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last edited 13 years ago by Bill Page |
Edit detail for SandBox Quaternion Algebra is Frobenius in Many Ways revision 18 of 32
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Editor: Bill Page
Time: 2011/05/19 23:26:40 GMT-7 |
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| Note: bi-algebra conditions | ||
changed: -Cartan-Killing Trace Form -\begin{axiom} - -( I I ) / _ -( Y Λ ) / _ -( Y I ) / _ - V - -\end{axiom} - Multiplication is Associative \begin{axiom} test( ( I Y ) / _ ( Y ) = _ ( Y I ) / _ ( Y ) ) \end{axiom} added: The Cartan-Killing Trace \begin{axiom} Ú:= ( Y Λ ) / _ ( Y I ) / _ V Ù:= ( Λ Y ) / _ ( I Y ) / _ V test(Ù=Ú) \end{axiom} forms a non-degenerate associative scalar product for Y \begin{axiom} Ũ := r*Ù test ( Y I ) / Ũ = ( I Y ) / Ũ determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim) \end{axiom} General Solution changed: -equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g); -eq1:=equate(d1,I); -eq2:=equate(d2,I); equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq); eq1:=equate(d1=I); eq2:=equate(d2=I); added: Co-associativity \begin{axiom} test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) ) \end{axiom} changed: -Χ := H := changed: - ( I Y ) = Χ ( I Y ) = H changed: - ( Y I ) = Χ - -\end{axiom} - -Bi-algebra -\begin{axiom} -)set output tex off -)set output algebra on -\end{axiom} -\begin{axiom} -bi1:= _ ( Y I ) = H \end{axiom} Bi-algebra conditions \begin{axiom} ΦΦ:= _ changed: - ( Y Y ); - -bi2:= _ - Y / _ - λ / _ - Y / _ - λ; - -test(bi1=bi2) -\end{axiom} -\begin{axiom} -)set output algebra off -)set output tex on -\end{axiom} ( Y Y ) ; test( ΦΦ=H ) test( ΦΦ=H/H ) solve(equate(ΦΦ=H),Ξ(sb('p,[i]), i,1..#Ñ)) \end{axiom} changed: -H:𝐋 := Φ:𝐋 := changed: -H0:𝐋 :=eval(H,ex1)$𝐋 -\end{axiom} Φ0:𝐋 :=eval(Φ,ex1)$𝐋 \end{axiom}
Quaternion Algebra Is Frobenius In Many WaysLinear operators over a 4-dimensional vector space representing quaternion algebra
Ref:
- http://arxiv.org/abs/1103.5113
$S_3$-permuted Frobenius Algebras
Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)
- http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories
Joachim Kock
- http://en.wikipedia.org/wiki/Frobenius_algebra
We need the Axiom LinearOperator library.
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(1) -> )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
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-- summation macro Σ(x,i, n)==reduce(+, [x for i in n])
Type: Void
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-- list macro Ξ(f,i, n)==[f for i in n]
Type: Void
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-- subscript macro sb == subscript
Type: Void
𝐋 is the domain of 4-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.
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dim:=4\begin{equation} \label{eq1}4\end{equation}
Type: PositiveInteger?
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macro ℒ == List
Type: Void
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macro ℂ == CaleyDickson
Type: Void
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macro ℚ == Expression Integer
Type: Void
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𝐋 := ClosedLinearOperator(OVAR ['1,\begin{equation*} \label{eq2}\hbox{\axiomType{ClosedLinearOperator?}\ } \left({{\hbox{\axiomType{OrderedVariableList?}\ } \left({\left[ 1, \: i , \: j , \: k \right]?}\right)}, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}}\right)\end{equation*}'i, 'j, 'k], ℚ)
Type: Type
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𝐞:ℒ 𝐋 := basisOut()\begin{equation*} \label{eq3}\left[{|_{\ 1}}, \:{|_{\ i}}, \:{|_{\ j}}, \:{|_{\ k}}\right]?\end{equation*}
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𝐝:ℒ 𝐋 := basisIn()\begin{equation*} \label{eq4}\left[{|^{\ 1}}, \:{|^{\ i}}, \:{|^{\ j}}, \:{|^{\ k}}\right]?\end{equation*}
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I:𝐋:=[1] -- identity for composition\begin{equation} \label{eq5}{|_{\ 1}^{\ 1}}+{|_{\ i}^{\ i}}+{|_{\ j}^{\ j}}+{|_{\ k}^{\ k}}\end{equation}
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X:𝐋:=[2,\begin{equation} \label{eq6}\begin{array}{@{}l} \displaystyle {|_{\ 1 \ 1}^{\ 1 \ 1}}+{|_{\ i \ 1}^{\ 1 \ i}}+{|_{\ j \ 1}^{\ 1 \ j}}+{|_{\ k \ 1}^{\ 1 \ k}}+{|_{\ 1 \ i}^{\ i \ 1}}+ \ \ \displaystyle {|_{\ i \ i}^{\ i \ i}}+{|_{\ j \ i}^{\ i \ j}}+{|_{\ k \ i}^{\ i \ k}}+{|_{\ 1 \ j}^{\ j \ 1}}+{|_{\ i \ j}^{\ j \ i}}+{|_{\ j \ j}^{\ j \ j}}+ \ \ \displaystyle {|_{\ k \ j}^{\ j \ k}}+{|_{\ 1 \ k}^{\ k \ 1}}+{|_{\ i \ k}^{\ k \ i}}+{|_{\ j \ k}^{\ k \ j}}+{|_{\ k \ k}^{\ k \ k}} \end{array} \end{equation}1] -- twist
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V:𝐋:=ev(1) -- evaluation\begin{equation} \label{eq7}{|^{\ 1 \ 1}}+{|^{\ i \ i}}+{|^{\ j \ j}}+{|^{\ k \ k}}\end{equation}
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Λ:𝐋:=co(1) -- co-evaluation\begin{equation} \label{eq8}{|_{\ 1 \ 1}}+{|_{\ i \ i}}+{|_{\ j \ j}}+{|_{\ k \ k}}\end{equation}
Now generate structure constants for Quaternion 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 and co-quaternions can be specified by Caley-Dickson parameters (q0 = -1, q1 = -1)
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--q0:=sb('q,
\begin{equation}
\label{eq9}1\end{equation}
Type: PositiveInteger?
