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Sedenion Algebra is Frobenius in just one way!

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

Ref:

We need the Axiom LinearOperator library.

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

Use the following macros for convenient notation

fricas
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
fricas
-- subscript
macro sb == subscript
Type: Void

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

fricas
dim:=16

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

\label{eq2}\hbox{\axiomType{LinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
fricas
ⅇ:List ℒ      := basisOut()

\label{eq3}\left[{|_{0}}, \:{|_{1}}, \:{|_{2}}, \:{|_{3}}, \:{|_{4}}, \:{|_{5}}, \:{|_{6}}, \:{|_{7}}, \:{|_{8}}, \:{|_{9}}, \:{|_{10}}, \:{|_{11}}, \:{|_{12}}, \:{|_{13}}, \:{|_{14}}, \:{|_{15}}\right](3)
Type: List(LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer)))
fricas
ⅆ:List ℒ      := basisIn()

\label{eq4}\begin{array}{@{}l}
\displaystyle
\left[{|_{\ }^{0}}, \:{|_{\ }^{1}}, \:{|_{\ }^{2}}, \:{|_{\ }^{3}}, \:{|_{\ }^{4}}, \:{|_{\ }^{5}}, \:{|_{\ }^{6}}, \:{|_{\ }^{7}}, \:{|_{\ }^{8}}, \:{|_{\ }^{9}}, \:{|_{\ }^{10}}, \:{|_{\ }^{11}}, \:{|_{\ }^{12}}, \:{|_{\ }^{13}}, \: \right.
\
\
\displaystyle
\left.{|_{\ }^{14}}, \:{|_{\ }^{15}}\right] 
(4)
Type: List(LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer)))
fricas
I:ℒ:=[1];   -- identity for composition
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
X:ℒ:=[2,1]; -- twist
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))

Now generate structure constants for Sedenion Algebra

The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Split-complex, co-quaternions, split-octonions and seneions can be specified by Caley-Dickson parameters

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

\label{eq5}1(5)
Type: PositiveInteger?
fricas
--q1:=sb('q,[1])
q1:=1  -- not co-quaternion

\label{eq6}1(6)
Type: PositiveInteger?
fricas
--q2:=sb('q,[2])
q2:=1  -- not split-octonion

\label{eq7}1(7)
Type: PositiveInteger?
fricas
--q3:=sb('q,[3])
q3:=1  -- not split-sedennion

\label{eq8}1(8)
Type: PositiveInteger?
fricas
QQ := ℂ(ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2),'l,q3);
Type: Type

Basis: Each B.i is a sedennion number

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

\label{eq9}\left[ 1, \: i , \: j , \:{ij}, \: k , \:{ik}, \:{jk}, \:{{ij}k}, \: l , \:{il}, \:{jl}, \:{{ij}l}, \:{kl}, \:{{ik}l}, \:{{jk}l}, \:{{{ij}k}l}\right](9)
Type: List(CaleyDickson(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,1),l,1))
fricas
-- Multiplication table:
M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]

\label{eq10}\left[ 
\begin{array}{cccccccccccccccc}
1 & i & j &{ij}& k &{ik}&{jk}&{{ij}k}& l &{il}&{jl}&{{ij}l}&{kl}&{{ik}l}&{{jk}l}&{{{ij}k}l}
\
i & - 1 & -{ij}& j &{- ik}& k &{{ij}k}& -{jk}&{- il}& l &{{ij}l}&{- jl}&{{ik}l}& -{kl}&{{-{ij}k}l}&{{jk}l}
\
j &{ij}& - 1 & - i & -{jk}&{-{ij}k}& k &{ik}&{- jl}&{-{ij}l}& l &{il}&{{jk}l}&{{{ij}k}l}& -{kl}&{{- ik}l}
\
{ij}& - j & i & - 1 &{-{ij}k}&{jk}&{- ik}& k &{-{ij}l}&{jl}&{- il}& l &{{{ij}k}l}&{-{jk}l}&{{ik}l}& -{kl}
\
k &{ik}&{jk}&{{ij}k}& - 1 & - i & - j & -{ij}& -{kl}&{{- ik}l}&{-{jk}l}&{{-{ij}k}l}& l &{il}&{jl}&{{ij}l}
\
{ik}& - k &{{ij}k}& -{jk}& i & - 1 &{ij}& - j &{{- ik}l}&{kl}&{{-{ij}k}l}&{{jk}l}&{- il}& l &{-{ij}l}&{jl}
\
{jk}&{-{ij}k}& - k &{ik}& j & -{ij}& - 1 & i &{-{jk}l}&{{{ij}k}l}&{kl}&{{- ik}l}&{- jl}&{{ij}l}& l &{- il}
\
{{ij}k}&{jk}&{- ik}& - k &{ij}& j & - i & - 1 &{{-{ij}k}l}&{-{jk}l}&{{ik}l}&{kl}&{-{ij}l}&{- jl}&{il}& l 
\
l &{il}&{jl}&{{ij}l}&{kl}&{{ik}l}&{{jk}l}&{{{ij}k}l}& - 1 & - i & - j & -{ij}& - k &{- ik}& -{jk}&{-{ij}k}
\
{il}& - l &{{ij}l}&{- jl}&{{ik}l}& -{kl}&{{-{ij}k}l}&{{jk}l}& i & - 1 &{ij}& - j &{ik}& - k &{-{ij}k}&{jk}
\
{jl}&{-{ij}l}& - l &{il}&{{jk}l}&{{{ij}k}l}& -{kl}&{{- ik}l}& j & -{ij}& - 1 & i &{jk}&{{ij}k}& - k &{- ik}
\
{{ij}l}&{jl}&{- il}& - l &{{{ij}k}l}&{-{jk}l}&{{ik}l}& -{kl}&{ij}& j & - i & - 1 &{{ij}k}& -{jk}&{ik}& - k 
\
{kl}&{{- ik}l}&{-{jk}l}&{{-{ij}k}l}& - l &{il}&{jl}&{{ij}l}& k &{- ik}& -{jk}&{-{ij}k}& - 1 & i & j &{ij}
\
{{ik}l}&{kl}&{{-{ij}k}l}&{{jk}l}&{- il}& - l &{-{ij}l}&{jl}&{ik}& k &{-{ij}k}&{jk}& - i & - 1 & -{ij}& j 
\
{{jk}l}&{{{ij}k}l}&{kl}&{{- ik}l}&{- jl}&{{ij}l}& - l &{- il}&{jk}&{{ij}k}& k &{- ik}& - j &{ij}& - 1 & - i 
\
{{{ij}k}l}&{-{jk}l}&{{ik}l}&{kl}&{-{ij}l}&{- jl}&{il}& - l &{{ij}k}& -{jk}&{ik}& k & -{ij}& - j & i & - 1 
(10)
Type: Matrix(CaleyDickson(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,1),l,1))
fricas
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real real real(x/y),M)
Type: Void
fricas
-- The result is a nested list
ѕ :=map(S,B)::List List List ℚ;
fricas
Compiling function S with type CaleyDickson(CaleyDickson(
      CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,1),l,1)
       -> Matrix(Expression(Integer))
Type: List(List(List(Expression(Integer))))
fricas
-- structure constants form a tensor operator
Y := Σ(Σ(Σ(ѕ(i)(k)(j)*ⅇ.i*ⅆ.j*ⅆ.k, i,1..dim), j,1..dim), k,1..dim);
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
arity Y

\label{eq11}2 \over 1(11)
Type: Prop(LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer)))
fricas
matrix [[(ⅇ.i*ⅇ.j)/Y for i in 1..dim] for j in 1..dim]

