Quaternion Algebra Is Frobenius In Many Ways
We need the Axiom LinearOperator library. fricas (1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY Use the following macros for convenient notation fricas -- summation macro Σ(x, Type: Void
fricas -- subscript and superscripts macro sb == subscript Type: Void
fricas macro sp == superscript 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. fricas dim:=4\begin{equation} \label{eq1}4\end{equation} Type: PositiveInteger?
fricas macro ℂ == CaleyDickson Type: Void
fricas macro ℚ == Expression Integer Type: Void
fricas 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*} Type: Type
fricas ⅇ:List ℒ := basisOut()\begin{equation*} \label{eq3}\left[{|_{\ 1}}, \:{|_{\ i}}, \:{|_{\ j}}, \:{|_{\ k}}\right]?\end{equation*} fricas ⅆ:List ℒ := basisIn()\begin{equation*} \label{eq4}\left[{|^{\ 1}}, \:{|^{\ i}}, \:{|^{\ j}}, \:{|^{\ k}}\right]?\end{equation*} fricas I:ℒ:=[1] -- identity for composition\begin{equation} \label{eq5}{|_{\ 1}^{\ 1}}+{|_{\ i}^{\ i}}+{|_{\ j}^{\ j}}+{|_{\ k}^{\ k}}\end{equation} fricas 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} fricas V:ℒ:=ev(1) -- evaluation\begin{equation} \label{eq7}{|^{\ 1 \ 1}}+{|^{\ i \ i}}+{|^{\ j \ j}}+{|^{\ k \ k}}\end{equation} fricas Λ:ℒ:=co(1) -- co-evaluation\begin{equation} \label{eq8}{|_{\ 1 \ 1}}+{|_{\ i \ i}}+{|_{\ j \ j}}+{|_{\ k \ k}}\end{equation} fricas equate(eq)==map((x, Type: Void
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 (i2, j2) fricas i2:=sp('i,
\begin{equation}
\label{eq9}i^{2}\end{equation}
Type: Symbol
fricas --i2:= -1 -- complex
j2:=sp('j,
\begin{equation}
\label{eq10}j^{2}\end{equation}
Type: Symbol
fricas --j2:= -1 -- quaternion k2:=-i2*j2; Type: Polynomial(Integer)
fricas QQ := ℂ(ℂ(ℚ, Type: Type
Basis: Each B.i is a quaternion number fricas B:List QQ := map(x +-> hyper x,\begin{equation*} \label{eq11}\left[ 1, \: i , \: j , \:{ij}\right]?\end{equation*} fricas -- Multiplication table: M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]\begin{equation*} \label{eq12}\left[ \begin{array}{cccc} 1 & i & j &{ij} \ i &{i^{2}}& -{ij}&{-{i^{2}}j} \ j &{ij}&{j^{2}}&{{j^{2}}i} \ {ij}&{{i^{2}}j}&{-{j^{2}}i}& -{{i^{2}}\ {j^{2}}} \end{array} \right]\end{equation*} fricas -- Function to divide the matrix entries by a basis element S(y) == map(x +-> real real(x/y), Type: Void
fricas -- The result is a nested list ѕ :=map(S, fricas Compiling function S with type CaleyDickson(CaleyDickson(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,\begin{equation} \label{eq14}\begin{array}{@{}l} \displaystyle {|_{\ 1}^{\ 1 \ 1}}+{{\frac{i^{2}}{\overline{i^{2}}}}\ {|_{\ i}^{\ 1 \ i}}}+{{\frac{j^{2}}{\overline{j^{2}}}}\ {|_{\ j}^{\ 1 \ j}}}+{|_{\ k}^{\ 1 \ k}}+ \ \ \displaystyle {{\frac{i^{2}}{\overline{i^{2}}}}\ {|_{\ i}^{\ i \ 1}}}+{{i^{2}}\ {|_{\ 1}^{\ i \ i}}}+{|_{\ k}^{\ i \ j}}+{{\frac{{i^{2}}\ {j^{2}}}{\overline{j^{2}}}}\ {|_{\ j}^{\ i \ k}}}+ \ \ \displaystyle {{\frac{j^{2}}{\overline{j^{2}}}}\ {|_{\ j}^{\ j \ 1}}}-{|_{\ k}^{\ j \ i}}+{{j^{2}}\ {|_{\ 1}^{\ j \ j}}}-{{\frac{{i^{2}}\ {j^{2}}}{\overline{i^{2}}}}\ {|_{\ i}^{\ j \ k}}}+ \ \ \displaystyle {|_{\ k}^{\ k \ 1}}-{{\frac{{i^{2}}\ {j^{2}}}{\overline{j^{2}}}}\ {|_{\ j}^{\ k \ i}}}+{{\frac{{i^{2}}\ {j^{2}}}{\overline{i^{2}}}}\ {|_{\ i}^{\ k \ j}}}- \ \ \displaystyle {{i^{2}}\ {j^{2}}\ {|_{\ 1}^{\ k \ k}}} \end{array} \end{equation} fricas arity Y\begin{equation} \label{eq15}\frac{{+}^{2}}{+}\end{equation} fricas matrix [[(ⅇ.i*ⅇ.j)/Y for i in 1..dim] for j in 1..dim]\begin{equation*} \label{eq16}\left[ \begin{array}{cccc} {|_{\ 1}}&{{\frac{i^{2}}{\overline{i^{2}}}}\ {|_{\ i}}}&{{\frac{j^{2}}{\overline{j^{2}}}}\ {|_{\ j}}}&{|_{\ k}} \ {{\frac{i^{2}}{\overline{i^{2}}}}\ {|_{\ i}}}&{{i^{2}}\ {|_{\ 1}}}& -{|_{\ k}}& -{{\frac{{i^{2}}\ {j^{2}}}{\overline{j^{2}}}}\ {|_{\ j}}} \ {{\frac{j^{2}}{\overline{j^{2}}}}\ {|_{\ j}}}&{|_{\ k}}&{{j^{2}}\ {|_{\ 1}}}&{{\frac{{i^{2}}\ {j^{2}}}{\overline{i^{2}}}}\ {|_{\ i}}} \ {|_{\ k}}&{{\frac{{i^{2}}\ {j^{2}}}{\overline{j^{2}}}}\ {|_{\ j}}}& -{{\frac{{i^{2}}\ {j^{2}}}{\overline{i^{2}}}}\ {|_{\ i}}}& -{{i^{2}}\ {j^{2}}\ {|_{\ 1}}} \end{array} \right]\end{equation*} Units fricas q:=ⅇ.1; i:=ⅇ.2; j:=ⅇ.3; k:=ⅇ.4; Multiplication of arbitrary quaternions $a$ and $b$ fricas a:=Σ(sb('a,
\begin{equation}
\label{eq17}{{a_{1}}\ {|_{\ 1}}}+{{a_{2}}\ {|_{\ i}}}+{{a_{3}}\ {|_{\ j}}}+{{a_{4}}\ {|_{\ k}}}\end{equation}
fricas b:=Σ(sb('b,
\begin{equation}
\label{eq18}{{b_{1}}\ {|_{\ 1}}}+{{b_{2}}\ {|_{\ i}}}+{{b_{3}}\ {|_{\ j}}}+{{b_{4}}\ {|_{\ k}}}\end{equation}