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q1:=sb('q,
\begin{equation}
\label{eq10}q_{1}\end{equation}
Type: Symbol
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--q1:=1 -- co-quaternion QQ := ℂ(ℂ(ℚ,'i, q0), 'j, q1);
Type: Type
Basis: Each B.i is a quaternion number
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B:ℒ QQ := map(x +-> hyper x,\begin{equation*} \label{eq11}\left[ 1, \: i , \: j , \:{ij}\right]?\end{equation*}1$SQMATRIX(dim, ℚ)::ℒ ℒ ℚ)
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-- Multiplication table: M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j,\begin{equation*} \label{eq12}\left[ \begin{array}{cccc} 1 & i & j &{ij} \ i & - 1 & -{ij}& j \ j &{ij}& -{q_{1}}&{-{q_{1}}i} \ {ij}& - j &{{q_{1}}i}& -{q_{1}} \end{array} \right]\end{equation*}i, 1..dim), j, 1..dim)
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-- Function to divide the matrix entries by a basis element S(y) == map(x +-> real real(x/y),M)
Type: Void
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-- The result is a nested list ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
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Compiling function S with type CaleyDickson(CaleyDickson(Expression(
Integer),Type: List(List(List(Expression(Integer))))
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-- structure constants form a tensor operator Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k,\begin{equation} \label{eq14}\begin{array}{@{}l} \displaystyle {|_{\ 1}^{\ 1 \ 1}}+{|_{\ i}^{\ 1 \ i}}+{{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}^{\ 1 \ j}}}+{|_{\ k}^{\ 1 \ k}}+{|_{\ i}^{\ i \ 1}}-{|_{\ 1}^{\ i \ i}}+ \ \ \displaystyle {|_{\ k}^{\ i \ j}}-{{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}^{\ i \ k}}}+{{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}^{\ j \ 1}}}-{|_{\ k}^{\ j \ i}}- \ \ \displaystyle {{q_{1}}\ {|_{\ 1}^{\ j \ j}}}+{{q_{1}}\ {|_{\ i}^{\ j \ k}}}+{|_{\ k}^{\ k \ 1}}+{{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}^{\ k \ i}}}- \ \ \displaystyle {{q_{1}}\ {|_{\ i}^{\ k \ j}}}-{{q_{1}}\ {|_{\ 1}^{\ k \ k}}} \end{array} \end{equation}i, 1..dim), j, 1..dim), k, 1..dim)
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arity Y\begin{equation} \label{eq15}\frac{{+}^{2}}{+}\end{equation}
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matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y,\begin{equation*} \label{eq16}\left[ \begin{array}{cccc} {|_{\ 1}}&{|_{\ i}}&{{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}}}&{|_{\ k}} \ {|_{\ i}}& -{|_{\ 1}}& -{|_{\ k}}&{{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}}} \ {{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}}}&{|_{\ k}}& -{{q_{1}}\ {|_{\ 1}}}& -{{q_{1}}\ {|_{\ i}}} \ {|_{\ k}}& -{{\frac{q_{1}}{\overline{q_{1}}}}\ {|_{\ j}}}&{{q_{1}}\ {|_{\ i}}}& -{{q_{1}}\ {|_{\ 1}}} \end{array} \right]\end{equation*}i, 1..dim), j, 1..dim)
Multiplication of arbitrary quaternions $a$ and $b$
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a:=Σ(sb('a,
\begin{equation}
\label{eq17}{{a_{1}}\ {|_{\ 1}}}+{{a_{2}}\ {|_{\ i}}}+{{a_{3}}\ {|_{\ j}}}+{{a_{4}}\ {|_{\ k}}}\end{equation}
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b:=Σ(sb('b,
\begin{equation}
\label{eq18}{{b_{1}}\ {|_{\ 1}}}+{{b_{2}}\ {|_{\ i}}}+{{b_{3}}\ {|_{\ j}}}+{{b_{4}}\ {|_{\ k}}}\end{equation}
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(a*b)/Y\begin{equation} \label{eq19}\begin{array}{@{}l} \displaystyle {{\left({{\left(-{{a_{4}}\ {b_{4}}}-{{a_{3}}\ {b_{3}}}\right)}\ {q_{1}}}-{{a_{2}}\ {b_{2}}}+{{a_{1}}\ {b_{1}}}\right)}\ {|_{\ 1}}}+ \ \ \displaystyle {{\left({{\left({{a_{3}}\ {b_{4}}}-{{a_{4}}\ {b_{3}}}\right)}\ {q_{1}}}+{{a_{1}}\ {b_{2}}}+{{a_{2}}\ {b_{1}}}\right)}\ {|_{\ i}}}+ \ \ \displaystyle {{\frac{{\left(-{{a_{2}}\ {b_{4}}}+{{a_{1}}\ {b_{3}}}+{{a_{4}}\ {b_{2}}}+{{a_{3}}\ {b_{1}}}\right)}\ {q_{1}}}{\overline{q_{1}}}}\ {|_{\ j}}}+ \ \ \displaystyle {{\left({{a_{1}}\ {b_{4}}}+{{a_{2}}\ {b_{3}}}-{{a_{3}}\ {b_{2}}}+{{a_{4}}\ {b_{1}}}\right)}\ {|_{\ k}}} \end{array} \end{equation}
Multiplication is Associative
