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	SandBox Quaternion Algebra is Frobenius in Many Ways
	↑	→ Snake Relation

SandBoxPauliAlgebra

Edit detail for SandBoxPauliAlgebra revision 7 of 7

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Editor: Bill Page Time: 2011/06/03 19:34:22 GMT-7
Note: general case

changed:
-Co-associativity
-\begin{axiom}
-test(
-  (  λ  ) / _
-  ( I λ ) = _
-  (  λ  ) / _
-  ( λ I ) )
-\end{axiom}
-
-\begin{axiom}
\begin{axiom}

The Pauli Algebra Cl(3) Is Frobenius In Many Ways

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

Ref:

http://arxiv.org/abs/1103.5113
-permuted Frobenius Algebras

Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)
http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories

Joachim Kock
http://en.wikipedia.org/wiki/Frobenius_algebra
http://en.wikipedia.org/wiki/Pauli_matrices
http://en.wikipedia.org/wiki/Clifford_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 and superscripts
macro sb == subscript

Type: Void

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macro sp == superscript

Type: Void

𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

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dim:=8

$\label{eq1}8$

(1)

Type: PositiveInteger?

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macro ℒ == List

Type: Void

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macro ℂ == CaleyDickson

Type: Void

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macro ℚ == Expression Integer

Type: Void

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𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ)

$\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } \left({{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[ 1, \: i , \: j , \: k , \: ij , \: ik , \: jk , \: ijk \right]}\right)}, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}}\right)$

(2)

Type: Type

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𝐞:ℒ 𝐋      := basisOut()

$\label{eq3}\left[{|_{\ 1}}, \:{|_{\ i}}, \:{|_{\ j}}, \:{|_{\ k}}, \:{|_{\ ij}}, \:{|_{\ ik}}, \:{|_{\ jk}}, \:{|_{\ ijk}}\right]$

(3)

Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))

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𝐝:ℒ 𝐋      := basisIn()

$\label{eq4}\left[{|^{\ 1}}, \:{|^{\ i}}, \:{|^{\ j}}, \:{|^{\ k}}, \:{|^{\ ij}}, \:{|^{\ ik}}, \:{|^{\ jk}}, \:{|^{\ ijk}}\right]$

(4)

Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))

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

$\label{eq5}{|_{\ 1}^{\ 1}}+{|_{\ i}^{\ i}}+{|_{\ j}^{\ j}}+{|_{\ k}^{\ k}}+{|_{\ ij}^{\ ij}}+{|_{\ ik}^{\ ik}}+{|_{\ jk}^{\ jk}}+{|_{\ ijk}^{\ ijk}}$

(5)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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X:𝐋:=[2,1] -- twist