\label{eq12}\left[ 
\begin{array}{cccccccccccccccc}
{|_{0}}&{|_{1}}&{|_{2}}&{|_{3}}&{|_{4}}&{|_{5}}&{|_{6}}&{|_{7}}&{|_{8}}&{|_{9}}&{|_{10}}&{|_{11}}&{|_{12}}&{|_{13}}&{|_{14}}&{|_{15}}
\
{|_{1}}& -{|_{0}}& -{|_{3}}&{|_{2}}& -{|_{5}}&{|_{4}}&{|_{7}}& -{|_{6}}& -{|_{9}}&{|_{8}}&{|_{11}}& -{|_{10}}&{|_{13}}& -{|_{12}}& -{|_{15}}&{|_{14}}
\
{|_{2}}&{|_{3}}& -{|_{0}}& -{|_{1}}& -{|_{6}}& -{|_{7}}&{|_{4}}&{|_{5}}& -{|_{10}}& -{|_{11}}&{|_{8}}&{|_{9}}&{|_{14}}&{|_{15}}& -{|_{12}}& -{|_{13}}
\
{|_{3}}& -{|_{2}}&{|_{1}}& -{|_{0}}& -{|_{7}}&{|_{6}}& -{|_{5}}&{|_{4}}& -{|_{11}}&{|_{10}}& -{|_{9}}&{|_{8}}&{|_{15}}& -{|_{14}}&{|_{13}}& -{|_{12}}
\
{|_{4}}&{|_{5}}&{|_{6}}&{|_{7}}& -{|_{0}}& -{|_{1}}& -{|_{2}}& -{|_{3}}& -{|_{12}}& -{|_{13}}& -{|_{14}}& -{|_{15}}&{|_{8}}&{|_{9}}&{|_{10}}&{|_{11}}
\
{|_{5}}& -{|_{4}}&{|_{7}}& -{|_{6}}&{|_{1}}& -{|_{0}}&{|_{3}}& -{|_{2}}& -{|_{13}}&{|_{12}}& -{|_{15}}&{|_{14}}& -{|_{9}}&{|_{8}}& -{|_{11}}&{|_{10}}
\
{|_{6}}& -{|_{7}}& -{|_{4}}&{|_{5}}&{|_{2}}& -{|_{3}}& -{|_{0}}&{|_{1}}& -{|_{14}}&{|_{15}}&{|_{12}}& -{|_{13}}& -{|_{10}}&{|_{1
1}}&{|_{8}}& -{|_{9}}
\
{|_{7}}&{|_{6}}& -{|_{5}}& -{|_{4}}&{|_{3}}&{|_{2}}& -{|_{1}}& -{|_{0}}& -{|_{15}}& -{|_{14}}&{|_{13}}&{|_{12}}& -{|_{11}}& -{|_{10}}&{|_{9}}&{|_{8}}
\
{|_{8}}&{|_{9}}&{|_{10}}&{|_{11}}&{|_{12}}&{|_{13}}&{|_{14}}&{|_{15}}& -{|_{0}}& -{|_{1}}& -{|_{2}}& -{|_{3}}& -{|_{4}}& -{|_{5}}& -{|_{6}}& -{|_{7}}
\
{|_{9}}& -{|_{8}}&{|_{11}}& -{|_{10}}&{|_{13}}& -{|_{12}}& -{|_{15}}&{|_{14}}&{|_{1}}& -{|_{0}}&{|_{3}}& -{|_{2}}&{|_{5}}& -{|_{4}}& -{|_{7}}&{|_{6}}
\
{|_{10}}& -{|_{11}}& -{|_{8}}&{|_{9}}&{|_{14}}&{|_{15}}& -{|_{12}}& -{|_{13}}&{|_{2}}& -{|_{3}}& -{|_{0}}&{|_{1}}&{|_{6}}&{|_{7}}& -{|_{4}}& -{|_{5}}
\
{|_{11}}&{|_{10}}& -{|_{9}}& -{|_{8}}&{|_{15}}& -{|_{14}}&{|_{13}}& -{|_{12}}&{|_{3}}&{|_{2}}& -{|_{1}}& -{|_{0}}&{|_{7}}& -{|_{6}}&{|_{5}}& -{|_{4}}
\
{|_{12}}& -{|_{13}}& -{|_{14}}& -{|_{15}}& -{|_{8}}&{|_{9}}&{|_{10}}&{|_{11}}&{|_{4}}& -{|_{5}}& -{|_{6}}& -{|_{7}}& -{|_{0}}&{|_{1}}&{|_{2}}&{|_{3}}
\
{|_{13}}&{|_{12}}& -{|_{15}}&{|_{14}}& -{|_{9}}& -{|_{8}}& -{|_{11}}&{|_{10}}&{|_{5}}&{|_{4}}& -{|_{7}}&{|_{6}}& -{|_{1}}& -{|_{0}}& -{|_{3}}&{|_{2}}
\
{|_{14}}&{|_{15}}&{|_{12}}& -{|_{13}}& -{|_{10}}&{|_{11}}& -{|_{8}}& -{|_{9}}&{|_{6}}&{|_{7}}&{|_{4}}& -{|_{5}}& -{|_{2}}&{|_{3}}& -{|_{0}}& -{|_{1}}
\
{|_{15}}& -{|_{14}}&{|_{13}}&{|_{12}}& -{|_{11}}& -{|_{10}}&{|_{9}}& -{|_{8}}&{|_{7}}& -{|_{6}}&{|_{5}}&{|_{4}}& -{|_{3}}& -{|_{2}}&{|_{1}}& -{|_{0}}
(12)
Type: Matrix(LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),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);
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),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{eq13}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (13)
(three-point function) is zero.

Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

fricas
ω:ℒ :=(Y*I)/U  - (I*Y)/U;
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))

Definition 2

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

We may consider the problem where multiplication Y is given, and look for all associative scalar products U = U(Y)

This problem can be solved using linear algebra.

fricas
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables,ravel U)::List Symbol);
Type: Matrix(Expression(Integer))
fricas
--u := transpose matrix [concat map(variables,ravel U)::List Symbol];
--J::OutputForm * u::OutputForm = 0
nrows(J),ncols(J)

\label{eq14}\left[{4096}, \:{256}\right](14)
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) )