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test( ( I Y ) / _ ( Y ) = _ ( Y I ) / _ ( Y ) )\begin{equation} \label{eq20} \mbox{\rm false} \end{equation}
Type: Boolean
A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$
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U:=Σ(Σ(script('u,
\begin{equation}
\label{eq21}\begin{array}{@{}l}
\displaystyle
{{u^{1, \: 1}}\ {|^{\ 1 \ 1}}}+{{u^{1, \: 2}}\ {|^{\ 1 \ i}}}+{{u^{1, \: 3}}\ {|^{\ 1 \ j}}}+{{u^{1, \: 4}}\ {|^{\ 1 \ k}}}+
\
\
\displaystyle
{{u^{2, \: 1}}\ {|^{\ i \ 1}}}+{{u^{2, \: 2}}\ {|^{\ i \ i}}}+{{u^{2, \: 3}}\ {|^{\ i \ j}}}+{{u^{2, \: 4}}\ {|^{\ i \ k}}}+{{u^{3, \: 1}}\ {|^{\ j \ 1}}}+
\
\
\displaystyle
{{u^{3, \: 2}}\ {|^{\ j \ i}}}+{{u^{3, \: 3}}\ {|^{\ j \ j}}}+{{u^{3, \: 4}}\ {|^{\ j \ k}}}+{{u^{4, \: 1}}\ {|^{\ k \ 1}}}+
\
\
\displaystyle
{{u^{4, \: 2}}\ {|^{\ k \ i}}}+{{u^{4, \: 3}}\ {|^{\ k \ j}}}+{{u^{4, \: 4}}\ {|^{\ k \ k}}}
\end{array}
\end{equation}
Definition 1
We say that the scalar product is associative if the tensor equation holds:
Y = Y
U U
In other words, if the (3,0)-tensor: $$ \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(4.82,0.92) \psbezier[linewidth=0.04]?(2.2,0.9)(2.2,0.1)(2.6,0.1)(2.6,0.9) \psline[linewidth=0.04cm]?(2.4,0.3)(2.4,-0.1) \psbezier[linewidth=0.04]?(2.4,-0.1)(2.4,-0.9)(3.0,-0.9)(3.0,-0.1) \psline[linewidth=0.04cm]?(3.0,-0.1)(3.0,0.9) \psbezier[linewidth=0.04]?(4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9) \psline[linewidth=0.04cm]?(4.6,0.3)(4.6,-0.1) \psbezier[linewidth=0.04]?(4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1) \psline[linewidth=0.04cm]?(4.0,-0.1)(4.0,0.9) \usefont{T1}{ptm}{m}{n} \rput(3.4948437,0.205){-} \psline[linewidth=0.04cm]?(0.6,-0.7)(0.6,0.9) \psbezier[linewidth=0.04]?(0.0,-0.1)(0.0,-0.9)(1.2,-0.9)(1.2,-0.1) \psline[linewidth=0.04cm]?(0.0,-0.1)(0.0,0.9) \psline[linewidth=0.04cm]?(1.2,-0.1)(1.2,0.9) \usefont{T1}{ptm}{m}{n} \rput(1.6948438,0.205){=} \end{pspicture} } $$
\begin{equation} \label{eq22} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \} \end{equation} (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.
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ω:𝐋 :=
( Y I ) /
U -
( I Y ) /
U
\begin{equation}
\label{eq23}\begin{array}{@{}l}
\displaystyle
{{\frac{{{u^{1, \: 3}}\ {\overline{q_{1}}}}-{{u^{1, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ 1 \ 1 \ j}}}+{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|^{\ 1 \ i \ 1}}}+
\
\
\displaystyle
{{\left({u^{2, \: 2}}+{u^{1, \: 1}}\right)}\ {|^{\ 1 \ i \ i}}}+{{\left({u^{2, \: 3}}-{u^{1, \: 4}}\right)}\ {|^{\ 1 \ i \ j}}}+
\
\
\displaystyle
{{\frac{{{u^{2, \: 4}}\ {\overline{q_{1}}}}+{{u^{1, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ 1 \ i \ k}}}+
\
\
\displaystyle
{{\frac{{\left({u^{3, \: 1}}-{u^{1, \: 3}}\right)}\ {q_{1}}}{\overline{q_{1}}}}\ {|^{\ 1 \ j \ 1}}}+
\
\
\displaystyle
{{\frac{{{u^{1, \: 4}}\ {\overline{q_{1}}}}+{{u^{3, \: 2}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ 1 \ j \ i}}}+
\
\
\displaystyle
{{\frac{{{u^{1, \: 1}}\ {q_{1}}\ {\overline{q_{1}}}}+{{u^{3, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ 1 \ j \ j}}}+ \
\
\displaystyle
{{\frac{-{{u^{1, \: 2}}\ {q_{1}}\ {\overline{q_{1}}}}+{{u^{3, \: 4}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ 1 \ j \ k}}}+
\
\
\displaystyle
{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\ {|^{\ 1 \ k \ 1}}}+{{\frac{{{u^{4, \: 2}}\ {\overline{q_{1}}}}-{{u^{1, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ 1 \ k \ i}}}+
\
\
\displaystyle
{{\left({{u^{1, \: 2}}\ {q_{1}}}+{u^{4, \: 3}}\right)}\ {|^{\ 1 \ k \ j}}}+{{\left({{u^{1, \: 1}}\ {q_{1}}}+{u^{4, \: 4}}\right)}\ {|^{\ 1 \ k \ k}}}+
\
\
\displaystyle
{{\frac{{{u^{2, \: 3}}\ {\overline{q_{1}}}}-{{u^{2, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ i \ 1 \ j}}}+{{\left(-{u^{2, \: 2}}-{u^{1, \: 1}}\right)}\ {|^{\ i \ i \ 1}}}+
\
\
\displaystyle
{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|^{\ i \ i \ i}}}+{{\left(-{u^{2, \: 4}}-{u^{1, \: 3}}\right)}\ {|^{\ i \ i \ j}}}+
\
\
\displaystyle