fricas (a*b)/Y\begin{equation} \label{eq19}\begin{array}{@{}l} \displaystyle {{\left(-{{i^{2}}\ {j^{2}}\ {a_{4}}\ {b_{4}}}+{{j^{2}}\ {a_{3}}\ {b_{3}}}+{{i^{2}}\ {a_{2}}\ {b_{2}}}+{{a_{1}}\ {b_{1}}}\right)}\ {|_{\ 1}}}+ \ \ \displaystyle {{\frac{-{{i^{2}}\ {j^{2}}\ {a_{3}}\ {b_{4}}}+{{i^{2}}\ {j^{2}}\ {a_{4}}\ {b_{3}}}+{{i^{2}}\ {a_{1}}\ {b_{2}}}+{{i^{2}}\ {a_{2}}\ {b_{1}}}}{\overline{i^{2}}}}\ {|_{\ i}}}+ \ \ \displaystyle {{\frac{{{i^{2}}\ {j^{2}}\ {a_{2}}\ {b_{4}}}+{{j^{2}}\ {a_{1}}\ {b_{3}}}-{{i^{2}}\ {j^{2}}\ {a_{4}}\ {b_{2}}}+{{j^{2}}\ {a_{3}}\ {b_{1}}}}{\overline{j^{2}}}}\ {|_{\ 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 fricas 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 $U = \{ u_{ij} \}$ fricas 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 1We 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. fricas ω:ℒ :=
( Y I ) /
U -
( I Y ) /
U
\begin{equation}
\label{eq23}\begin{array}{@{}l}
\displaystyle
{{\frac{{{u^{1, \: 2}}\ {\overline{i^{2}}}}-{{i^{2}}\ {u^{1, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ 1 \ 1 \ i}}}+
\
\
\displaystyle
{{\frac{{{u^{1, \: 3}}\ {\overline{j^{2}}}}-{{j^{2}}\ {u^{1, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ 1 \ 1 \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {u^{2, \: 1}}}-{{i^{2}}\ {u^{1, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ 1 \ i \ 1}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {u^{1, \: 1}}\ {\overline{i^{2}}}}+{{i^{2}}\ {u^{2, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ 1 \ i \ i}}}+
\
\
\displaystyle
{{\frac{-{{u^{1, \: 4}}\ {\overline{i^{2}}}}+{{i^{2}}\ {u^{2, \: 3}}}}{\overline{i^{2}}}}\ {|^{\ 1 \ i \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {u^{2, \: 4}}\ {\overline{j^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{1, \: 3}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ 1 \ i \ k}}}+
\
\
\displaystyle
{{\frac{{{j^{2}}\ {u^{3, \: 1}}}-{{j^{2}}\ {u^{1, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ 1 \ j \ 1}}}+
\
\
\displaystyle
{{\frac{{{u^{1, \: 4}}\ {\overline{j^{2}}}}+{{j^{2}}\ {u^{3, \: 2}}}}{\overline{j^{2}}}}\ {|^{\ 1 \ j \ i}}}+
\
\
\displaystyle
{{\frac{-{{j^{2}}\ {u^{1, \: 1}}\ {\overline{j^{2}}}}+{{j^{2}}\ {u^{3, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ 1 \ j \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {j^{2}}\ {u^{1, \: 2}}\ {\overline{j^{2}}}}+{{j^{2}}\ {u^{3, \: 4}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ 1 \ j \ k}}}+
\
\
\displaystyle
{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\ {|^{\ 1 \ k \ 1}}}+{{\frac{{{u^{4, \: 2}}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{1, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ 1 \ k \ i}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 3}}\ {\overline{i^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{1, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ 1 \ k \ j}}}+
\
\
\displaystyle
{{\left({u^{4, \: 4}}+{{i^{2}}\ {j^{2}}\ {u^{1, \: 1}}}\right)}\ {|^{\ 1 \ k \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{2, \: 1}}\ {\overline{i^{2}}}}+{{i^{2}}\ {u^{2, \: 1}}}}{\overline{i^{2}}}}\ {|^{\ i \ 1 \ 1}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {u^{2, \: 3}}\ {\overline{j^{2}}}}-{{j^{2}}\ {u^{2, \: 3}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ i \ 1 \ j}}}+
\
\
\displaystyle
{{\frac{-{{u^{2, \: 4}}\ {\overline{i^{2}}}}+{{i^{2}}\ {u^{2, \: 4}}}}{\overline{i^{2}}}}\ {|^{\ i \ 1 \ k}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {u^{1, \: 1}}\ {\overline{i^{2}}}}-{{i^{2}}\ {u^{2, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ i \ i \ 1}}}+
\
\
\displaystyle
{{\left(-{{i^{2}}\ {u^{2, \: 1}}}+{{i^{2}}\ {u^{1, \: 2}}}\right)}\ {|^{\ i \ i \ i}}}+{{\left(-{u^{2, \: 4}}+{{i^{2}}\ {u^{1, \: 3}}}\right)}\ {|^{\ i \ i \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {u^{1, \: 4}}\ {\overline{j^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{2, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ i \ i \ k}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 1}}\ {\overline{j^{2}}}}-{{j^{2}}\ {u^{2, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ i \ j \ 1}}}+{{\left({u^{4, \: 2}}+{u^{2, \: 4}}\right)}\ {|^{\ i \ j \ i}}}+
\
\
\displaystyle