\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{{u^{1, \: 1}}= -{p_{1}}}, \:{{u^{1, \: 2}}= 0}, \:{{u^{1, \: 3}}= 0}, \:{{u^{1, \: 4}}= 0}, \:{{u^{1, \: 5}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{1, \: 6}}= 0}, \:{{u^{1, \: 7}}= 0}, \:{{u^{1, \: 8}}= 0}, \:{{u^{1, \: 9}}= 0}, \:{{u^{1, \:{10}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{1, \:{11}}}= 0}, \:{{u^{1, \:{12}}}= 0}, \:{{u^{1, \:{13}}}= 0}, \:{{u^{1, \:{14}}}= 0}, \:{{u^{1, \:{15}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{1, \:{16}}}= 0}, \:{{u^{2, \: 1}}= 0}, \:{{u^{2, \: 2}}={p_{1}}}, \:{{u^{2, \: 3}}= 0}, \:{{u^{2, \: 4}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{2, \: 5}}= 0}, \:{{u^{2, \: 6}}= 0}, \:{{u^{2, \: 7}}= 0}, \:{{u^{2, \: 8}}= 0}, \:{{u^{2, \: 9}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{2, \:{10}}}= 0}, \:{{u^{2, \:{11}}}= 0}, \:{{u^{2, \:{12}}}= 0}, \:{{u^{2, \:{13}}}= 0}, \:{{u^{2, \:{14}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{2, \:{15}}}= 0}, \:{{u^{2, \:{16}}}= 0}, \:{{u^{3, \: 1}}= 0}, \:{{u^{3, \: 2}}= 0}, \:{{u^{3, \: 3}}={p_{1}}}, \: \right.
\
\
\displaystyle
\left.{{u^{3, \: 4}}= 0}, \:{{u^{3, \: 5}}= 0}, \:{{u^{3, \: 6}}= 0}, \:{{u^{3, \: 7}}= 0}, \:{{u^{3, \: 8}}= 0}, \:{{u^{3, \: 9}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{3, \:{10}}}= 0}, \:{{u^{3, \:{11}}}= 0}, \:{{u^{3, \:{12}}}= 0}, \:{{u^{3, \:{13}}}= 0}, \:{{u^{3, \:{14}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{3, \:{15}}}= 0}, \:{{u^{3, \:{16}}}= 0}, \:{{u^{4, \: 1}}= 0}, \:{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{4, \: 4}}={p_{1}}}, \:{{u^{4, \: 5}}= 0}, \:{{u^{4, \: 6}}= 0}, \:{{u^{4, \: 7}}= 0}, \:{{u^{4, \: 8}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{4, \: 9}}= 0}, \:{{u^{4, \:{10}}}= 0}, \:{{u^{4, \:{1
1}}}= 0}, \:{{u^{4, \:{12}}}= 0}, \:{{u^{4, \:{13}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{4, \:{14}}}= 0}, \:{{u^{4, \:{15}}}= 0}, \:{{u^{4, \:{16}}}= 0}, \:{{u^{5, \: 1}}= 0}, \:{{u^{5, \: 2}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{5, \: 3}}= 0}, \:{{u^{5, \: 4}}= 0}, \:{{u^{5, \: 5}}={p_{1}}}, \:{{u^{5, \: 6}}= 0}, \:{{u^{5, \: 7}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{5, \: 8}}= 0}, \:{{u^{5, \: 9}}= 0}, \:{{u^{5, \:{1
0}}}= 0}, \:{{u^{5, \:{11}}}= 0}, \:{{u^{5, \:{12}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{5, \:{13}}}= 0}, \:{{u^{5, \:{14}}}= 0}, \:{{u^{5, \:{15}}}= 0}, \:{{u^{5, \:{16}}}= 0}, \:{{u^{6, \: 1}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{6, \: 2}}= 0}, \:{{u^{6, \: 3}}= 0}, \:{{u^{6, \: 4}}= 0}, \:{{u^{6, \: 5}}= 0}, \:{{u^{6, \: 6}}={p_{1}}}, \: \right.
\
\
\displaystyle
\left.{{u^{6, \: 7}}= 0}, \:{{u^{6, \: 8}}= 0}, \:{{u^{6, \: 9}}= 0}, \:{{u^{6, \:{10}}}= 0}, \:{{u^{6, \:{11}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{6, \:{12}}}= 0}, \:{{u^{6, \:{13}}}= 0}, \:{{u^{6, \:{14}}}= 0}, \:{{u^{6, \:{15}}}= 0}, \:{{u^{6, \:{16}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{7, \: 1}}= 0}, \:{{u^{7, \: 2}}= 0}, \:{{u^{7, \: 3}}= 0}, \:{{u^{7, \: 4}}= 0}, \:{{u^{7, \: 5}}= 0}, \:{{u^{7, \: 6}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{7, \: 7}}={p_{1}}}, \:{{u^{7, \: 8}}= 0}, \:{{u^{7, \: 9}}= 0}, \:{{u^{7, \:{10}}}= 0}, \:{{u^{7, \:{11}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{7, \:{12}}}= 0}, \:{{u^{7, \:{13}}}= 0}, \:{{u^{7, \:{14}}}= 0}, \:{{u^{7, \:{15}}}= 0}, \:{{u^{7, \:{16}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{8, \: 1}}= 0}, \:{{u^{8, \: 2}}= 0}, \:{{u^{8, \: 3}}= 0}, \:{{u^{8, \: 4}}= 0}, \:{{u^{8, \: 5}}= 0}, \:{{u^{8, \: 6}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{8, \: 7}}= 0}, \:{{u^{8, \: 8}}={p_{1}}}, \:{{u^{8, \: 9}}= 0}, \:{{u^{8, \:{10}}}= 0}, \:{{u^{8, \:{11}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{8, \:{12}}}= 0}, \:{{u^{8, \:{13}}}= 0}, \:{{u^{8, \:{14}}}= 0}, \:{{u^{8, \:{15}}}= 0}, \:{{u^{8, \:{16}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{9, \: 1}}= 0}, \:{{u^{9, \: 2}}= 0}, \:{{u^{9, \: 3}}= 0}, \:{{u^{9, \: 4}}= 0}, \:{{u^{9, \: 5}}= 0}, \:{{u^{9, \: 6}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{9, \: 7}}= 0}, \:{{u^{9, \: 8}}= 0}, \:{{u^{9, \: 9}}={p_{1}}}, \:{{u^{9, \:{10}}}= 0}, \:{{u^{9, \:{11}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{9, \:{12}}}= 0}, \:{{u^{9, \:{13}}}= 0}, \:{{u^{9, \:{14}}}= 0}, \:{{u^{9, \:{15}}}= 0}, \:{{u^{9, \:{16}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{10}, \: 1}}= 0}, \:{{u^{{10}, \: 2}}= 0}, \:{{u^{{1
0}, \: 3}}= 0}, \:{{u^{{10}, \: 4}}= 0}, \:{{u^{{10}, \: 5}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{10}, \: 6}}= 0}, \:{{u^{{10}, \: 7}}= 0}, \:{{u^{{1
0}, \: 8}}= 0}, \:{{u^{{10}, \: 9}}= 0}, \:{{u^{{10}, \:{10}}}={p_{1}}}, \: \right.
\
\
\displaystyle
\left.{{u^{{10}, \:{11}}}= 0}, \:{{u^{{10}, \:{12}}}= 0}, \:{{u^{{10}, \:{13}}}= 0}, \:{{u^{{10}, \:{14}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{10}, \:{15}}}= 0}, \:{{u^{{10}, \:{16}}}= 0}, \:{{u^{{11}, \: 1}}= 0}, \:{{u^{{11}, \: 2}}= 0}, \:{{u^{{11}, \: 3}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{11}, \: 4}}= 0}, \:{{u^{{11}, \: 5}}= 0}, \:{{u^{{1
1}, \: 6}}= 0}, \:{{u^{{11}, \: 7}}= 0}, \:{{u^{{11}, \: 8}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{11}, \: 9}}= 0}, \:{{u^{{11}, \:{10}}}= 0}, \:{{u^{{11}, \:{11}}}={p_{1}}}, \:{{u^{{11}, \:{12}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{11}, \:{13}}}= 0}, \:{{u^{{11}, \:{14}}}= 0}, \:{{u^{{11}, \:{15}}}= 0}, \:{{u^{{11}, \:{16}}}= 0}, \:{{u^{{12}, \: 1}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{{12}, \: 2}}= 0}, \:{{u^{{12}, \: 3}}= 0}, \:{{u^{{12}, \: 4}}= 0}, \:{{u^{{12}, \: 5}}= 0}, \:{{u^{{12}, \: 6}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{12}, \: 7}}= 0}, \:{{u^{{12}, \: 8}}= 0}, \:{{u^{{1
2}, \: 9}}= 0}, \:{{u^{{12}, \:{10}}}= 0}, \:{{u^{{12}, \:{11}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{12}, \:{12}}}={p_{1}}}, \:{{u^{{12}, \:{13}}}= 0}, \:{{u^{{12}, \:{14}}}= 0}, \:{{u^{{12}, \:{15}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{12}, \:{16}}}= 0}, \:{{u^{{13}, \: 1}}= 0}, \:{{u^{{13}, \: 2}}= 0}, \:{{u^{{13}, \: 3}}= 0}, \:{{u^{{13}, \: 4}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{13}, \: 5}}= 0}, \:{{u^{{13}, \: 6}}= 0}, \:{{u^{{1
3}, \: 7}}= 0}, \:{{u^{{13}, \: 8}}= 0}, \:{{u^{{13}, \: 9}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{13}, \:{10}}}= 0}, \:{{u^{{13}, \:{11}}}= 0}, \:{{u^{{13}, \:{12}}}= 0}, \:{{u^{{13}, \:{13}}}={p_{1}}}, \: \right.
\
\
\displaystyle
\left.{{u^{{13}, \:{14}}}= 0}, \:{{u^{{13}, \:{15}}}= 0}, \:{{u^{{13}, \:{16}}}= 0}, \:{{u^{{14}, \: 1}}= 0}, \:{{u^{{14}, \: 2}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{14}, \: 3}}= 0}, \:{{u^{{14}, \: 4}}= 0}, \:{{u^{{1
4}, \: 5}}= 0}, \:{{u^{{14}, \: 6}}= 0}, \:{{u^{{14}, \: 7}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{14}, \: 8}}= 0}, \:{{u^{{14}, \: 9}}= 0}, \:{{u^{{1
4}, \:{10}}}= 0}, \:{{u^{{14}, \:{11}}}= 0}, \:{{u^{{14}, \:{1
2}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{14}, \:{13}}}= 0}, \:{{u^{{14}, \:{14}}}={p_{1}}}, \:{{u^{{14}, \:{15}}}= 0}, \:{{u^{{14}, \:{16}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{15}, \: 1}}= 0}, \:{{u^{{15}, \: 2}}= 0}, \:{{u^{{1
5}, \: 3}}= 0}, \:{{u^{{15}, \: 4}}= 0}, \:{{u^{{15}, \: 5}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{15}, \: 6}}= 0}, \:{{u^{{15}, \: 7}}= 0}, \:{{u^{{1
5}, \: 8}}= 0}, \:{{u^{{15}, \: 9}}= 0}, \:{{u^{{15}, \:{10}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{15}, \:{11}}}= 0}, \:{{u^{{15}, \:{12}}}= 0}, \:{{u^{{15}, \:{13}}}= 0}, \:{{u^{{15}, \:{14}}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{15}, \:{15}}}={p_{1}}}, \:{{u^{{15}, \:{16}}}= 0}, \:{{u^{{16}, \: 1}}= 0}, \:{{u^{{16}, \: 2}}= 0}, \:{{u^{{16}, \: 3}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{16}, \: 4}}= 0}, \:{{u^{{16}, \: 5}}= 0}, \:{{u^{{1
6}, \: 6}}= 0}, \:{{u^{{16}, \: 7}}= 0}, \:{{u^{{16}, \: 8}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{{16}, \: 9}}= 0}, \:{{u^{{16}, \:{10}}}= 0}, \:{{u^{{16}, \:{11}}}= 0}, \:{{u^{{16}, \:{12}}}= 0}, \:{{u^{{16}, \:{13}}}= 0}, \right.
\
\
\displaystyle
\left.\:{{u^{{16}, \:{14}}}= 0}, \:{{u^{{16}, \:{15}}}= 0}, \:{{u^{{16}, \:{16}}}={p_{1}}}\right] 
(15)
Type: List(Equation(Expression(Integer)))