{{\frac{-{{u^{1, \: 4}}\ {\overline{q_{1}}}}+{{u^{2, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ i \ i \ k}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 1}}\ {\overline{q_{1}}}}-{{u^{2, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ i \ j \ 1}}}+{{\left({u^{4, \: 2}}+{u^{2, \: 4}}\right)}\ {|^{\ i \ j \ i}}}+
\
\
\displaystyle
{{\left({{u^{2, \: 1}}\ {q_{1}}}+{u^{4, \: 3}}\right)}\ {|^{\ i \ j \ j}}}+{{\left(-{{u^{2, \: 2}}\ {q_{1}}}+{u^{4, \: 4}}\right)}\ {|^{\ i \ j \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{2, \: 4}}\ {\overline{q_{1}}}}-{{u^{3, \: 1}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ i \ k \ 1}}}+
\
\
\displaystyle
{{\frac{{\left(-{u^{3, \: 2}}-{u^{2, \: 3}}\right)}\ {q_{1}}}{\overline{q_{1}}}}\ {|^{\ i \ k \ i}}}+
\
\
\displaystyle
{{\frac{{{u^{2, \: 2}}\ {q_{1}}\ {\overline{q_{1}}}}-{{u^{3, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ i \ k \ j}}}+ \
\
\displaystyle
{{\frac{{{u^{2, \: 1}}\ {q_{1}}\ {\overline{q_{1}}}}-{{u^{3, \: 4}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ i \ k \ k}}}+ \
\
\displaystyle
{{\frac{-{{u^{3, \: 1}}\ {\overline{q_{1}}}}+{{u^{3, \: 1}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ j \ 1 \ 1}}}+
\
\
\displaystyle
{{\frac{-{{u^{3, \: 2}}\ {\overline{q_{1}}}}+{{u^{3, \: 2}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ j \ 1 \ i}}}+
\
\
\displaystyle
{{\frac{-{{u^{3, \: 4}}\ {\overline{q_{1}}}}+{{u^{3, \: 4}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ j \ 1 \ k}}}+
\
\
\displaystyle
{{\left(-{u^{4, \: 1}}-{u^{3, \: 2}}\right)}\ {|^{\ j \ i \ 1}}}+{{\left(-{u^{4, \: 2}}+{u^{3, \: 1}}\right)}\ {|^{\ j \ i \ i}}}+
\
\
\displaystyle
{{\left(-{u^{4, \: 3}}-{u^{3, \: 4}}\right)}\ {|^{\ j \ i \ j}}}+
\
\
\displaystyle
{{\frac{-{{u^{4, \: 4}}\ {\overline{q_{1}}}}+{{u^{3, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ j \ i \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{1, \: 1}}\ {q_{1}}\ {\overline{q_{1}}}}-{{u^{3, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ j \ j \ 1}}}+
\
\
\displaystyle
{{\left(-{{u^{1, \: 2}}\ {q_{1}}}+{u^{3, \: 4}}\right)}\ {|^{\ j \ j \ i}}}+{{\left({u^{3, \: 1}}-{u^{1, \: 3}}\right)}\ {q_{1}}\ {|^{\ j \ j \ j}}}+
\
\
\displaystyle
{{\left(-{u^{3, \: 2}}-{u^{1, \: 4}}\right)}\ {q_{1}}\ {|^{\ j \ j \ k}}}+{{\left({{u^{2, \: 1}}\ {q_{1}}}-{u^{3, \: 4}}\right)}\ {|^{\ j \ k \ 1}}}+
\
\
\displaystyle
{{\frac{{{u^{2, \: 2}}\ {q_{1}}\ {\overline{q_{1}}}}-{{u^{3, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ j \ k \ i}}}+ \
\
\displaystyle
{{\left({u^{3, \: 2}}+{u^{2, \: 3}}\right)}\ {q_{1}}\ {|^{\ j \ k \ j}}}+{{\left({u^{3, \: 1}}+{u^{2, \: 4}}\right)}\ {q_{1}}\ {|^{\ j \ k \ k}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 3}}\ {\overline{q_{1}}}}-{{u^{4, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ k \ 1 \ j}}}+
\
\
\displaystyle
{{\frac{-{{u^{4, \: 2}}\ {\overline{q_{1}}}}+{{u^{3, \: 1}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ k \ i \ 1}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 1}}\ {\overline{q_{1}}}}+{{u^{3, \: 2}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ k \ i \ i}}}+
\
\
\displaystyle
{{\frac{-{{u^{4, \: 4}}\ {\overline{q_{1}}}}+{{u^{3, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ k \ i \ j}}}+
\
\
\displaystyle
{{\frac{{\left({u^{4, \: 3}}+{u^{3, \: 4}}\right)}\ {q_{1}}}{\overline{q_{1}}}}\ {|^{\ k \ i \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{2, \: 1}}\ {q_{1}}\ {\overline{q_{1}}}}-{{u^{4, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ k \ j \ 1}}}+
\
\
\displaystyle
{{\left(-{{u^{2, \: 2}}\ {q_{1}}}+{u^{4, \: 4}}\right)}\ {|^{\ k \ j \ i}}}+{{\left({u^{4, \: 1}}-{u^{2, \: 3}}\right)}\ {q_{1}}\ {|^{\ k \ j \ j}}}+
\
\
\displaystyle
{{\left(-{u^{4, \: 2}}-{u^{2, \: 4}}\right)}\ {q_{1}}\ {|^{\ k \ j \ k}}}+{{\left(-{{u^{1, \: 1}}\ {q_{1}}}-{u^{4, \: 4}}\right)}\ {|^{\ k \ k \ 1}}}+
\
\
\displaystyle
{{\frac{-{{u^{1, \: 2}}\ {q_{1}}\ {\overline{q_{1}}}}-{{u^{4, \: 3}}\ {q_{1}}}}{\overline{q_{1}}}}\ {|^{\ k \ k \ i}}}+
\
\
\displaystyle
{{\left({u^{4, \: 2}}-{u^{1, \: 3}}\right)}\ {q_{1}}\ {|^{\ k \ k \ j}}}+{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\ {q_{1}}\ {|^{\ k \ k \ k}}}
\end{array}
\end{equation}
Definition 2
An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.