{{\left({u^{4, \: 3}}-{{j^{2}}\ {u^{2, \: 1}}}\right)}\ {|^{\ i \ j \ j}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 4}}\ {\overline{i^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{2, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ i \ j \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{2, \: 4}}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{3, \: 1}}}}{\overline{j^{2}}}}\ {|^{\ i \ k \ 1}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {j^{2}}\ {u^{3, \: 2}}}+{{i^{2}}\ {j^{2}}\ {u^{2, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ i \ k \ i}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {j^{2}}\ {u^{2, \: 2}}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{3, \: 3}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ i \ k \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {j^{2}}\ {u^{2, \: 1}}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{3, \: 4}}}}{\overline{j^{2}}}}\ {|^{\ i \ k \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{3, \: 1}}\ {\overline{j^{2}}}}+{{j^{2}}\ {u^{3, \: 1}}}}{\overline{j^{2}}}}\ {|^{\ j \ 1 \ 1}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {u^{3, \: 2}}\ {\overline{j^{2}}}}+{{j^{2}}\ {u^{3, \: 2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ j \ 1 \ i}}}+
\
\
\displaystyle
{{\frac{-{{u^{3, \: 4}}\ {\overline{j^{2}}}}+{{j^{2}}\ {u^{3, \: 4}}}}{\overline{j^{2}}}}\ {|^{\ j \ 1 \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{4, \: 1}}\ {\overline{i^{2}}}}-{{i^{2}}\ {u^{3, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ j \ i \ 1}}}+
\
\
\displaystyle
{{\left(-{u^{4, \: 2}}-{{i^{2}}\ {u^{3, \: 1}}}\right)}\ {|^{\ j \ i \ i}}}+{{\left(-{u^{4, \: 3}}-{u^{3, \: 4}}\right)}\ {|^{\ j \ i \ j}}}+
\
\
\displaystyle
{{\frac{-{{u^{4, \: 4}}\ {\overline{j^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{3, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ j \ i \ k}}}+
\
\
\displaystyle
{{\frac{{{j^{2}}\ {u^{1, \: 1}}\ {\overline{j^{2}}}}-{{j^{2}}\ {u^{3, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ j \ j \ 1}}}+
\
\
\displaystyle
{{\left({u^{3, \: 4}}+{{j^{2}}\ {u^{1, \: 2}}}\right)}\ {|^{\ j \ j \ i}}}+{{\left(-{{j^{2}}\ {u^{3, \: 1}}}+{{j^{2}}\ {u^{1, \: 3}}}\right)}\ {|^{\ j \ j \ j}}}+
\
\
\displaystyle
{{\frac{{{j^{2}}\ {u^{1, \: 4}}\ {\overline{i^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{3, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ j \ j \ k}}}+
\
\
\displaystyle
{{\frac{-{{u^{3, \: 4}}\ {\overline{i^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{2, \: 1}}}}{\overline{i^{2}}}}\ {|^{\ j \ k \ 1}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {j^{2}}\ {u^{2, \: 2}}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{3, \: 3}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ j \ k \ i}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {j^{2}}\ {u^{3, \: 2}}}-{{i^{2}}\ {j^{2}}\ {u^{2, \: 3}}}}{\overline{i^{2}}}}\ {|^{\ j \ k \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {j^{2}}\ {u^{3, \: 1}}\ {\overline{i^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{2, \: 4}}}}{\overline{i^{2}}}}\ {|^{\ j \ k \ k}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 2}}\ {\overline{i^{2}}}}-{{i^{2}}\ {u^{4, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ k \ 1 \ i}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 3}}\ {\overline{j^{2}}}}-{{j^{2}}\ {u^{4, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ k \ 1 \ j}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {u^{4, \: 2}}\ {\overline{j^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{3, \: 1}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ k \ i \ 1}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {u^{4, \: 1}}\ {\overline{j^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{3, \: 2}}}}{\overline{j^{2}}}}\ {|^{\ k \ i \ i}}}+
\
\
\displaystyle
{{\frac{-{{u^{4, \: 4}}\ {\overline{j^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{3, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ k \ i \ j}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {j^{2}}\ {u^{4, \: 3}}}-{{i^{2}}\ {j^{2}}\ {u^{3, \: 4}}}}{\overline{j^{2}}}}\ {|^{\ k \ i \ k}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {j^{2}}\ {u^{2, \: 1}}\ {\overline{j^{2}}}}-{{j^{2}}\ {u^{4, \: 3}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ k \ j \ 1}}}+
\
\
\displaystyle
{{\frac{{{u^{4, \: 4}}\ {\overline{i^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{2, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ k \ j \ i}}}+
\
\
\displaystyle
{{\frac{-{{j^{2}}\ {u^{4, \: 1}}\ {\overline{i^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{2, \: 3}}}}{\overline{i^{2}}}}\ {|^{\ k \ j \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}\ {j^{2}}\ {u^{4, \: 2}}}+{{i^{2}}\ {j^{2}}\ {u^{2, \: 4}}}}{\overline{i^{2}}}}\ {|^{\ k \ j \ k}}}+
\
\
\displaystyle
{{\left(-{u^{4, \: 4}}-{{i^{2}}\ {j^{2}}\ {u^{1, \: 1}}}\right)}\ {|^{\ k \ k \ 1}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {j^{2}}\ {u^{1, \: 2}}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {u^{4, \: 3}}}}{\overline{j^{2}}}}\ {|^{\ k \ k \ i}}}+
