^{\  7 \  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  7 \  0}^{\  7 \  4 \  4}}}+{{q_{2}}\  �� \ {|_{\  7 \  1}^{\  7 \  4 \  5}}}+ 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  7 \  2}^{\  7 \  4 \  6}}}+{{q_{2}}\  �� \ {|_{\  7 \  3}^{\  7 \  4 \  7}}}+{�� \ {|_{\  7 \  5}^{\  7 \  5 \  0}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  5 \  1}}}-{�� \ {|_{\  7 \  7}^{\  7 \  5 \  2}}}+{�� \ {|_{\  7 \  6}^{\  7 \  5 \  3}}}-{{q_{2}}\  �� \ {|_{\  7 \  1}^{\  7 \  5 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  7 \  0}^{\  7 \  5 \  5}}}-{�� \ {\overline{q_{2}}}\ {|_{\  7 \  3}^{\  7 \  5 \  6}}}+{�� \ {\overline{q_{2}}}\ {|_{\  7 \  2}^{\  7 \  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7 \  6}^{\  7 \  6 \  0}}}+{�� \ {|_{\  7 \  7}^{\  7 \  6 \  1}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  6 \  2}}}-{�� \ {|_{\  7 \  5}^{\  7 \  6 \  3}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  7 \  2}^{\  7 \  6 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  7 \  3}^{\  7 \  6 \  5}}}-{{q_{2}}\  �� \ {|_{\  7 \  0}^{\  7 \  6 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  7 \  1}^{\  7 \  6 \  7}}}+{�� \ {|_{\  7 \  7}^{\  7 \  7 \  0}}}-{�� \ {|_{\  7 \  6}^{\  7 \  7 \  1}}}+{�� \ {|_{\  7 \  5}^{\  7 \  7 \  2}}}+ \
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  7 \  4}^{\  7 \  7 \  3}}}-{{q_{2}}\  �� \ {|_{\  7 \  3}^{\  7 \  7 \  4}}}-{�� \ {\overline{q_{2}}}\ {|_{\  7 \  2}^{\  7 \  7 \  5}}}+ 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  7 \  1}^{\  7 \  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  7 \  0}^{\  7 \  7 \  7}}}
(31)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
test(Hr = H )

\label{eq32} \mbox{\rm false} (32)
Type: Boolean
fricas
Hl := (I,λ)/(Y,I);
2 + + - -- + 2 + arity warning: ---- 3 0 + - -- 0 2 +
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
test( Hl = H )

\label{eq33} \mbox{\rm false} (33)
Type: Boolean
fricas
test( Hr = Hl )

\label{eq34} \mbox{\rm false} (34)
Type: Boolean

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{eq35}|_{\  0}(35)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
test
         i     /
         λ     =    Ω
0 0 - - + 0 arity warning: --- + 0 - - + 0

\label{eq36} \mbox{\rm false} (36)
Type: Boolean

Handle and handle element

fricas
Φ:ℒ :=
         λ     /
         X     /
         Y
2 0 + - -- 0 2 + arity warning: ---- 2 0 + - -- 0 2 +