The Cartan-Killing Trace
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Ú:=
( Y Λ ) / _
( Y I ) / _
V
\begin{equation}
\label{eq24}\begin{array}{@{}l}
\displaystyle
{{\frac{{3 \ {\overline{q_{1}}}}+{q_{1}}}{\overline{q_{1}}}}\ {|^{\ 1 \ 1}}}+{{\frac{-{3 \ {\overline{q_{1}}}}-{q_{1}}}{\overline{q_{1}}}}\ {|^{\ i \ i}}}+
\
\
\displaystyle
{{\frac{-{3 \ {q_{1}}\ {\overline{q_{1}}}}-{{q_{1}}^{2}}}{\overline{q_{1}}}}\ {|^{\ j \ j}}}+{{\frac{-{3 \ {q_{1}}\ {\overline{q_{1}}}}-{{q_{1}}^{2}}}{\overline{q_{1}}}}\ {|^{\ k \ k}}}
\end{array}
\end{equation}
fricas
Ù:=
( Λ Y ) / _
( I Y ) / _
V
\begin{equation}
\label{eq25}\begin{array}{@{}l}
\displaystyle
{{\frac{{3 \ {\overline{q_{1}}}}+{q_{1}}}{\overline{q_{1}}}}\ {|^{\ 1 \ 1}}}+{{\frac{-{3 \ {\overline{q_{1}}}}-{q_{1}}}{\overline{q_{1}}}}\ {|^{\ i \ i}}}+
\
\
\displaystyle
{{\frac{-{3 \ {q_{1}}\ {\overline{q_{1}}}}-{{q_{1}}^{2}}}{\overline{q_{1}}}}\ {|^{\ j \ j}}}+{{\frac{-{3 \ {q_{1}}\ {\overline{q_{1}}}}-{{q_{1}}^{2}}}{\overline{q_{1}}}}\ {|^{\ k \ k}}}
\end{array}
\end{equation}
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test(Ù=Ú)\begin{equation} \label{eq26} \mbox{\rm true} \end{equation}
Type: Boolean
forms a non-degenerate associative scalar product for Y
fricas
Ũ := r*Ù\begin{equation} \label{eq27}\begin{array}{@{}l} \displaystyle {{\frac{{3 \ r \ {\overline{q_{1}}}}+{{q_{1}}\ r}}{\overline{q_{1}}}}\ {|^{\ 1 \ 1}}}+{{\frac{-{3 \ r \ {\overline{q_{1}}}}-{{q_{1}}\ r}}{\overline{q_{1}}}}\ {|^{\ i \ i}}}+ \ \ \displaystyle {{\frac{-{3 \ {q_{1}}\ r \ {\overline{q_{1}}}}-{{{q_{1}}^{2}}\ r}}{\overline{q_{1}}}}\ {|^{\ j \ j}}}+ \ \ \displaystyle {{\frac{-{3 \ {q_{1}}\ r \ {\overline{q_{1}}}}-{{{q_{1}}^{2}}\ r}}{\overline{q_{1}}}}\ {|^{\ k \ k}}} \end{array} \end{equation}
fricas
test
( Y I ) /
Ũ =
( I Y ) /
Ũ
\begin{equation}
\label{eq28} \mbox{\rm false} \end{equation}
Type: Boolean
fricas
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ),\begin{equation} \label{eq29}\frac{-{{81}\ {{q_{1}}^{2}}\ {{r}^{4}}\ {{\overline{q_{1}}}^{4}}}-{{108}\ {{q_{1}}^{3}}\ {{r}^{4}}\ {{\overline{q_{1}}}^{3}}}-{{5 4}\ {{q_{1}}^{4}}\ {{r}^{4}}\ {{\overline{q_{1}}}^{2}}}-{{12}\ {{q_{1}}^{5}}\ {{r}^{4}}\ {\overline{q_{1}}}}-{{{q_{1}}^{6}}\ {{r}^{4}}}}{{\overline{q_{1}}}^{4}}\end{equation}j, 1..dim), i, 1..dim)
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)::ℒ Symbol);
Type: Matrix(Expression(Integer))
fricas
u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
Type: Matrix(Polynomial(Integer))
fricas
J::OutputForm * u::OutputForm = 0\begin{equation*} \label{eq30}\begin{array}{@{}l} \displaystyle {{\left[ \begin{array}{cccccccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{\frac{{\overline{q_{1}}}-{q_{1}}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 \ {q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 \ 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ {q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 &{\frac{{\overline{q_{1}}}-{q_{1}}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ - 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & - 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & - 1 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{-{\overline{q_{1}}}+{q_{1}}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{-{\overline{q_{1}}}+{q_{1}}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{-{\overline{q_{1}}}+{q_{1}}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & - 1 \ -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 \ 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}&{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{{\overline{q_{1}}}-{q_{1}}}{\overline{q_{1}}}}& 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & - 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 & 0 & 0 & - 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 & 0 &{\frac{q_{1}}{\overline{q_{1}}}}& 0 \ 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 \ 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 \ -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 \ 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{q_{1}}{\overline{q_{1}}}}& 0 \ 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 \ 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 \end{array} \right]}\ {\left[ \begin{array}{c} {u^{1, \: 1}} \ {u^{1, \: 2}} \ {u^{1, \: 3}} \ {u^{1, \: 4}} \ {u^{2, \: 1}} \ {u^{2, \: 2}} \ {u^{2, \: 3}} \ {u^{2, \: 4}} \ {u^{3, \: 1}} \ {u^{3, \: 2}} \ {u^{3, \: 3}} \ {u^{3, \: 4}} \ {u^{4, \: 1}} \ {u^{4, \: 2}} \ {u^{4, \: 3}} \ {u^{4, \: 4}} \end{array} \right]}}= \ \ \displaystyle 0 \end{array} \end{equation*}
Type: Equation(OutputForm?)
fricas
nrows(J),\begin{equation*} \label{eq31}\left[{64}, \:{16}\right]?\end{equation*}ncols(J)
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:
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Ñ:=nullSpace(J)\begin{equation*} \label{eq32}\begin{array}{@{}l} \displaystyle \left[ \left[ -{\frac{1}{q_{1}}}, \: 0, \: 0, \: 0, \: 0, \:{\frac{1}{q_{1}}}, \: 0, \: 0, \: 0, \: 0, \:{\frac{\overline{q_{1}}}{q_{1}}}, \: 0, \: 0, \: 0, \: 0, \right. \ \ \displaystyle \left.\: 1 \right] \right] \end{array} \end{equation*}
Type: List(Vector(Expression(Integer)))
fricas
ℰ:=map((x,\begin{equation*} \label{eq33}\begin{array}{@{}l} \displaystyle \left[{{u^{1, \: 1}}= -{\frac{p_{1}}{q_{1}}}}, \:{{u^{1, \: 2}}= 0}, \:{{u^{1, \: 3}}= 0}, \:{{u^{1, \: 4}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{2, \: 1}}= 0}, \:{{u^{2, \: 2}}={\frac{p_{1}}{q_{1}}}}, \:{{u^{2, \: 3}}= 0}, \:{{u^{2, \: 4}}= 0}, \:{{u^{3, \: 1}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{3, \: 2}}= 0}, \:{{u^{3, \: 3}}={\frac{{p_{1}}\ {\overline{q_{1}}}}{q_{1}}}}, \:{{u^{3, \: 4}}= 0}, \:{{u^{4, \: 1}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \:{{u^{4, \: 4}}={p_{1}}}\right] \end{array} \end{equation*}y)+->x=y, concat map(variables, ravel U), entries Σ(sb('p, [i])*Ñ.i, i, 1..#Ñ) )
Type: List(Equation(Expression(Integer)))
This defines a family of pre-Frobenius algebras:
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zero? eval(ω,\begin{equation} \label{eq34} \mbox{\rm true} \end{equation}ℰ)
Type: Boolean
In general the pairing is not symmetric!