\
\
\displaystyle
{{\frac{-{{i^{2}}\ {j^{2}}\ {u^{1, \: 3}}\ {\overline{i^{2}}}}-{{i^{2}}\ {j^{2}}\ {u^{4, \: 2}}}}{\overline{i^{2}}}}\ {|^{\ k \ k \ j}}}+
\
\
\displaystyle
{{\left({{i^{2}}\ {j^{2}}\ {u^{4, \: 1}}}-{{i^{2}}\ {j^{2}}\ {u^{1, \: 4}}}\right)}\ {|^{\ k \ k \ k}}}
\end{array}
\end{equation}
Definition 2An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?. The Cartan-Killing Trace fricas Ú:=
( Y Λ ) / _
( Y I ) / _
V
\begin{equation}
\label{eq24}\begin{array}{@{}l}
\displaystyle
{{\frac{{{\left({2 \ {\overline{i^{2}}}}+{i^{2}}\right)}\ {\overline{j^{2}}}}+{{j^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ 1 \ 1}}}+
\
\
\displaystyle
{{\frac{{{\left({2 \ {i^{2}}\ {\overline{i^{2}}}}+{{i^{2}}^{2}}\right)}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ i \ i}}}+
\
\
\displaystyle
{{\frac{{{\left({2 \ {j^{2}}\ {\overline{i^{2}}}}+{{i^{2}}\ {j^{2}}}\right)}\ {\overline{j^{2}}}}+{{{j^{2}}^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ j \ j}}}+
\
\
\displaystyle
{{\frac{{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {\overline{i^{2}}}}-{{{i^{2}}^{2}}\ {j^{2}}}\right)}\ {\overline{j^{2}}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ k \ k}}}
\end{array}
\end{equation}
fricas Ù:=
( Λ Y ) / _
( I Y ) / _
V
\begin{equation}
\label{eq25}\begin{array}{@{}l}
\displaystyle
{{\frac{{{\left({2 \ {\overline{i^{2}}}}+{i^{2}}\right)}\ {\overline{j^{2}}}}+{{j^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ 1 \ 1}}}+
\
\
\displaystyle
{{\frac{{{\left({2 \ {i^{2}}\ {\overline{i^{2}}}}+{{i^{2}}^{2}}\right)}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ i \ i}}}+
\
\
\displaystyle
{{\frac{{{\left({2 \ {j^{2}}\ {\overline{i^{2}}}}+{{i^{2}}\ {j^{2}}}\right)}\ {\overline{j^{2}}}}+{{{j^{2}}^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ j \ j}}}+
\
\
\displaystyle
{{\frac{{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {\overline{i^{2}}}}-{{{i^{2}}^{2}}\ {j^{2}}}\right)}\ {\overline{j^{2}}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ k \ k}}}
\end{array}
\end{equation}
fricas 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 := Ù\begin{equation} \label{eq27}\begin{array}{@{}l} \displaystyle {{\frac{{{\left({2 \ {\overline{i^{2}}}}+{i^{2}}\right)}\ {\overline{j^{2}}}}+{{j^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ 1 \ 1}}}+ \ \ \displaystyle {{\frac{{{\left({2 \ {i^{2}}\ {\overline{i^{2}}}}+{{i^{2}}^{2}}\right)}\ {\overline{j^{2}}}}+{{i^{2}}\ {j^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ i \ i}}}+ \ \ \displaystyle {{\frac{{{\left({2 \ {j^{2}}\ {\overline{i^{2}}}}+{{i^{2}}\ {j^{2}}}\right)}\ {\overline{j^{2}}}}+{{{j^{2}}^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ j \ j}}}+ \ \ \displaystyle {{\frac{{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {\overline{i^{2}}}}-{{{i^{2}}^{2}}\ {j^{2}}}\right)}\ {\overline{j^{2}}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {\overline{i^{2}}}}}{{\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|^{\ 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 [[retract((ⅇ.i * ⅇ.j)/Ũ) for j in 1..dim] for i in 1..dim]\begin{equation} \label{eq29}\frac{{{\left(-{{16}\ {{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{\overline{i^{2}}}^{4}}}-{{32}\ {{i^{2}}^{3}}\ {{j^{2}}^{2}}\ {{\overline{i^{2}}}^{3}}}-{{24}\ {{i^{2}}^{4}}\ {{j^{2}}^{2}}\ {{\overline{i^{2}}}^{2}}}-{8 \ {{i^{2}}^{5}}\ {{j^{2}}^{2}}\ {\overline{i^{2}}}}-{{{i^{2}}^{6}}\ {{j^{2}}^{2}}}\right)}\ {{\overline{j^{2}}}^{4}}}+{{\left(-{{32}\ {{i^{2}}^{2}}\ {{j^{2}}^{3}}\ {{\overline{i^{2}}}^{4}}}-{{48}\ {{i^{2}}^{3}}\ {{j^{2}}^{3}}\ {{\overline{i^{2}}}^{3}}}-{{24}\ {{i^{2}}^{4}}\ {{j^{2}}^{3}}\ {{\overline{i^{2}}}^{2}}}-{4 \ {{i^{2}}^{5}}\ {{j^{2}}^{3}}\ {\overline{i^{2}}}}\right)}\ {{\overline{j^{2}}}^{3}}}+{{\left(-{{24}\ {{i^{2}}^{2}}\ {{j^{2}}^{4}}\ {{\overline{i^{2}}}^{4}}}-{{24}\ {{i^{2}}^{3}}\ {{j^{2}}^{4}}\ {{\overline{i^{2}}}^{3}}}-{6 \ {{i^{2}}^{4}}\ {{j^{2}}^{4}}\ {{\overline{i^{2}}}^{2}}}\right)}\ {{\overline{j^{2}}}^{2}}}+{{\left(-{8 \ {{i^{2}}^{2}}\ {{j^{2}}^{5}}\ {{\overline{i^{2}}}^{4}}}-{4 \ {{i^{2}}^{3}}\ {{j^{2}}^{5}}\ {{\overline{i^{2}}}^{3}}}\right)}\ {\overline{j^{2}}}}-{{{i^{2}}^{2}}\ {{j^{2}}^{6}}\ {{\overline{i^{2}}}^{4}}}}{{{\overline{i^{2}}}^{4}}\ {{\overline{j^{2}}}^{4}}}\end{equation} 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 Type: Matrix(Expression(Integer))
fricas nrows(J),\begin{equation*} \label{eq30}\left[{64}, \:{16}\right]?\end{equation*} Type: Tuple(PositiveInteger?)