\label{eq37}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\  0}^{\  0 \  0}}}+{�� \ {|_{\  1}^{\  0 \  1}}}+{�� \ {|_{\  2}^{\  0 \  2}}}+{�� \ {|_{\  3}^{\  0 \  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  0 \  4}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5}^{\  0 \  5}}}+{�� \ {|_{\  6}^{\  0 \  6}}}+{�� \ {|_{\  7}^{\  0 \  7}}}+{�� \ {|_{\  1}^{\  1 \  0}}}-{�� \ {|_{\  0}^{\  1 \  1}}}-{�� \ {|_{\  3}^{\  1 \  2}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2}^{\  1 \  3}}}-{�� \ {|_{\  5}^{\  1 \  4}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  1 \  5}}}+{�� \ {|_{\  7}^{\  1 \  6}}}-{�� \ {|_{\  6}^{\  1 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  2}^{\  2 \  0}}}+{�� \ {|_{\  3}^{\  2 \  1}}}-{�� \ {|_{\  0}^{\  2 \  2}}}-{�� \ {|_{\  1}^{\  2 \  3}}}-{�� \ {|_{\  6}^{\  2 \  4}}}-{�� \ {|_{\  7}^{\  2 \  5}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  2 \  6}}}+{�� \ {|_{\  5}^{\  2 \  7}}}+{�� \ {|_{\  3}^{\  3 \  0}}}-{�� \ {|_{\  2}^{\  3 \  1}}}+{�� \ {|_{\  1}^{\  3 \  2}}}- 
\
\
\displaystyle
{�� \ {|_{\  0}^{\  3 \  3}}}-{�� \ {|_{\  7}^{\  3 \  4}}}+{�� \ {|_{\  6}^{\  3 \  5}}}-{�� \ {|_{\  5}^{\  3 \  6}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  3 \  7}}}+ 
\
\
\displaystyle
{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  4 \  0}}}+{�� \ {|_{\  5}^{\  4 \  1}}}+{�� \ {|_{\  6}^{\  4 \  2}}}+{�� \ {|_{\  7}^{\  4 \  3}}}-{{q_{2}}\  �� \ {|_{\  0}^{\  4 \  4}}}- 
\
\
\displaystyle
{{q_{2}}\  �� \ {|_{\  1}^{\  4 \  5}}}-{{q_{2}}\  �� \ {|_{\  2}^{\  4 \  6}}}-{{q_{2}}\  �� \ {|_{\  3}^{\  4 \  7}}}+{�� \ {|_{\  5}^{\  5 \  0}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  5 \  1}}}+ 
\
\
\displaystyle
{�� \ {|_{\  7}^{\  5 \  2}}}-{�� \ {|_{\  6}^{\  5 \  3}}}+{{q_{2}}\  �� \ {|_{\  1}^{\  5 \  4}}}-{{q_{2}}\  �� \ {|_{\  0}^{\  5 \  5}}}+{�� \ {\overline{q_{2}}}\ {|_{\  3}^{\  5 \  6}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  2}^{\  5 \  7}}}+{�� \ {|_{\  6}^{\  6 \  0}}}-{�� \ {|_{\  7}^{\  6 \  1}}}-{{{{q_{2}}\  ��}\over{\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}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6}^{\  7 \  1}}}-{�� \ {|_{\  5}^{\  7 \  2}}}-{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  7 \  3}}}+{{q_{2}}\  �� \ {|_{\  3}^{\  7 \  4}}}+{�� \ {\overline{q_{2}}}\ {|_{\  2}^{\  7 \  5}}}- 
\
\
\displaystyle
{�� \ {\overline{q_{2}}}\ {|_{\  1}^{\  7 \  6}}}-{{q_{2}}\  �� \ {|_{\  0}^{\  7 \  7}}}
(37)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
Φ1:=     i     /
         Φ
0 + - - + + arity warning: ---- 2 0 + - -- 0 +

\label{eq38}{�� \ {|_{\  0}^{\  0}}}+{�� \ {|_{\  1}^{\  1}}}+{�� \ {|_{\  2}^{\  2}}}+{�� \ {|_{\  3}^{\  3}}}+{{{{q_{2}}\  ��}\over{\overline{q_{2}}}}\ {|_{\  4}^{\  4}}}+{�� \ {|_{\  5}^{\  5}}}+{�� \ {|_{\  6}^{\  6}}}+{�� \ {|_{\  7}^{\  7}}}(38)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

Definition 5

Co-unit
  i 
  U
  

fricas
ι:ℒ:=
    (    i I    ) /
    (     Ų     )
+ 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +

\label{eq39}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\  0 \  0}^{\  0}}}+{�� \ {|_{\  0 \  1}^{\  1}}}+{�� \ {|_{\  0 \  2}^{\  2}}}+{�� \ {|_{\  0 \  3}^{\  3}}}+{�� \ {|_{\  0 \  4}^{\  4}}}+{�� \ {|_{\  0 \  5}^{\  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  0 \  6}^{\  6}}}+{�� \ {|_{\  0 \  7}^{\  7}}}
(39)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))

Y=U
ι  
fricas
test
        Y    /
        ι       = Ų

\label{eq40} \mbox{\rm false} (40)
Type: Boolean

For example:

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

\label{eq41}\left[{{q_{2}}= 1}, \:{{p_{1}}= 1}\right](41)
Type: List(Equation(Polynomial(Integer)))
fricas
Ų0:ℒ  :=eval(Ų,ex1)