This defines a family of Frobenius algebras:

fricas
zero? eval(ω,ℰ)

\label{eq16} \mbox{\rm true} (16)
Type: Boolean

The pairing is necessarily diagonal!

fricas
Ų:ℒ := eval(U,ℰ)

\label{eq17}\begin{array}{@{}l}
\displaystyle
-{{p_{1}}\ {|_{\ }^{0 \  0}}}+{{p_{1}}\ {|_{\ }^{1 \  1}}}+{{p_{1}}\ {|_{\ }^{2 \  2}}}+{{p_{1}}\ {|_{\ }^{3 \  3}}}+{{p_{1}}\ {|_{\ }^{4 \  4}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{5 \  5}}}+{{p_{1}}\ {|_{\ }^{6 \  6}}}+{{p_{1}}\ {|_{\ }^{7 \  7}}}+{{p_{1}}\ {|_{\ }^{8 \  8}}}+{{p_{1}}\ {|_{\ }^{9 \  9}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{{10}\ {10}}}}+{{p_{1}}\ {|_{\ }^{{11}\ {11}}}}+{{p_{1}}\ {|_{\ }^{{12}\ {12}}}}+{{p_{1}}\ {|_{\ }^{{13}\ {13}}}}+{{p_{1}}\ {|_{\ }^{{14}\ {14}}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{{15}\ {15}}}}
(17)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
matrix [[(ⅇ.i ⅇ.j)/Ų for i in 1..dim] for j in 1..dim]

\label{eq18}\left[ 
\begin{array}{cccccccccccccccc}
-{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}& 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{p_{1}}
(18)
Type: Matrix(LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer)))

The scalar product must be non-degenerate:

fricas
Ů:=determinant [[retract((ⅇ.i * ⅇ.j)/Ų) for j in 1..dim] for i in 1..dim]

\label{eq19}-{{p_{1}}^{16}}(19)
Type: Expression(Integer)
fricas
factor Ů

\label{eq20}-{{p_{1}}^{16}}(20)
Type: Factored(Expression(Integer))

Definition 3

Co-pairing

Solve the Snake Relation as a system of linear equations.

fricas
Um:=matrix [[(ⅇ.i*ⅇ.j)/Ų for i in 1..dim] for j in 1..dim];
Type: Matrix(LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer)))
fricas
mU:=transpose inverse map(retract,Um);
Type: Matrix(Expression(Integer))
fricas
Ω:=Σ(Σ(mU(i,j)*(ⅇ.i*ⅇ.j), i,1..dim), j,1..dim)

\label{eq21}\begin{array}{@{}l}
\displaystyle
-{{1 \over{p_{1}}}\ {|_{0 \  0}}}+{{1 \over{p_{1}}}\ {|_{1 \  1}}}+{{1 \over{p_{1}}}\ {|_{2 \  2}}}+{{1 \over{p_{1}}}\ {|_{3 \  3}}}+{{1 \over{p_{1}}}\ {|_{4 \  4}}}+{{1 \over{p_{1}}}\ {|_{5 \  5}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{6 \  6}}}+{{1 \over{p_{1}}}\ {|_{7 \  7}}}+{{1 \over{p_{1}}}\ {|_{8 \  8}}}+{{1 \over{p_{1}}}\ {|_{9 \  9}}}+{{1 \over{p_{1}}}\ {|_{{10}\ {10}}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\ {11}}}}+{{1 \over{p_{1}}}\ {|_{{1
2}\ {12}}}}+{{1 \over{p_{1}}}\ {|_{{13}\ {13}}}}+{{1 \over{p_{1}}}\ {|_{{14}\ {14}}}}+{{1 \over{p_{1}}}\ {|_{{15}\ {15}}}}
(21)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
matrix [[Ω/(ⅆ.i*ⅆ.j) for i in 1..dim] for j in 1..dim]

\label{eq22}\left[ 
\begin{array}{cccccccccccccccc}
-{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}& 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{1 \over{p_{1}}}
(22)
Type: Matrix(LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer)))

Check "dimension" and the snake relations.

fricas
d:ℒ:=
       Ω    /
       X    /
       Ų

\label{eq23}16(23)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
test
    (    I Ω     )  /
    (     Ų I    )  =  I

\label{eq24} \mbox{\rm true} (24)
Type: Boolean
fricas
test
    (     Ω I    )  /
    (    I Ų     )  =  I

\label{eq25} \mbox{\rm true} (25)
Type: Boolean

Definition 4

Co-algebra

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

Too slow:

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

fricas
λ:= (I,Ω) / (Y,I)