The matrix If the null space of the fricas Ñ:=nullSpace(J)\begin{equation*} \label{eq31}\begin{array}{@{}l} \displaystyle \left[ \left[ -{\frac{1}{{i^{2}}\ {j^{2}}}}, \: 0, \: 0, \: 0, \: 0, \: -{\frac{\overline{i^{2}}}{{i^{2}}\ {j^{2}}}}, \: 0, \: 0, \: 0, \: 0, \: -{\frac{\overline{j^{2}}}{{i^{2}}\ {j^{2}}}}, \right. \ \ \displaystyle \left.\: 0, \: 0, \: 0, \: 0, \: 1 \right] \right] \end{array} \end{equation*} Type: List(Vector(Expression(Integer)))
fricas ℰ:=map((x,\begin{equation*} \label{eq32}\begin{array}{@{}l} \displaystyle \left[{{u^{1, \: 1}}= -{\frac{p_{1}}{{i^{2}}\ {j^{2}}}}}, \:{{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}}\ {\overline{i^{2}}}}{{i^{2}}\ {j^{2}}}}}, \:{{u^{2, \: 3}}= 0}, \:{{u^{2, \: 4}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{3, \: 1}}= 0}, \:{{u^{3, \: 2}}= 0}, \:{{u^{3, \: 3}}= -{\frac{{p_{1}}\ {\overline{j^{2}}}}{{i^{2}}\ {j^{2}}}}}, \:{{u^{3, \: 4}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{4, \: 1}}= 0}, \:{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \:{{u^{4, \: 4}}={p_{1}}}\right] \end{array} \end{equation*} Type: List(Equation(Expression(Integer)))
This defines a family of pre-Frobenius algebras: fricas zero? eval(ω,\begin{equation} \label{eq33} \mbox{\rm true} \end{equation} Type: Boolean
fricas Ų:ℒ := eval(U,\begin{equation} \label{eq34}\begin{array}{@{}l} \displaystyle -{{\frac{p_{1}}{{i^{2}}\ {j^{2}}}}\ {|^{\ 1 \ 1}}}-{{\frac{{p_{1}}\ {\overline{i^{2}}}}{{i^{2}}\ {j^{2}}}}\ {|^{\ i \ i}}}- \ \ \displaystyle {{\frac{{p_{1}}\ {\overline{j^{2}}}}{{i^{2}}\ {j^{2}}}}\ {|^{\ j \ j}}}+{{p_{1}}\ {|^{\ k \ k}}} \end{array} \end{equation} Frobenius Form (co-unit) fricas d:=ε1*ⅆ.1+εi*ⅆ.2+εj*ⅆ.3+εk*ⅆ.4\begin{equation*} \label{eq35}{�� 1 \ {|^{\ 1}}}+{�� i \ {|^{\ i}}}+{�� j \ {|^{\ j}}}+{�� k \ {|^{\ k}}}\end{equation*} fricas equate(d=
( q I ) / _
Ų )
fricas Compiling function equate with type Equation(ClosedLinearOperator(
OrderedVariableList([1,Type: List(Equation(Expression(Integer)))
Express scalar product in terms of Frobenius form fricas Ξ:=solve(%, In general the pairing is not symmetric! fricas u1:=matrix [[retract((ⅇ.i ⅇ.j)/Ų) for i in 1..dim] for j in 1..dim]\begin{equation*} \label{eq37}\left[ \begin{array}{cccc} -{\frac{p_{1}}{{i^{2}}\ {j^{2}}}}& 0 & 0 & 0 \ 0 & -{\frac{{p_{1}}\ {\overline{i^{2}}}}{{i^{2}}\ {j^{2}}}}& 0 & 0 \ 0 & 0 & -{\frac{{p_{1}}\ {\overline{j^{2}}}}{{i^{2}}\ {j^{2}}}}& 0 \ 0 & 0 & 0 &{p_{1}} \end{array} \right]\end{equation*} Type: Matrix(Expression(Integer))
The scalar product must be non-degenerate: fricas Ů:=determinant u1\begin{equation} \label{eq38}-{\frac{{{p_{1}}^{4}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}{{{i^{2}}^{3}}\ {{j^{2}}^{3}}}}\end{equation} Type: Expression(Integer)
fricas factor(numer Ů)/factor(denom Ů)\begin{equation} \label{eq39}-{\frac{{{p_{1}}^{4}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}{{{i^{2}}^{3}}\ {{j^{2}}^{3}}}}\end{equation} Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,
Cartan-Killing is a special case fricas ck:=solve(equate(eval(Ũ, Frobenius scalar product of "vector" quaternions $a$ and $b$ fricas a:=sb('a,
\begin{equation}
\label{eq40}{{a_{1}}\ {|_{\ i}}}+{{a_{2}}\ {|_{\ j}}}\end{equation}
fricas b:=sb('b,
\begin{equation}
\label{eq41}{{b_{1}}\ {|_{\ i}}}+{{b_{2}}\ {|_{\ j}}}\end{equation}
fricas (a,\begin{equation} \label{eq42}\frac{-{{{a_{2}}^{2}}\ {p_{1}}\ {\overline{j^{2}}}}-{{{a_{1}}^{2}}\ {p_{1}}\ {\overline{i^{2}}}}}{{i^{2}}\ {j^{2}}}\end{equation} fricas (b,\begin{equation} \label{eq43}\frac{-{{{b_{2}}^{2}}\ {p_{1}}\ {\overline{j^{2}}}}-{{{b_{1}}^{2}}\ {p_{1}}\ {\overline{i^{2}}}}}{{i^{2}}\ {j^{2}}}\end{equation} fricas (a,\begin{equation} \label{eq44}\frac{-{{a_{2}}\ {b_{2}}\ {p_{1}}\ {\overline{j^{2}}}}-{{a_{1}}\ {b_{1}}\ {p_{1}}\ {\overline{i^{2}}}}}{{i^{2}}\ {j^{2}}}\end{equation} Definition 3Co-scalar product Solve the Snake Relation as a system of linear equations. fricas Ω:ℒ:=Σ(Σ(script('u,
\begin{equation}
\label{eq45}\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 ΩX:=Ω/X; fricas UXΩ:=(I*ΩX)/(Ų*I); fricas ΩXU:=(ΩX*I)/(I*Ų); fricas eq1:=equate(UXΩ=I); Type: List(Equation(Expression(Integer)))
fricas eq2:=equate(ΩXU=I); Type: List(Equation(Expression(Integer)))
fricas snake:=solve(concat(eq1, Type: List(List(Equation(Expression(Integer))))
fricas if #snake ~= 1 then error "no solution" Type: Void