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Ų:𝐋 := eval(U,\begin{equation} \label{eq35}-{{\frac{p_{1}}{q_{1}}}\ {|^{\ 1 \ 1}}}+{{\frac{p_{1}}{q_{1}}}\ {|^{\ i \ i}}}+{{\frac{{p_{1}}\ {\overline{q_{1}}}}{q_{1}}}\ {|^{\ j \ j}}}+{{p_{1}}\ {|^{\ k \ k}}}\end{equation}ℰ)
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matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų,\begin{equation*} \label{eq36}\left[ \begin{array}{cccc} -{\frac{p_{1}}{q_{1}}}& 0 & 0 & 0 \ 0 &{\frac{p_{1}}{q_{1}}}& 0 & 0 \ 0 & 0 &{\frac{{p_{1}}\ {\overline{q_{1}}}}{q_{1}}}& 0 \ 0 & 0 & 0 &{p_{1}} \end{array} \right]\end{equation*}i, 1..dim), j, 1..dim)
This is the most general form of the "dot product" of two quaternions
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(a*b)/Ų\begin{equation} \label{eq37}\frac{{{a_{3}}\ {b_{3}}\ {p_{1}}\ {\overline{q_{1}}}}+{{a_{4}}\ {b_{4}}\ {p_{1}}\ {q_{1}}}+{{\left({{a_{2}}\ {b_{2}}}-{{a_{1}}\ {b_{1}}}\right)}\ {p_{1}}}}{q_{1}}\end{equation}
fricas
(a*a)/Ų\begin{equation} \label{eq38}\frac{{{{a_{3}}^{2}}\ {p_{1}}\ {\overline{q_{1}}}}+{{{a_{4}}^{2}}\ {p_{1}}\ {q_{1}}}+{{\left({{a_{2}}^{2}}-{{a_{1}}^{2}}\right)}\ {p_{1}}}}{q_{1}}\end{equation}
The scalar product must be non-degenerate:
fricas
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų),\begin{equation} \label{eq39}-{\frac{{{p_{1}}^{4}}\ {\overline{q_{1}}}}{{q_{1}}^{3}}}\end{equation}j, 1..dim), i, 1..dim)
Type: Expression(Integer)
fricas
factor Ů\begin{equation} \label{eq40}-{\frac{{{p_{1}}^{4}}\ {\overline{q_{1}}}}{{q_{1}}^{3}}}\end{equation}
Type: Factored(Expression(Integer))
Definition 3
Co-pairing
Solve the [Snake Relation]? as a system of linear equations.
fricas
Ω:𝐋:=Σ(Σ(script('u,
\begin{equation}
\label{eq41}\begin{array}{@{}l}
\displaystyle
{{u_{1, \: 1}}\ {|_{\ 1 \ 1}}}+{{u_{1, \: 2}}\ {|_{\ 1 \ i}}}+{{u_{1, \: 3}}\ {|_{\ 1 \ j}}}+{{u_{1, \: 4}}\ {|_{\ 1 \ k}}}+
\
\
\displaystyle
{{u_{2, \: 1}}\ {|_{\ i \ 1}}}+{{u_{2, \: 2}}\ {|_{\ i \ i}}}+{{u_{2, \: 3}}\ {|_{\ i \ j}}}+{{u_{2, \: 4}}\ {|_{\ i \ k}}}+{{u_{3, \: 1}}\ {|_{\ j \ 1}}}+
\
\
\displaystyle
{{u_{3, \: 2}}\ {|_{\ j \ i}}}+{{u_{3, \: 3}}\ {|_{\ j \ j}}}+{{u_{3, \: 4}}\ {|_{\ j \ k}}}+{{u_{4, \: 1}}\ {|_{\ k \ 1}}}+
\
\
\displaystyle
{{u_{4, \: 2}}\ {|_{\ k \ i}}}+{{u_{4, \: 3}}\ {|_{\ k \ j}}}+{{u_{4, \: 4}}\ {|_{\ k \ k}}}
\end{array}
\end{equation}
fricas
d1:=(I*Ω)/(Ų*I);
fricas
d2:=(Ω*I)/(I*Ų);
fricas
equate(eq)==map((x,y)+->(x=y), ravel lhs eq, ravel rhs eq);
Type: Void
fricas
eq1:=equate(d1=I);
fricas
Compiling function equate with type Equation(ClosedLinearOperator(
OrderedVariableList([1,Type: List(Equation(Expression(Integer)))
fricas
eq2:=equate(d2=I);
Type: List(Equation(Expression(Integer)))
fricas
snake:=solve(concat(eq1,eq2), concat Ξ(Ξ(script('u, [[i, j]]), i, 1..dim), j, 1..dim));
Type: List(List(Equation(Expression(Integer))))
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if #snake ~= 1 then error "no solution"
Type: Void
fricas
Ω:=eval(Ω,\begin{equation} \label{eq42}\begin{array}{@{}l} \displaystyle -{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ 1}}}+{{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ i}}}+{{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ j}}}+ \ \ \displaystyle {{\frac{1}{p_{1}}}\ {|_{\ k \ k}}} \end{array} \end{equation}snake(1))
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matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j),\begin{equation*} \label{eq43}\left[ \begin{array}{cccc} -{\frac{q_{1}}{p_{1}}}& 0 & 0 & 0 \ 0 &{\frac{q_{1}}{p_{1}}}& 0 & 0 \ 0 & 0 &{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}& 0 \ 0 & 0 & 0 &{\frac{1}{p_{1}}} \end{array} \right]\end{equation*}i, 1..dim), j, 1..dim)
Check "dimension" and the snake relations.