\label{eq26}\begin{array}{@{}l}
\displaystyle
-{{1 \over{p_{1}}}\ {|_{0 \  0}^{0}}}+{{1 \over{p_{1}}}\ {|_{1 \  1}^{0}}}+{{1 \over{p_{1}}}\ {|_{2 \  2}^{0}}}+{{1 \over{p_{1}}}\ {|_{3 \  3}^{0}}}+{{1 \over{p_{1}}}\ {|_{4 \  4}^{0}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \  5}^{0}}}+{{1 \over{p_{1}}}\ {|_{6 \  6}^{0}}}+{{1 \over{p_{1}}}\ {|_{7 \  7}^{0}}}+{{1 \over{p_{1}}}\ {|_{8 \  8}^{0}}}+{{1 \over{p_{1}}}\ {|_{9 \  9}^{0}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{10}\ {10}}^{0}}}+{{1 \over{p_{1}}}\ {|_{{11}\ {11}}^{0}}}+{{1 \over{p_{1}}}\ {|_{{12}\ {12}}^{0}}}+{{1 \over{p_{1}}}\ {|_{{13}\ {13}}^{0}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{14}\ {14}}^{0}}}+{{1 \over{p_{1}}}\ {|_{{15}\ {15}}^{0}}}-{{1 \over{p_{1}}}\ {|_{0 \  1}^{1}}}-{{1 \over{p_{1}}}\ {|_{1 \  0}^{1}}}-{{1 \over{p_{1}}}\ {|_{2 \  3}^{1}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \  2}^{1}}}-{{1 \over{p_{1}}}\ {|_{4 \  5}^{1}}}+{{1 \over{p_{1}}}\ {|_{5 \  4}^{1}}}+{{1 \over{p_{1}}}\ {|_{6 \  7}^{1}}}-{{1 \over{p_{1}}}\ {|_{7 \  6}^{1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{8 \  9}^{1}}}+{{1 \over{p_{1}}}\ {|_{9 \  8}^{1}}}+{{1 \over{p_{1}}}\ {|_{{10}\ {11}}^{1}}}-{{1 \over{p_{1}}}\ {|_{{11}\ {10}}^{1}}}+{{1 \over{p_{1}}}\ {|_{{12}\ {13}}^{1}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{13}\ {12}}^{1}}}-{{1 \over{p_{1}}}\ {|_{{14}\ {15}}^{1}}}+{{1 \over{p_{1}}}\ {|_{{15}\ {14}}^{1}}}-{{1 \over{p_{1}}}\ {|_{0 \  2}^{2}}}+{{1 \over{p_{1}}}\ {|_{1 \  3}^{2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{2 \  0}^{2}}}-{{1 \over{p_{1}}}\ {|_{3 \  1}^{2}}}-{{1 \over{p_{1}}}\ {|_{4 \  6}^{2}}}-{{1 \over{p_{1}}}\ {|_{5 \  7}^{2}}}+{{1 \over{p_{1}}}\ {|_{6 \  4}^{2}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \  5}^{2}}}-{{1 \over{p_{1}}}\ {|_{8 \ {10}}^{2}}}-{{1 \over{p_{1}}}\ {|_{9 \ {11}}^{2}}}+{{1 \over{p_{1}}}\ {|_{{10}\  8}^{2}}}+{{1 \over{p_{1}}}\ {|_{{11}\  9}^{2}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{12}\ {14}}^{2}}}+{{1 \over{p_{1}}}\ {|_{{13}\ {15}}^{2}}}-{{1 \over{p_{1}}}\ {|_{{14}\ {12}}^{2}}}-{{1 \over{p_{1}}}\ {|_{{15}\ {13}}^{2}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{0 \  3}^{3}}}-{{1 \over{p_{1}}}\ {|_{1 \  2}^{3}}}+{{1 \over{p_{1}}}\ {|_{2 \  1}^{3}}}-{{1 \over{p_{1}}}\ {|_{3 \  0}^{3}}}-{{1 \over{p_{1}}}\ {|_{4 \  7}^{3}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \  6}^{3}}}-{{1 \over{p_{1}}}\ {|_{6 \  5}^{3}}}+{{1 \over{p_{1}}}\ {|_{7 \  4}^{3}}}-{{1 \over{p_{1}}}\ {|_{8 \ {11}}^{3}}}+{{1 \over{p_{1}}}\ {|_{9 \ {10}}^{3}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{10}\  9}^{3}}}+{{1 \over{p_{1}}}\ {|_{{11}\  8}^{3}}}+{{1 \over{p_{1}}}\ {|_{{12}\ {15}}^{3}}}-{{1 \over{p_{1}}}\ {|_{{13}\ {14}}^{3}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{14}\ {13}}^{3}}}-{{1 \over{p_{1}}}\ {|_{{15}\ {12}}^{3}}}-{{1 \over{p_{1}}}\ {|_{0 \  4}^{4}}}+{{1 \over{p_{1}}}\ {|_{1 \  5}^{4}}}+{{1 \over{p_{1}}}\ {|_{2 \  6}^{4}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \  7}^{4}}}-{{1 \over{p_{1}}}\ {|_{4 \  0}^{4}}}-{{1 \over{p_{1}}}\ {|_{5 \  1}^{4}}}-{{1 \over{p_{1}}}\ {|_{6 \  2}^{4}}}-{{1 \over{p_{1}}}\ {|_{7 \  3}^{4}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{8 \ {12}}^{4}}}-{{1 \over{p_{1}}}\ {|_{9 \ {13}}^{4}}}-{{1 \over{p_{1}}}\ {|_{{10}\ {14}}^{4}}}-{{1 \over{p_{1}}}\ {|_{{11}\ {15}}^{4}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{12}\  8}^{4}}}+{{1 \over{p_{1}}}\ {|_{{13}\  9}^{4}}}+{{1 \over{p_{1}}}\ {|_{{14}\ {10}}^{4}}}+{{1 \over{p_{1}}}\ {|_{{15}\ {11}}^{4}}}-{{1 \over{p_{1}}}\ {|_{0 \  5}^{5}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{1 \  4}^{5}}}+{{1 \over{p_{1}}}\ {|_{2 \  7}^{5}}}-{{1 \over{p_{1}}}\ {|_{3 \  6}^{5}}}+{{1 \over{p_{1}}}\ {|_{4 \  1}^{5}}}-{{1 \over{p_{1}}}\ {|_{5 \  0}^{5}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{6 \  3}^{5}}}-{{1 \over{p_{1}}}\ {|_{7 \  2}^{5}}}-{{1 \over{p_{1}}}\ {|_{8 \ {13}}^{5}}}+{{1 \over{p_{1}}}\ {|_{9 \ {12}}^{5}}}-{{1 \over{p_{1}}}\ {|_{{10}\ {15}}^{5}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\ {14}}^{5}}}-{{1 \over{p_{1}}}\ {|_{{12}\  9}^{5}}}+{{1 \over{p_{1}}}\ {|_{{13}\  8}^{5}}}-{{1 \over{p_{1}}}\ {|_{{14}\ {11}}^{5}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\ {10}}^{5}}}-{{1 \over{p_{1}}}\ {|_{0 \  6}^{6}}}-{{1 \over{p_{1}}}\ {|_{1 \  7}^{6}}}-{{1 \over{p_{1}}}\ {|_{2 \  4}^{6}}}+{{1 \over{p_{1}}}\ {|_{3 \  5}^{6}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{4 \  2}^{6}}}-{{1 \over{p_{1}}}\ {|_{5 \  3}^{6}}}-{{1 \over{p_{1}}}\ {|_{6 \  0}^{6}}}+{{1 \over{p_{1}}}\ {|_{7 \  1}^{6}}}-{{1 \over{p_{1}}}\ {|_{8 \ {14}}^{6}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{9 \ {15}}^{6}}}+{{1 \over{p_{1}}}\ {|_{{10}\ {12}}^{6}}}-{{1 \over{p_{1}}}\ {|_{{11}\ {13}}^{6}}}-{{1 \over{p_{1}}}\ {|_{{12}\ {10}}^{6}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{13}\ {11}}^{6}}}+{{1 \over{p_{1}}}\ {|_{{14}\  8}^{6}}}-{{1 \over{p_{1}}}\ {|_{{15}\  9}^{6}}}-{{1 \over{p_{1}}}\ {|_{0 \  7}^{7}}}+{{1 \over{p_{1}}}\ {|_{1 \  6}^{7}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{2 \  5}^{7}}}-{{1 \over{p_{1}}}\ {|_{3 \  4}^{7}}}+{{1 \over{p_{1}}}\ {|_{4 \  3}^{7}}}+{{1 \over{p_{1}}}\ {|_{5 \  2}^{7}}}-{{1 \over{p_{1}}}\ {|_{6 \  1}^{7}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \  0}^{7}}}-{{1 \over{p_{1}}}\ {|_{8 \ {15}}^{7}}}-{{1 \over{p_{1}}}\ {|_{9 \ {14}}^{7}}}+{{1 \over{p_{1}}}\ {|_{{10}\ {13}}^{7}}}+{{1 \over{p_{1}}}\ {|_{{11}\ {12}}^{7}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{12}\ {11}}^{7}}}-{{1 \over{p_{1}}}\ {|_{{13}\ {10}}^{7}}}+{{1 \over{p_{1}}}\ {|_{{14}\  9}^{7}}}+{{1 \over{p_{1}}}\ {|_{{15}\  8}^{7}}}-{{1 \over{p_{1}}}\ {|_{0 \  8}^{8}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{1 \  9}^{8}}}+{{1 \over{p_{1}}}\ {|_{2 \ {10}}^{8}}}+{{1 \over{p_{1}}}\ {|_{3 \ {11}}^{8}}}+{{1 \over{p_{1}}}\ {|_{4 \ {12}}^{8}}}+{{1 \over{p_{1}}}\ {|_{5 \ {13}}^{8}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{6 \ {14}}^{8}}}+{{1 \over{p_{1}}}\ {|_{7 \ {15}}^{8}}}-{{1 \over{p_{1}}}\ {|_{8 \  0}^{8}}}-{{1 \over{p_{1}}}\ {|_{9 \  1}^{8}}}-{{1 \over{p_{1}}}\ {|_{{10}\  2}^{8}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\  3}^{8}}}-{{1 \over{p_{1}}}\ {|_{{12}\  4}^{8}}}-{{1 \over{p_{1}}}\ {|_{{13}\  5}^{8}}}-{{1 \over{p_{1}}}\ {|_{{14}\  6}^{8}}}-{{1 \over{p_{1}}}\ {|_{{15}\  7}^{8}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{0 \  9}^{9}}}-{{1 \over{p_{1}}}\ {|_{1 \  8}^{9}}}+{{1 \over{p_{1}}}\ {|_{2 \ {11}}^{9}}}-{{1 \over{p_{1}}}\ {|_{3 \ {10}}^{9}}}+{{1 \over{p_{1}}}\ {|_{4 \ {13}}^{9}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{5 \ {12}}^{9}}}-{{1 \over{p_{1}}}\ {|_{6 \ {15}}^{9}}}+{{1 \over{p_{1}}}\ {|_{7 \ {14}}^{9}}}+{{1 \over{p_{1}}}\ {|_{8 \  1}^{9}}}-{{1 \over{p_{1}}}\ {|_{9 \  0}^{9}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{10}\  3}^{9}}}-{{1 \over{p_{1}}}\ {|_{{11}\  2}^{9}}}+{{1 \over{p_{1}}}\ {|_{{12}\  5}^{9}}}-{{1 \over{p_{1}}}\ {|_{{13}\  4}^{9}}}-{{1 \over{p_{1}}}\ {|_{{14}\  7}^{9}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\  6}^{9}}}-{{1 \over{p_{1}}}\ {|_{0 \ {10}}^{10}}}-{{1 \over{p_{1}}}\ {|_{1 \ {11}}^{10}}}-{{1 \over{p_{1}}}\ {|_{2 \  8}^{10}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \  9}^{10}}}+{{1 \over{p_{1}}}\ {|_{4 \ {14}}^{10}}}+{{1 \over{p_{1}}}\ {|_{5 \ {15}}^{10}}}-{{1 \over{p_{1}}}\ {|_{6 \ {12}}^{10}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \ {13}}^{10}}}+{{1 \over{p_{1}}}\ {|_{8 \  2}^{10}}}-{{1 \over{p_{1}}}\ {|_{9 \  3}^{10}}}-{{1 \over{p_{1}}}\ {|_{{10}\  0}^{10}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\  1}^{10}}}+{{1 \over{p_{1}}}\ {|_{{12}\  6}^{10}}}+{{1 \over{p_{1}}}\ {|_{{13}\  7}^{10}}}-{{1 \over{p_{1}}}\ {|_{{14}\  4}^{10}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\  5}^{10}}}-{{1 \over{p_{1}}}\ {|_{0 \ {11}}^{11}}}+{{1 \over{p_{1}}}\ {|_{1 \ {10}}^{11}}}-{{1 \over{p_{1}}}\ {|_{2 \  9}^{11}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \  8}^{11}}}+{{1 \over{p_{1}}}\ {|_{4 \ {15}}^{11}}}-{{1 \over{p_{1}}}\ {|_{5 \ {14}}^{11}}}+{{1 \over{p_{1}}}\ {|_{6 \ {13}}^{11}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \ {12}}^{11}}}+{{1 \over{p_{1}}}\ {|_{8 \  3}^{11}}}+{{1 \over{p_{1}}}\ {|_{9 \  2}^{11}}}-{{1 \over{p_{1}}}\ {|_{{10}\  1}^{11}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\  0}^{11}}}+{{1 \over{p_{1}}}\ {|_{{12}\  7}^{11}}}-{{1 \over{p_{1}}}\ {|_{{13}\  6}^{11}}}+{{1 \over{p_{1}}}\ {|_{{14}\  5}^{11}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\  4}^{11}}}-{{1 \over{p_{1}}}\ {|_{0 \ {12}}^{12}}}-{{1 \over{p_{1}}}\ {|_{1 \ {13}}^{12}}}-{{1 \over{p_{1}}}\ {|_{2 \ {14}}^{12}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \ {15}}^{12}}}-{{1 \over{p_{1}}}\ {|_{4 \  8}^{12}}}+{{1 \over{p_{1}}}\ {|_{5 \  9}^{12}}}+{{1 \over{p_{1}}}\ {|_{6 \ {10}}^{12}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \ {11}}^{12}}}+{{1 \over{p_{1}}}\ {|_{8 \  4}^{12}}}-{{1 \over{p_{1}}}\ {|_{9 \  5}^{12}}}-{{1 \over{p_{1}}}\ {|_{{10}\  6}^{12}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\  7}^{12}}}-{{1 \over{p_{1}}}\ {|_{{12}\  0}^{12}}}+{{1 \over{p_{1}}}\ {|_{{13}\  1}^{12}}}+{{1 \over{p_{1}}}\ {|_{{14}\  2}^{12}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\  3}^{12}}}-{{1 \over{p_{1}}}\ {|_{0 \ {13}}^{13}}}+{{1 \over{p_{1}}}\ {|_{1 \ {12}}^{13}}}-{{1 \over{p_{1}}}\ {|_{2 \ {15}}^{13}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \ {14}}^{13}}}-{{1 \over{p_{1}}}\ {|_{4 \  9}^{13}}}-{{1 \over{p_{1}}}\ {|_{5 \  8}^{13}}}-{{1 \over{p_{1}}}\ {|_{6 \ {11}}^{13}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \ {10}}^{13}}}+{{1 \over{p_{1}}}\ {|_{8 \  5}^{13}}}+{{1 \over{p_{1}}}\ {|_{9 \  4}^{13}}}-{{1 \over{p_{1}}}\ {|_{{10}\  7}^{13}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\  6}^{13}}}-{{1 \over{p_{1}}}\ {|_{{12}\  1}^{13}}}-{{1 \over{p_{1}}}\ {|_{{13}\  0}^{13}}}-{{1 \over{p_{1}}}\ {|_{{14}\  3}^{13}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\  2}^{13}}}-{{1 \over{p_{1}}}\ {|_{0 \ {14}}^{14}}}+{{1 \over{p_{1}}}\ {|_{1 \ {15}}^{14}}}+{{1 \over{p_{1}}}\ {|_{2 \ {12}}^{14}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \ {13}}^{14}}}-{{1 \over{p_{1}}}\ {|_{4 \ {10}}^{14}}}+{{1 \over{p_{1}}}\ {|_{5 \ {11}}^{14}}}-{{1 \over{p_{1}}}\ {|_{6 \  8}^{14}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \  9}^{14}}}+{{1 \over{p_{1}}}\ {|_{8 \  6}^{14}}}+{{1 \over{p_{1}}}\ {|_{9 \  7}^{14}}}+{{1 \over{p_{1}}}\ {|_{{10}\  4}^{14}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\  5}^{14}}}-{{1 \over{p_{1}}}\ {|_{{12}\  2}^{14}}}+{{1 \over{p_{1}}}\ {|_{{13}\  3}^{14}}}-{{1 \over{p_{1}}}\ {|_{{14}\  0}^{14}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\  1}^{14}}}-{{1 \over{p_{1}}}\ {|_{0 \ {15}}^{15}}}-{{1 \over{p_{1}}}\ {|_{1 \ {14}}^{15}}}+{{1 \over{p_{1}}}\ {|_{2 \ {13}}^{15}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{3 \ {12}}^{15}}}-{{1 \over{p_{1}}}\ {|_{4 \ {11}}^{15}}}-{{1 \over{p_{1}}}\ {|_{5 \ {10}}^{15}}}+{{1 \over{p_{1}}}\ {|_{6 \  9}^{15}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{7 \  8}^{15}}}+{{1 \over{p_{1}}}\ {|_{8 \  7}^{15}}}-{{1 \over{p_{1}}}\ {|_{9 \  6}^{15}}}+{{1 \over{p_{1}}}\ {|_{{10}\  5}^{15}}}+ 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{11}\  4}^{15}}}-{{1 \over{p_{1}}}\ {|_{{12}\  3}^{15}}}-{{1 \over{p_{1}}}\ {|_{{13}\  2}^{15}}}+{{1 \over{p_{1}}}\ {|_{{14}\  1}^{15}}}- 
\
\
\displaystyle
{{1 \over{p_{1}}}\ {|_{{15}\  0}^{15}}}
(26)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
test( (Ω,I) / (I,Y) = λ )