\label{eq42}��(42)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
Ω0:ℒ  :=eval(Ω,ex1)$ℒ

\label{eq43}��(43)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
λ0:ℒ  :=eval(λ,ex1)$ℒ

\label{eq44}��(44)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
H0:ℒ :=eval(H,ex1)$ℒ

\label{eq45}\begin{array}{@{}l}
\displaystyle
{�� \ {|_{\  0}^{\  0 \  0}}}+{�� \ {|_{\  1}^{\  0 \  1}}}+{�� \ {|_{\  2}^{\  0 \  2}}}+{�� \ {|_{\  3}^{\  0 \  3}}}+{�� \ {|_{\  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}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5}^{\  1 \  4}}}-{�� \ {|_{\  4}^{\  1 \  5}}}-{�� \ {|_{\  7}^{\  1 \  6}}}+{�� \ {|_{\  6}^{\  1 \  7}}}+{�� \ {|_{\  2}^{\  2 \  0}}}-{�� \ {|_{\  3}^{\  2 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  0}^{\  2 \  2}}}+{�� \ {|_{\  1}^{\  2 \  3}}}+{�� \ {|_{\  6}^{\  2 \  4}}}+{�� \ {|_{\  7}^{\  2 \  5}}}-{�� \ {|_{\  4}^{\  2 \  6}}}-{�� \ {|_{\  5}^{\  2 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  3}^{\  3 \  0}}}+{�� \ {|_{\  2}^{\  3 \  1}}}-{�� \ {|_{\  1}^{\  3 \  2}}}-{�� \ {|_{\  0}^{\  3 \  3}}}+{�� \ {|_{\  7}^{\  3 \  4}}}-{�� \ {|_{\  6}^{\  3 \  5}}}+ 
\
\
\displaystyle
{�� \ {|_{\  5}^{\  3 \  6}}}-{�� \ {|_{\  4}^{\  3 \  7}}}+{�� \ {|_{\  4}^{\  4 \  0}}}-{�� \ {|_{\  5}^{\  4 \  1}}}-{�� \ {|_{\  6}^{\  4 \  2}}}-{�� \ {|_{\  7}^{\  4 \  3}}}- 
\
\
\displaystyle
{�� \ {|_{\  0}^{\  4 \  4}}}+{�� \ {|_{\  1}^{\  4 \  5}}}+{�� \ {|_{\  2}^{\  4 \  6}}}+{�� \ {|_{\  3}^{\  4 \  7}}}+{�� \ {|_{\  5}^{\  5 \  0}}}+{�� \ {|_{\  4}^{\  5 \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  7}^{\  5 \  2}}}+{�� \ {|_{\  6}^{\  5 \  3}}}-{�� \ {|_{\  1}^{\  5 \  4}}}-{�� \ {|_{\  0}^{\  5 \  5}}}-{�� \ {|_{\  3}^{\  5 \  6}}}+{�� \ {|_{\  2}^{\  5 \  7}}}+ 
\
\
\displaystyle
{�� \ {|_{\  6}^{\  6 \  0}}}+{�� \ {|_{\  7}^{\  6 \  1}}}+{�� \ {|_{\  4}^{\  6 \  2}}}-{�� \ {|_{\  5}^{\  6 \  3}}}-{�� \ {|_{\  2}^{\  6 \  4}}}+{�� \ {|_{\  3}^{\  6 \  5}}}- 
\
\
\displaystyle
{�� \ {|_{\  0}^{\  6 \  6}}}-{�� \ {|_{\  1}^{\  6 \  7}}}+{�� \ {|_{\  7}^{\  7 \  0}}}-{�� \ {|_{\  6}^{\  7 \  1}}}+{�� \ {|_{\  5}^{\  7 \  2}}}+{�� \ {|_{\  4}^{\  7 \  3}}}- 
\
\
\displaystyle
{�� \ {|_{\  3}^{\  7 \  4}}}-{�� \ {|_{\  2}^{\  7 \  5}}}+{�� \ {|_{\  1}^{\  7 \  6}}}-{�� \ {|_{\  0}^{\  7 \  7}}}
(45)
Type: ClosedLinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7]),Expression(Integer))




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