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d:𝐋:=
Ω /
X /
Ų
\begin{equation}
\label{eq44}4\end{equation}
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test
( I Ω ) /
( Ų I ) = I
\begin{equation}
\label{eq45} \mbox{\rm true} \end{equation}
Type: Boolean
fricas
test
( Ω I ) /
( I Ų ) = I
\begin{equation}
\label{eq46} \mbox{\rm true} \end{equation}
Type: Boolean
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
fricas
W:=(Y,\begin{equation} \label{eq47}\begin{array}{@{}l} \displaystyle -{{\frac{p_{1}}{q_{1}}}\ {|^{\ 1 \ 1 \ 1}}}+{{\frac{p_{1}}{q_{1}}}\ {|^{\ 1 \ i \ i}}}+{{p_{1}}\ {|^{\ 1 \ j \ j}}}+ \ \ \displaystyle {{p_{1}}\ {|^{\ 1 \ k \ k}}}+{{\frac{p_{1}}{q_{1}}}\ {|^{\ i \ 1 \ i}}}+{{\frac{p_{1}}{q_{1}}}\ {|^{\ i \ i \ 1}}}+{{p_{1}}\ {|^{\ i \ j \ k}}}- \ \ \displaystyle {{p_{1}}\ {|^{\ i \ k \ j}}}+{{p_{1}}\ {|^{\ j \ 1 \ j}}}-{{p_{1}}\ {|^{\ j \ i \ k}}}+{{p_{1}}\ {|^{\ j \ j \ 1}}}+{{p_{1}}\ {|^{\ j \ k \ i}}}+ \ \ \displaystyle {{p_{1}}\ {|^{\ k \ 1 \ k}}}+{{p_{1}}\ {|^{\ k \ i \ j}}}-{{p_{1}}\ {|^{\ k \ j \ i}}}+{{p_{1}}\ {|^{\ k \ k \ 1}}} \end{array} \end{equation}I)/Ų
fricas
λ:=(Ω,\begin{equation} \label{eq48}\begin{array}{@{}l} \displaystyle -{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ 1}}}+{{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ i}^{\ 1}}}+ \ \ \displaystyle {{\frac{{q_{1}}^{2}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ j}^{\ 1}}}+{{\frac{1}{p_{1}}}\ {|_{\ k \ k}^{\ 1}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ i}^{\ i}}}- \ \ \displaystyle {{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ 1}^{\ i}}}-{{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ k}^{\ i}}}+{{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ j}^{\ i}}}- \ \ \displaystyle {{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ 1 \ j}^{\ j}}}+{{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ k}^{\ j}}}- \ \ \displaystyle {{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ 1}^{\ j}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ i}^{\ j}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ k}^{\ k}}}- \ \ \displaystyle {{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ j}^{\ k}}}+{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ i}^{\ k}}}- \ \ \displaystyle {{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ 1}^{\ k}}} \end{array} \end{equation}I, Ω)/(I, W, I)
fricas
test
( I Ω ) /
( Y I ) = λ
\begin{equation}
\label{eq49} \mbox{\rm true} \end{equation}
Type: Boolean
fricas
test
( Ω I ) /
( I Y ) = λ
\begin{equation}
\label{eq50} \mbox{\rm true} \end{equation}
Type: Boolean
Co-associativity
fricas
test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) )\begin{equation} \label{eq51} \mbox{\rm false} \end{equation}
Type: Boolean
Frobenius Condition
fricas
H :=
Y /
λ
\begin{equation}
\label{eq52}\begin{array}{@{}l}
\displaystyle
-{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ 1 \ 1}}}+{{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ i}^{\ 1 \ 1}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ j}^{\ 1 \ 1}}}+{{\frac{1}{p_{1}}}\ {|_{\ k \ k}^{\ 1 \ 1}}}-
\
\
\displaystyle
{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ i}^{\ 1 \ i}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ 1}^{\ 1 \ i}}}-
\
\
\displaystyle
{{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ k}^{\ 1 \ i}}}+{{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ j}^{\ 1 \ i}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ 1 \ j}^{\ 1 \ j}}}+{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ k}^{\ 1 \ j}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ 1}^{\ 1 \ j}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ i}^{\ 1 \ j}}}-
\
\
\displaystyle
{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ k}^{\ 1 \ k}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ j}^{\ 1 \ k}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ i}^{\ 1 \ k}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ 1}^{\ 1 \ k}}}-
\
\
\displaystyle
{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ i}^{\ i \ 1}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ 1}^{\ i \ 1}}}-
\
\
\displaystyle
{{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ k}^{\ i \ 1}}}+{{\frac{q_{1}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ j}^{\ i \ 1}}}+
\
\
\displaystyle
{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ i \ i}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ i \ i}^{\ i \ i}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ j}^{\ i \ i}}}-{{\frac{1}{p_{1}}}\ {|_{\ k \ k}^{\ i \ i}}}-
\
\
\displaystyle
{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ k}^{\ i \ j}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ j}^{\ i \ j}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ i}^{\ i \ j}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ 1}^{\ i \ j}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ 1 \ j}^{\ i \ k}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ k}^{\ i \ k}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ 1}^{\ i \ k}}}+{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ i}^{\ i \ k}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ 1 \ j}^{\ j \ 1}}}+{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ k}^{\ j \ 1}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ 1}^{\ j \ 1}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ i}^{\ j \ 1}}}+