\label{eq27} \mbox{\rm true} (27)
Type: Boolean

Frobenius Condition

Like Octonion algebra Sedenion algebra also fails the Frobenius Condition!

Too slow to complete here:

  \begin{axiom}

  Χ := Y / λ ;

  Χr := (λ,I)/(I,Y)
  test(Χr = Χ )

  Χl := (I,λ)/(Y,I);
  --test( Χl = Χ )
  test( Χr = Χl )

  \end{axiom}

Perhaps this is not too surprising since like Octonion Seden algebra is non-associative (in fact also non-alternative). Nevertheless Sedenions are "Frobenius" in a more general sense just because there is a non-degenerate associative pairing.

i = Unit of the algebra

fricas
i:=ⅇ.1

\label{eq28}|_{0}(28)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
test
         i     /
         λ     =    Ω

\label{eq29} \mbox{\rm true} (29)
Type: Boolean

Handle

fricas
H:ℒ :=
         λ     /
         X     /
         Y

\label{eq30}\begin{array}{@{}l}
\displaystyle
-{{{16}\over{p_{1}}}\ {|_{0}^{0}}}+{{{12}\over{p_{1}}}\ {|_{1}^{1}}}+{{{12}\over{p_{1}}}\ {|_{2}^{2}}}+{{{12}\over{p_{1}}}\ {|_{3}^{3}}}+{{{12}\over{p_{1}}}\ {|_{4}^{4}}}+{{{12}\over{p_{1}}}\ {|_{5}^{5}}}+ 
\
\
\displaystyle
{{{12}\over{p_{1}}}\ {|_{6}^{6}}}+{{{12}\over{p_{1}}}\ {|_{7}^{7}}}+{{{12}\over{p_{1}}}\ {|_{8}^{8}}}+{{{12}\over{p_{1}}}\ {|_{9}^{9}}}+{{{12}\over{p_{1}}}\ {|_{10}^{10}}}+{{{12}\over{p_{1}}}\ {|_{11}^{11}}}+ 
\
\
\displaystyle
{{{12}\over{p_{1}}}\ {|_{12}^{12}}}+{{{12}\over{p_{1}}}\ {|_{1
3}^{13}}}+{{{12}\over{p_{1}}}\ {|_{14}^{14}}}+{{{12}\over{p_{1}}}\ {|_{15}^{15}}}
(30)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))

Definition 5

Co-unit
  i 
  U
  

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

\label{eq31}-{{p_{1}}\ {|_{\ }^{0}}}(31)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))

Y=U
ι  
fricas
test
        Y    /
        ι       = Ų

\label{eq32} \mbox{\rm true} (32)
Type: Boolean

For example:

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

\label{eq33}\left[{{q_{3}}= 1}, \:{{p_{1}}= 1}\right](33)
Type: List(Equation(Polynomial(Integer)))
fricas
Ų0:ℒ  :=eval(Ų,ex1)

\label{eq34}\begin{array}{@{}l}
\displaystyle
-{|_{\ }^{0 \  0}}+{|_{\ }^{1 \  1}}+{|_{\ }^{2 \  2}}+{|_{\ }^{3 \  3}}+{|_{\ }^{4 \  4}}+{|_{\ }^{5 \  5}}+{|_{\ }^{6 \  6}}+{|_{\ }^{7 \  7}}+ 
\
\
\displaystyle
{|_{\ }^{8 \  8}}+{|_{\ }^{9 \  9}}+{|_{\ }^{{10}\ {10}}}+{|_{\ }^{{11}\ {11}}}+{|_{\ }^{{12}\ {12}}}+{|_{\ }^{{13}\ {13}}}+{|_{\ }^{{14}\ {14}}}+ 
\
\
\displaystyle
{|_{\ }^{{15}\ {15}}}
(34)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
Ω0:ℒ  :=eval(Ω,ex1)$ℒ

\label{eq35}\begin{array}{@{}l}
\displaystyle
-{|_{0 \  0}}+{|_{1 \  1}}+{|_{2 \  2}}+{|_{3 \  3}}+{|_{4 \  4}}+{|_{5 \  5}}+{|_{6 \  6}}+{|_{7 \  7}}+{|_{8 \  8}}+ 
\
\
\displaystyle
{|_{9 \  9}}+{|_{{10}\ {10}}}+{|_{{11}\ {11}}}+{|_{{12}\ {12}}}+{|_{{13}\ {13}}}+{|_{{14}\ {14}}}+{|_{{15}\ {15}}}
(35)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
λ0:ℒ  :=eval(λ,ex1)$ℒ