fricas Ω:=eval(Ω,\begin{equation} \label{eq46}\begin{array}{@{}l} \displaystyle -{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ 1}}}-{{\frac{{i^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}}}\ {|_{\ i \ i}}}- \ \ \displaystyle {{\frac{{i^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{j^{2}}}}}\ {|_{\ j \ j}}}+{{\frac{1}{p_{1}}}\ {|_{\ k \ k}}} \end{array} \end{equation} fricas ΩX:=Ω/X; The common demoninator is $1/\sqrt{\mathring{U}}$ fricas squareFreePart factor denom Ů / squareFreePart factor numer Ů Check "dimension" and the snake relations. fricas O:ℒ:=
Ω /
Ų
\begin{equation}
\label{eq47}4\end{equation}
fricas test
( I ΩX ) /
( Ų I ) = I
\begin{equation}
\label{eq48} \mbox{\rm true} \end{equation}
Type: Boolean
fricas test
( ΩX I ) /
( I Ų ) = I
\begin{equation}
\label{eq49} \mbox{\rm true} \end{equation}
Type: Boolean
Cartan-Killing co-scalar fricas eval(Ω, Definition 4Co-algebra Compute the "three-point" function and use it to define co-multiplication. fricas W:=
(Y I) /
Ų
\begin{equation}
\label{eq50}\begin{array}{@{}l}
\displaystyle
-{{\frac{p_{1}}{{i^{2}}\ {j^{2}}}}\ {|^{\ 1 \ 1 \ 1}}}-{{\frac{p_{1}}{j^{2}}}\ {|^{\ 1 \ i \ i}}}-{{\frac{p_{1}}{i^{2}}}\ {|^{\ 1 \ j \ j}}}+
\
\
\displaystyle
{{p_{1}}\ {|^{\ 1 \ k \ k}}}-{{\frac{p_{1}}{j^{2}}}\ {|^{\ i \ 1 \ i}}}-{{\frac{p_{1}}{j^{2}}}\ {|^{\ i \ i \ 1}}}+{{p_{1}}\ {|^{\ i \ j \ k}}}-
\
\
\displaystyle
{{p_{1}}\ {|^{\ i \ k \ j}}}-{{\frac{p_{1}}{i^{2}}}\ {|^{\ j \ 1 \ j}}}-{{p_{1}}\ {|^{\ j \ i \ k}}}-{{\frac{p_{1}}{i^{2}}}\ {|^{\ j \ j \ 1}}}+
\
\
\displaystyle
{{p_{1}}\ {|^{\ j \ k \ i}}}+{{p_{1}}\ {|^{\ k \ 1 \ k}}}+{{p_{1}}\ {|^{\ k \ i \ j}}}-{{p_{1}}\ {|^{\ k \ j \ i}}}+{{p_{1}}\ {|^{\ k \ k \ 1}}}
\end{array}
\end{equation}
fricas λ:= ( ΩX I ΩX ) / ( I W I )\begin{equation} \label{eq51}\begin{array}{@{}l} \displaystyle -{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ 1}}}-{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ i}^{\ 1}}}- \ \ \displaystyle {{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ j}^{\ 1}}}+{{\frac{1}{p_{1}}}\ {|_{\ k \ k}^{\ 1}}}- \ \ \displaystyle {{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}}}\ {|_{\ 1 \ i}^{\ i}}}-{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}}}\ {|_{\ i \ 1}^{\ i}}}+ \ \ \displaystyle {{\frac{{i^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{j^{2}}}}}\ {|_{\ j \ k}^{\ i}}}-{{\frac{{i^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{j^{2}}}}}\ {|_{\ k \ j}^{\ i}}}- \ \ \displaystyle {{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{j^{2}}}}}\ {|_{\ 1 \ j}^{\ j}}}-{{\frac{{i^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}}}\ {|_{\ i \ k}^{\ j}}}- \ \ \displaystyle {{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{j^{2}}}}}\ {|_{\ j \ 1}^{\ j}}}+{{\frac{{i^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}}}\ {|_{\ k \ i}^{\ j}}}- \ \ \displaystyle {{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ k}^{\ k}}}-{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ j}^{\ k}}}+ \ \ \displaystyle {{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ i}^{\ k}}}-{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ k \ 1}^{\ k}}} \end{array} \end{equation} Cartan-Killing co-multiplication fricas eval(λ, fricas test
( I ΩX ) /
( Y I ) = λ
\begin{equation}
\label{eq52} \mbox{\rm true} \end{equation}
Type: Boolean
fricas test
( ΩX I ) /
( I Y ) = λ
\begin{equation}
\label{eq53} \mbox{\rm true} \end{equation}
Type: Boolean
Co-associativity fricas test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) )\begin{equation} \label{eq54} \mbox{\rm false} \end{equation} Type: Boolean
fricas test
q /
λ = ΩX
\begin{equation}
\label{eq55} \mbox{\rm false} \end{equation}
Type: Boolean
Frobenius Condition (fork) fricas H :=
Y /
λ
\begin{equation}
\label{eq56}\begin{array}{@{}l}
\displaystyle
-{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ 1 \ 1}}}-{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ i}^{\ 1 \ 1}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ j}^{\ 1 \ 1}}}+{{\frac{1}{p_{1}}}\ {|_{\ k \ k}^{\ 1 \ 1}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{3}}\ {j^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ 1 \ i}^{\ 1 \ i}}}-{{\frac{{{i^{2}}^{3}}\ {j^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ 1}^{\ 1 \ i}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ k}^{\ 1 \ i}}}-{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ j}^{\ 1 \ i}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ 1 \ j}^{\ 1 \ j}}}-{{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ k}^{\ 1 \ j}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ 1}^{\ 1 \ j}}}+{{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ i}^{\ 1 \ j}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ k}^{\ 1 \ k}}}-{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ j}^{\ 1 \ k}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ i}^{\ 1 \ k}}}-{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ k \ 1}^{\ 1 \ k}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{3}}\ {j^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ 1 \ i}^{\ i \ 1}}}-{{\frac{{{i^{2}}^{3}}\ {j^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ 1}^{\ i \ 1}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ k}^{\ i \ 1}}}-{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ j}^{\ i \ 1}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ i \ i}}}-{{\frac{{{i^{2}}^{3}}\ {j^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ i}^{\ i \ i}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ j}^{\ i \ i}}}+{{\frac{i^{2}}{p_{1}}}\ {|_{\ k \ k}^{\ i \ i}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ k}^{\ i \ j}}}-{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ j}^{\ i \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ i}^{\ i \ j}}}-{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ k \ 1}^{\ i \ j}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ 1 \ j}^{\ i \ k}}}-{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ k}^{\ i \ k}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ 1}^{\ i \ k}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ i}^{\ i \ k}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ 1 \ j}^{\ j \ 1}}}-{{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ k}^{\ j \ 1}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ 1}^{\ j \ 1}}}+{{\frac{{i^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ i}^{\ j \ 1}}}+
\
\
\displaystyle
{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ k}^{\ j \ i}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ j}^{\ j \ i}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ i}^{\ j \ i}}}+{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ k \ 1}^{\ j \ i}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {{j^{2}}^{2}}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ j \ j}}}-{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ i}^{\ j \ j}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ j}^{\ j \ j}}}+{{\frac{j^{2}}{p_{1}}}\ {|_{\ k \ k}^{\ j \ j}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{3}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ 1 \ i}^{\ j \ k}}}+{{\frac{{{i^{2}}^{3}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ 1}^{\ j \ k}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ k}^{\ j \ k}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ j}^{\ j \ k}}}-{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ 1 \ k}^{\ k \ 1}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ j}^{\ k \ 1}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ i}^{\ k \ 1}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ k \ 1}^{\ k \ 1}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ 1 \ j}^{\ k \ i}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ i \ k}^{\ k \ i}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ 1}^{\ k \ i}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ i}^{\ k \ i}}}-{{\frac{{{i^{2}}^{3}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ 1 \ i}^{\ k \ j}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{3}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ 1}^{\ k \ j}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ j \ k}^{\ k \ j}}}-
\
\
\displaystyle
{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k \ j}^{\ k \ j}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{2}}}{p_{1}}}\ {|_{\ 1 \ 1}^{\ k \ k}}}+
\
\
\displaystyle
{{\frac{{{i^{2}}^{3}}\ {{j^{2}}^{2}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}}}\ {|_{\ i \ i}^{\ k \ k}}}+{{\frac{{{i^{2}}^{2}}\ {{j^{2}}^{3}}}{{p_{1}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j \ j}^{\ k \ k}}}-
\
\
\displaystyle
{{\frac{{i^{2}}\ {j^{2}}}{p_{1}}}\ {|_{\ k \ k}^{\ k \ k}}}
\end{array}
\end{equation}
fricas test
( λ I ) /
( I Y ) = H
\begin{equation}
\label{eq57} \mbox{\rm false} \end{equation}
Type: Boolean
fricas test
( I λ ) /
( Y I ) = H
\begin{equation}
\label{eq58} \mbox{\rm false} \end{equation}
Type: Boolean
The Cartan-Killing form makes H of the Frobenius condition idempotent fricas test( eval(H, But it is not unique. E.g. other idempots fricas h1:=map(numer, Handle and handle element fricas Φ :=
λ /
Y
\begin{equation}
\label{eq59}\begin{array}{@{}l}
\displaystyle
{{\frac{{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {{\overline{i^{2}}}^{2}}}-{{{i^{2}}^{3}}\ {j^{2}}}\right)}\ {{\overline{j^{2}}}^{2}}}-{{i^{2}}\ {{j^{2}}^{3}}\ {{\overline{i^{2}}}^{2}}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ 1}^{\ 1}}}+