\
\
\displaystyle
{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ k}^{\ j \ i}}}+{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ j}^{\ j \ i}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ i}^{\ j \ i}}}+{{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ 1}^{\ j \ i}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ j \ j}}}-{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ i \ i}^{\ j \ j}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ j}^{\ j \ j}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ k}^{\ j \ j}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ 1 \ i}^{\ j \ k}}}-{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ i \ 1}^{\ j \ k}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ k}^{\ j \ k}}}+{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ j}^{\ j \ k}}}-
\
\
\displaystyle
{{\frac{q_{1}}{p_{1}}}\ {|_{\ 1 \ k}^{\ k \ 1}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ j}^{\ k \ 1}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ i}^{\ k \ 1}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ 1}^{\ k \ 1}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ 1 \ j}^{\ k \ i}}}+{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i \ k}^{\ k \ i}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ 1}^{\ k \ i}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ i}^{\ k \ i}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ 1 \ i}^{\ k \ j}}}+{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ i \ 1}^{\ k \ j}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ j \ k}^{\ k \ j}}}-{{\frac{{q_{1}}^{2}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k \ j}^{\ k \ j}}}+
\
\
\displaystyle
{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ k \ k}}}-{{\frac{{q_{1}}^{2}}{p_{1}}}\ {|_{\ i \ i}^{\ k \ k}}}-
\
\
\displaystyle
{{\frac{{q_{1}}^{3}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j \ j}^{\ k \ k}}}-{{\frac{q_{1}}{p_{1}}}\ {|_{\ k \ k}^{\ k \ k}}}
\end{array}
\end{equation}
fricas
test
( λ I ) /
( I Y ) = H
\begin{equation}
\label{eq53} \mbox{\rm false} \end{equation}
Type: Boolean
fricas
test
( I λ ) /
( Y I ) = H
\begin{equation}
\label{eq54} \mbox{\rm false} \end{equation}
Type: Boolean
Bi-algebra conditions
fricas
ΦΦ:= _ ( λ λ ) / _ ( I X I ) / _ ( Y Y ) ;
fricas
test( ΦΦ=H )\begin{equation} \label{eq55} \mbox{\rm false} \end{equation}
Type: Boolean
fricas
test( ΦΦ=H/H )\begin{equation} \label{eq56} \mbox{\rm false} \end{equation}
Type: Boolean
fricas
solve(equate(ΦΦ=H),\begin{equation*} \label{eq57}\left[ \right]?\end{equation*}Ξ(sb('p, [i]), i, 1..#Ñ))
Type: List(List(Equation(Expression(Integer))))
i = Unit of the algebra
fricas
i:=𝐞.1\begin{equation} \label{eq58}|_{\ 1}\end{equation}
fricas
test
i /
λ = Ω
\begin{equation}
\label{eq59} \mbox{\rm false} \end{equation}
Type: Boolean
Handle
fricas
Φ:𝐋 :=
λ /
X /
Y
\begin{equation}
\label{eq60}\begin{array}{@{}l}
\displaystyle
{{\frac{-{3 \ {q_{1}}\ {{\overline{q_{1}}}^{2}}}-{{q_{1}}^{3}}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ 1}^{\ 1}}}+
\
\
\displaystyle
{{\frac{-{2 \ {q_{1}}\ {\overline{q_{1}}}}+{2 \ {{q_{1}}^{2}}}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ i}^{\ i}}}+
\
\
\displaystyle
{{\frac{{2 \ {{q_{1}}^{2}}\ {\overline{q_{1}}}}-{2 \ {{q_{1}}^{3}}}}{{p_{1}}\ {{\overline{q_{1}}}^{2}}}}\ {|_{\ j}^{\ j}}}+
\
\
\displaystyle
{{\frac{-{2 \ {q_{1}}\ {\overline{q_{1}}}}+{2 \ {{q_{1}}^{2}}}}{{p_{1}}\ {\overline{q_{1}}}}}\ {|_{\ k}^{\ k}}}
\end{array}
\end{equation}
Definition 5
i U
fricas
ι:𝐋:=
( i I ) /
Ų
\begin{equation}
\label{eq61}-{{\frac{p_{1}}{q_{1}}}\ {|^{\ 1}}}\end{equation}
Y=U ι
fricas
test
Y /
ι = Ų
\begin{equation}
\label{eq62} \mbox{\rm false} \end{equation}
Type: Boolean
For example:
fricas
ex1:=[q[1]=1,\begin{equation*} \label{eq63}\left[{{q_{1}}= 1}, \:{{p_{1}}= 1}, \:{{p_{2}}= 1}, \:{{p_{3}}= 1}, \:{{p_{4}}= 1}\right]?\end{equation*}p[1]=1, p[2]=1, p[3]=1, p[4]=1]
Type: List(Equation(Polynomial(Integer)))
fricas
Ų0:𝐋 :=eval(Ų,\begin{equation} \label{eq64}-{|^{\ 1 \ 1}}+{|^{\ i \ i}}+{|^{\ j \ j}}+{|^{\ k \ k}}\end{equation}ex1)
fricas
Ω0:𝐋 :=eval(Ω,\begin{equation} \label{eq65}-{|_{\ 1 \ 1}}+{|_{\ i \ i}}+{|_{\ j \ j}}+{|_{\ k \ k}}\end{equation}ex1)$𝐋
fricas
λ0:𝐋 :=eval(λ,\begin{equation} \label{eq66}\begin{array}{@{}l} \displaystyle -{|_{\ 1 \ 1}^{\ 1}}+{|_{\ i \ i}^{\ 1}}+{|_{\ j \ j}^{\ 1}}+{|_{\ k \ k}^{\ 1}}-{|_{\ 1 \ i}^{\ i}}-{|_{\ i \ 1}^{\ i}}- \ \ \displaystyle {|_{\ j \ k}^{\ i}}+{|_{\ k \ j}^{\ i}}-{|_{\ 1 \ j}^{\ j}}+{|_{\ i \ k}^{\ j}}-{|_{\ j \ 1}^{\ j}}-{|_{\ k \ i}^{\ j}}-{|_{\ 1 \ k}^{\ k}}- \ \ \displaystyle {|_{\ i \ j}^{\ k}}+{|_{\ j \ i}^{\ k}}-{|_{\ k \ 1}^{\ k}} \end{array} \end{equation}ex1)$𝐋
fricas
Φ0:𝐋 :=eval(Φ,\begin{equation} \label{eq67}-{4 \ {|_{\ 1}^{\ 1}}}\end{equation}ex1)$𝐋
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