\label{eq36}\begin{array}{@{}l}
\displaystyle
-{|_{0 \  0}^{0}}+{|_{1 \  1}^{0}}+{|_{2 \  2}^{0}}+{|_{3 \  3}^{0}}+{|_{4 \  4}^{0}}+{|_{5 \  5}^{0}}+{|_{6 \  6}^{0}}+ 
\
\
\displaystyle
{|_{7 \  7}^{0}}+{|_{8 \  8}^{0}}+{|_{9 \  9}^{0}}+{|_{{10}\ {1
0}}^{0}}+{|_{{11}\ {11}}^{0}}+{|_{{12}\ {12}}^{0}}+{|_{{13}\ {1
3}}^{0}}+ 
\
\
\displaystyle
{|_{{14}\ {14}}^{0}}+{|_{{15}\ {15}}^{0}}-{|_{0 \  1}^{1}}-{|_{1 \  0}^{1}}-{|_{2 \  3}^{1}}+{|_{3 \  2}^{1}}-{|_{4 \  5}^{1}}+ 
\
\
\displaystyle
{|_{5 \  4}^{1}}+{|_{6 \  7}^{1}}-{|_{7 \  6}^{1}}-{|_{8 \  9}^{1}}+{|_{9 \  8}^{1}}+{|_{{10}\ {11}}^{1}}-{|_{{11}\ {10}}^{1}}+ 
\
\
\displaystyle
{|_{{12}\ {13}}^{1}}-{|_{{13}\ {12}}^{1}}-{|_{{14}\ {15}}^{1}}+{|_{{15}\ {14}}^{1}}-{|_{0 \  2}^{2}}+{|_{1 \  3}^{2}}-{|_{2 \  0}^{2}}- 
\
\
\displaystyle
{|_{3 \  1}^{2}}-{|_{4 \  6}^{2}}-{|_{5 \  7}^{2}}+{|_{6 \  4}^{2}}+{|_{7 \  5}^{2}}-{|_{8 \ {10}}^{2}}-{|_{9 \ {11}}^{2}}+ \
\
\displaystyle
{|_{{10}\  8}^{2}}+{|_{{11}\  9}^{2}}+{|_{{12}\ {14}}^{2}}+{|_{{13}\ {15}}^{2}}-{|_{{14}\ {12}}^{2}}-{|_{{15}\ {13}}^{2}}- \
\
\displaystyle
{|_{0 \  3}^{3}}-{|_{1 \  2}^{3}}+{|_{2 \  1}^{3}}-{|_{3 \  0}^{3}}-{|_{4 \  7}^{3}}+{|_{5 \  6}^{3}}-{|_{6 \  5}^{3}}+{|_{7 \  4}^{3}}- 
\
\
\displaystyle
{|_{8 \ {11}}^{3}}+{|_{9 \ {10}}^{3}}-{|_{{10}\  9}^{3}}+{|_{{1
1}\  8}^{3}}+{|_{{12}\ {15}}^{3}}-{|_{{13}\ {14}}^{3}}+ 
\
\
\displaystyle
{|_{{14}\ {13}}^{3}}-{|_{{15}\ {12}}^{3}}-{|_{0 \  4}^{4}}+{|_{1 \  5}^{4}}+{|_{2 \  6}^{4}}+{|_{3 \  7}^{4}}-{|_{4 \  0}^{4}}- 
\
\
\displaystyle
{|_{5 \  1}^{4}}-{|_{6 \  2}^{4}}-{|_{7 \  3}^{4}}-{|_{8 \ {1
2}}^{4}}-{|_{9 \ {13}}^{4}}-{|_{{10}\ {14}}^{4}}-{|_{{11}\ {1
5}}^{4}}+ 
\
\
\displaystyle
{|_{{12}\  8}^{4}}+{|_{{13}\  9}^{4}}+{|_{{14}\ {10}}^{4}}+{|_{{15}\ {11}}^{4}}-{|_{0 \  5}^{5}}-{|_{1 \  4}^{5}}+{|_{2 \  7}^{5}}- 
\
\
\displaystyle
{|_{3 \  6}^{5}}+{|_{4 \  1}^{5}}-{|_{5 \  0}^{5}}+{|_{6 \  3}^{5}}-{|_{7 \  2}^{5}}-{|_{8 \ {13}}^{5}}+{|_{9 \ {12}}^{5}}- \
\
\displaystyle
{|_{{10}\ {15}}^{5}}+{|_{{11}\ {14}}^{5}}-{|_{{12}\  9}^{5}}+{|_{{13}\  8}^{5}}-{|_{{14}\ {11}}^{5}}+{|_{{15}\ {10}}^{5}}- 
\
\
\displaystyle
{|_{0 \  6}^{6}}-{|_{1 \  7}^{6}}-{|_{2 \  4}^{6}}+{|_{3 \  5}^{6}}+{|_{4 \  2}^{6}}-{|_{5 \  3}^{6}}-{|_{6 \  0}^{6}}+{|_{7 \  1}^{6}}- 
\
\
\displaystyle
{|_{8 \ {14}}^{6}}+{|_{9 \ {15}}^{6}}+{|_{{10}\ {12}}^{6}}-{|_{{11}\ {13}}^{6}}-{|_{{12}\ {10}}^{6}}+{|_{{13}\ {11}}^{6}}+ \
\
\displaystyle
{|_{{14}\  8}^{6}}-{|_{{15}\  9}^{6}}-{|_{0 \  7}^{7}}+{|_{1 \  6}^{7}}-{|_{2 \  5}^{7}}-{|_{3 \  4}^{7}}+{|_{4 \  3}^{7}}+ 
\
\
\displaystyle
{|_{5 \  2}^{7}}-{|_{6 \  1}^{7}}-{|_{7 \  0}^{7}}-{|_{8 \ {1
5}}^{7}}-{|_{9 \ {14}}^{7}}+{|_{{10}\ {13}}^{7}}+{|_{{11}\ {1
2}}^{7}}- 
\
\
\displaystyle
{|_{{12}\ {11}}^{7}}-{|_{{13}\ {10}}^{7}}+{|_{{14}\  9}^{7}}+{|_{{15}\  8}^{7}}-{|_{0 \  8}^{8}}+{|_{1 \  9}^{8}}+{|_{2 \ {1
0}}^{8}}+ 
\
\
\displaystyle
{|_{3 \ {11}}^{8}}+{|_{4 \ {12}}^{8}}+{|_{5 \ {13}}^{8}}+{|_{6 \ {14}}^{8}}+{|_{7 \ {15}}^{8}}-{|_{8 \  0}^{8}}-{|_{9 \  1}^{8}}- 
\
\
\displaystyle
{|_{{10}\  2}^{8}}-{|_{{11}\  3}^{8}}-{|_{{12}\  4}^{8}}-{|_{{1
3}\  5}^{8}}-{|_{{14}\  6}^{8}}-{|_{{15}\  7}^{8}}-{|_{0 \  9}^{9}}- 
\
\
\displaystyle
{|_{1 \  8}^{9}}+{|_{2 \ {11}}^{9}}-{|_{3 \ {10}}^{9}}+{|_{4 \ {13}}^{9}}-{|_{5 \ {12}}^{9}}-{|_{6 \ {15}}^{9}}+{|_{7 \ {1
4}}^{9}}+ 
\
\
\displaystyle
{|_{8 \  1}^{9}}-{|_{9 \  0}^{9}}+{|_{{10}\  3}^{9}}-{|_{{11}\  2}^{9}}+{|_{{12}\  5}^{9}}-{|_{{13}\  4}^{9}}-{|_{{14}\  7}^{9}}+ 
\
\
\displaystyle
{|_{{15}\  6}^{9}}-{|_{0 \ {10}}^{10}}-{|_{1 \ {11}}^{10}}-{|_{2 \  8}^{10}}+{|_{3 \  9}^{10}}+{|_{4 \ {14}}^{10}}+ 
\
\
\displaystyle
{|_{5 \ {15}}^{10}}-{|_{6 \ {12}}^{10}}-{|_{7 \ {13}}^{10}}+{|_{8 \  2}^{10}}-{|_{9 \  3}^{10}}-{|_{{10}\  0}^{10}}+ 
\
\
\displaystyle
{|_{{11}\  1}^{10}}+{|_{{12}\  6}^{10}}+{|_{{13}\  7}^{10}}-{|_{{14}\  4}^{10}}-{|_{{15}\  5}^{10}}-{|_{0 \ {11}}^{11}}+ 
\
\
\displaystyle
{|_{1 \ {10}}^{11}}-{|_{2 \  9}^{11}}-{|_{3 \  8}^{11}}+{|_{4 \ {15}}^{11}}-{|_{5 \ {14}}^{11}}+{|_{6 \ {13}}^{11}}- 
\
\
\displaystyle
{|_{7 \ {12}}^{11}}+{|_{8 \  3}^{11}}+{|_{9 \  2}^{11}}-{|_{{1
0}\  1}^{11}}-{|_{{11}\  0}^{11}}+{|_{{12}\  7}^{11}}- 
\
\
\displaystyle
{|_{{13}\  6}^{11}}+{|_{{14}\  5}^{11}}-{|_{{15}\  4}^{11}}-{|_{0 \ {12}}^{12}}-{|_{1 \ {13}}^{12}}-{|_{2 \ {14}}^{12}}- 
\
\
\displaystyle
{|_{3 \ {15}}^{12}}-{|_{4 \  8}^{12}}+{|_{5 \  9}^{12}}+{|_{6 \ {10}}^{12}}+{|_{7 \ {11}}^{12}}+{|_{8 \  4}^{12}}- 
\
\
\displaystyle
{|_{9 \  5}^{12}}-{|_{{10}\  6}^{12}}-{|_{{11}\  7}^{12}}-{|_{{12}\  0}^{12}}+{|_{{13}\  1}^{12}}+{|_{{14}\  2}^{12}}+ 
\
\
\displaystyle
{|_{{15}\  3}^{12}}-{|_{0 \ {13}}^{13}}+{|_{1 \ {12}}^{13}}-{|_{2 \ {15}}^{13}}+{|_{3 \ {14}}^{13}}-{|_{4 \  9}^{13}}- 
\
\
\displaystyle
{|_{5 \  8}^{13}}-{|_{6 \ {11}}^{13}}+{|_{7 \ {10}}^{13}}+{|_{8 \  5}^{13}}+{|_{9 \  4}^{13}}-{|_{{10}\  7}^{13}}+ 
\
\
\displaystyle
{|_{{11}\  6}^{13}}-{|_{{12}\  1}^{13}}-{|_{{13}\  0}^{13}}-{|_{{14}\  3}^{13}}+{|_{{15}\  2}^{13}}-{|_{0 \ {14}}^{14}}+ 
\
\
\displaystyle
{|_{1 \ {15}}^{14}}+{|_{2 \ {12}}^{14}}-{|_{3 \ {13}}^{14}}-{|_{4 \ {10}}^{14}}+{|_{5 \ {11}}^{14}}-{|_{6 \  8}^{14}}- 
\
\
\displaystyle
{|_{7 \  9}^{14}}+{|_{8 \  6}^{14}}+{|_{9 \  7}^{14}}+{|_{{10}\  4}^{14}}-{|_{{11}\  5}^{14}}-{|_{{12}\  2}^{14}}+ 
\
\
\displaystyle
{|_{{13}\  3}^{14}}-{|_{{14}\  0}^{14}}-{|_{{15}\  1}^{14}}-{|_{0 \ {15}}^{15}}-{|_{1 \ {14}}^{15}}+{|_{2 \ {13}}^{15}}+ 
\
\
\displaystyle
{|_{3 \ {12}}^{15}}-{|_{4 \ {11}}^{15}}-{|_{5 \ {10}}^{15}}+{|_{6 \  9}^{15}}-{|_{7 \  8}^{15}}+{|_{8 \  7}^{15}}- 
\
\
\displaystyle
{|_{9 \  6}^{15}}+{|_{{10}\  5}^{15}}+{|_{{11}\  4}^{15}}-{|_{{12}\  3}^{15}}-{|_{{13}\  2}^{15}}+{|_{{14}\  1}^{15}}- 
\
\
\displaystyle
{|_{{15}\  0}^{15}}
(36)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))
fricas
H0:ℒ :=eval(H,ex1)$ℒ

\label{eq37}\begin{array}{@{}l}
\displaystyle
-{{16}\ {|_{0}^{0}}}+{{12}\ {|_{1}^{1}}}+{{12}\ {|_{2}^{2}}}+{{12}\ {|_{3}^{3}}}+{{12}\ {|_{4}^{4}}}+{{12}\ {|_{5}^{5}}}+ \
\
\displaystyle
{{12}\ {|_{6}^{6}}}+{{12}\ {|_{7}^{7}}}+{{12}\ {|_{8}^{8}}}+{{1
2}\ {|_{9}^{9}}}+{{12}\ {|_{10}^{10}}}+{{12}\ {|_{11}^{11}}}+ 
\
\
\displaystyle
{{12}\ {|_{12}^{12}}}+{{12}\ {|_{13}^{13}}}+{{12}\ {|_{14}^{1
4}}}+{{12}\ {|_{15}^{15}}}
(37)
Type: LinearOperator(OrderedVariableList?([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]),Expression(Integer))




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