\
\
\displaystyle
{{\frac{-{2 \ {{i^{2}}^{3}}\ {j^{2}}\ {\overline{j^{2}}}}-{2 \ {{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {\overline{i^{2}}}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}\ {\overline{j^{2}}}}}\ {|_{\ i}^{\ i}}}+
\
\
\displaystyle
{{\frac{-{2 \ {{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {\overline{j^{2}}}}-{2 \ {i^{2}}\ {{j^{2}}^{3}}\ {\overline{i^{2}}}}}{{p_{1}}\ {\overline{i^{2}}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ j}^{\ j}}}+
\
\
\displaystyle
{{\frac{-{2 \ {i^{2}}\ {j^{2}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}-{2 \ {{i^{2}}^{2}}\ {{j^{2}}^{2}}}}{{p_{1}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}}\ {|_{\ k}^{\ k}}}
\end{array}
\end{equation}
fricas Φ1 := q /
Φ
\begin{equation}
\label{eq60}{\frac{{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {{\overline{i^{2}}}^{2}}}-{{{i^{2}}^{3}}\ {j^{2}}}\right)}\ {{\overline{j^{2}}}^{2}}}-{{i^{2}}\ {{j^{2}}^{3}}\ {{\overline{i^{2}}}^{2}}}}{{p_{1}}\ {{\overline{i^{2}}}^{2}}\ {{\overline{j^{2}}}^{2}}}}\ {|_{\ 1}}\end{equation}
The Cartan-Killing form makes Φ into the identity fricas test( eval(Φ, but it can be the identity in many ways. For example, fricas solve(equate(eval(Φ,\begin{equation*} \label{eq61}\left[ \right]?\end{equation*} Type: List(List(Equation(Expression(Integer))))
If handle is identity then fork is idempotent but the converse is not true fricas Φ1:=map(numer, Figure 12 fricas φφ:= _ ( Ω Ω ) / _ ( X I I ) / _ ( I X I ) / _ ( I I X ) / _ ( Y Y ); fricas φφ1:=map((x:ℚ):ℚ+->numer x,\begin{equation} \label{eq62}\begin{array}{@{}l} \displaystyle {{\left({{\left({2 \ {{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{\overline{i^{2}}}^{2}}}+{{{i^{2}}^{4}}\ {{j^{2}}^{2}}}\right)}\ {{\overline{j^{2}}}^{2}}}+{{{i^{2}}^{2}}\ {{j^{2}}^{4}}\ {{\overline{i^{2}}}^{2}}}\right)}\ {|_{\ 1 \ 1}}}+ \ \ \displaystyle {{\left({2 \ {{i^{2}}^{4}}\ {{j^{2}}^{2}}\ {\overline{j^{2}}}}+{2 \ {{i^{2}}^{3}}\ {{j^{2}}^{3}}\ {\overline{i^{2}}}}\right)}\ {|_{\ i \ i}}}+ \ \ \displaystyle {{\left({2 \ {{i^{2}}^{3}}\ {{j^{2}}^{3}}\ {\overline{j^{2}}}}+{2 \ {{i^{2}}^{2}}\ {{j^{2}}^{4}}\ {\overline{i^{2}}}}\right)}\ {|_{\ j \ j}}}+ \ \ \displaystyle {{\left(-{2 \ {i^{2}}\ {j^{2}}\ {\overline{i^{2}}}\ {\overline{j^{2}}}}-{2 \ {{i^{2}}^{2}}\ {{j^{2}}^{2}}}\right)}\ {|_{\ k \ k}}} \end{array} \end{equation} fricas φφ2:=denom(ravel(φφ).1)\begin{equation} \label{eq63}{{p_{1}}^{2}}\ {{\overline{i^{2}}}^{2}}\ {{\overline{j^{2}}}^{2}}\end{equation} Type: SparseMultivariatePolynomial?(Integer,
fricas test(φφ=(1/φφ2)*φφ1)\begin{equation} \label{eq64} \mbox{\rm false} \end{equation} Type: Boolean
For Cartan-Killing this is just the co-scalar fricas test(eval(φφ, Bi-algebra conditions fricas ΦΦ:= _ ( λ λ ) / _ ( I I X ) / _ ( I X I ) / _ ( I I X ) / _ ( Y Y ) ; fricas test((q,\begin{equation} \label{eq65} \mbox{\rm false} \end{equation} Type: Boolean
fricas test(eval(ΦΦ, Theorem 8.3 fricas u2:=map(retract,\begin{equation*} \label{eq66}\left[ \begin{array}{cccc} �� 1 & �� i & �� j & �� k \ �� i &{{i^{2}}\ �� 1}& �� k &{{i^{2}}\ �� j} \ �� j & - �� k &{{j^{2}}\ �� 1}& -{{j^{2}}\ �� i} \ �� k & -{{i^{2}}\ �� j}&{{j^{2}}\ �� i}& -{{i^{2}}\ {j^{2}}\ �� 1} \end{array} \right]\end{equation*} Type: Matrix(Expression(Integer))
fricas test(u2=transpose(u1))\begin{equation} \label{eq67} \mbox{\rm false} \end{equation} Type: Boolean
fricas Ů2 := -retract( k2*(q/d)^2 + j2*(i/d)^2 + i2*(j/d)^2 - (k/d)^2 )^2\begin{equation*} \label{eq68}\begin{array}{@{}l} \displaystyle -{{�� k}^{4}}+{{\left({2 \ {i^{2}}\ {{�� j}^{2}}}+{2 \ {j^{2}}\ {{�� i}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {{�� 1}^{2}}}\right)}\ {{�� k}^{2}}}-{{{i^{2}}^{2}}\ {{�� j}^{4}}}+ \ \ \displaystyle {{\left(-{2 \ {i^{2}}\ {j^{2}}\ {{�� i}^{2}}}+{2 \ {{i^{2}}^{2}}\ {j^{2}}\ {{�� 1}^{2}}}\right)}\ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ {{�� i}^{4}}}+{2 \ {i^{2}}\ {{j^{2}}^{2}}\ {{�� 1}^{2}}\ {{�� i}^{2}}}- \ \ \displaystyle {{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{�� 1}^{4}}} \end{array} \end{equation*} Type: Expression(Integer)
fricas factor(numer Ů2)/factor(denom Ů2)\begin{equation*} \label{eq69}-{{\left({{�� k}^{2}}-{{i^{2}}\ {{�� j}^{2}}}-{{j^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� 1}^{2}}}\right)}^{2}}\end{equation*} Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,
fricas test(Ů=Ů2)\begin{equation} \label{eq70} \mbox{\rm false} \end{equation} Type: Boolean
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