$\label{eq6}\begin{array}{@{}l} \displaystyle {|_{\ 1 \ 1}^{\ 1 \ 1}}+{|_{\ i \ 1}^{\ 1 \ i}}+{|_{\ j \ 1}^{\ 1 \ j}}+{|_{\ k \ 1}^{\ 1 \ k}}+{|_{\ ij \ 1}^{\ 1 \ ij}}+ \ \ \displaystyle {|_{\ ik \ 1}^{\ 1 \ ik}}+{|_{\ jk \ 1}^{\ 1 \ jk}}+{|_{\ ijk \ 1}^{\ 1 \ ijk}}+{|_{\ 1 \ i}^{\ i \ 1}}+{|_{\ i \ i}^{\ i \ i}}+ \ \ \displaystyle {|_{\ j \ i}^{\ i \ j}}+{|_{\ k \ i}^{\ i \ k}}+{|_{\ ij \ i}^{\ i \ ij}}+{|_{\ ik \ i}^{\ i \ ik}}+{|_{\ jk \ i}^{\ i \ jk}}+ \ \ \displaystyle {|_{\ ijk \ i}^{\ i \ ijk}}+{|_{\ 1 \ j}^{\ j \ 1}}+{|_{\ i \ j}^{\ j \ i}}+{|_{\ j \ j}^{\ j \ j}}+{|_{\ k \ j}^{\ j \ k}}+ \ \ \displaystyle {|_{\ ij \ j}^{\ j \ ij}}+{|_{\ ik \ j}^{\ j \ ik}}+{|_{\ jk \ j}^{\ j \ jk}}+{|_{\ ijk \ j}^{\ j \ ijk}}+ \ \ \displaystyle {|_{\ 1 \ k}^{\ k \ 1}}+{|_{\ i \ k}^{\ k \ i}}+{|_{\ j \ k}^{\ k \ j}}+{|_{\ k \ k}^{\ k \ k}}+{|_{\ ij \ k}^{\ k \ ij}}+ \ \ \displaystyle {|_{\ ik \ k}^{\ k \ ik}}+{|_{\ jk \ k}^{\ k \ jk}}+{|_{\ ijk \ k}^{\ k \ ijk}}+{|_{\ 1 \ ij}^{\ ij \ 1}}+ \ \ \displaystyle {|_{\ i \ ij}^{\ ij \ i}}+{|_{\ j \ ij}^{\ ij \ j}}+{|_{\ k \ ij}^{\ ij \ k}}+{|_{\ ij \ ij}^{\ ij \ ij}}+ \ \ \displaystyle {|_{\ ik \ ij}^{\ ij \ ik}}+{|_{\ jk \ ij}^{\ ij \ jk}}+{|_{\ ijk \ ij}^{\ ij \ ijk}}+{|_{\ 1 \ ik}^{\ ik \ 1}}+ \ \ \displaystyle {|_{\ i \ ik}^{\ ik \ i}}+{|_{\ j \ ik}^{\ ik \ j}}+{|_{\ k \ ik}^{\ ik \ k}}+{|_{\ ij \ ik}^{\ ik \ ij}}+ \ \ \displaystyle {|_{\ ik \ ik}^{\ ik \ ik}}+{|_{\ jk \ ik}^{\ ik \ jk}}+{|_{\ ijk \ ik}^{\ ik \ ijk}}+{|_{\ 1 \ jk}^{\ jk \ 1}}+ \ \ \displaystyle {|_{\ i \ jk}^{\ jk \ i}}+{|_{\ j \ jk}^{\ jk \ j}}+{|_{\ k \ jk}^{\ jk \ k}}+{|_{\ ij \ jk}^{\ jk \ ij}}+ \ \ \displaystyle {|_{\ ik \ jk}^{\ jk \ ik}}+{|_{\ jk \ jk}^{\ jk \ jk}}+{|_{\ ijk \ jk}^{\ jk \ ijk}}+ \ \ \displaystyle {|_{\ 1 \ ijk}^{\ ijk \ 1}}+{|_{\ i \ ijk}^{\ ijk \ i}}+{|_{\ j \ ijk}^{\ ijk \ j}}+{|_{\ k \ ijk}^{\ ijk \ k}}+ \ \ \displaystyle {|_{\ ij \ ijk}^{\ ijk \ ij}}+{|_{\ ik \ ijk}^{\ ijk \ ik}}+{|_{\ jk \ ijk}^{\ ijk \ jk}}+ \ \ \displaystyle {|_{\ ijk \ ijk}^{\ ijk \ ijk}}$

(6)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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V:𝐋:=ev(1) -- evaluation

$\label{eq7}{|^{\ 1 \ 1}}+{|^{\ i \ i}}+{|^{\ j \ j}}+{|^{\ k \ k}}+{|^{\ ij \ ij}}+{|^{\ ik \ ik}}+{|^{\ jk \ jk}}+{|^{\ ijk \ ijk}}$

(7)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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Λ:𝐋:=co(1) -- co-evaluation

$\label{eq8}{|_{\ 1 \ 1}}+{|_{\ i \ i}}+{|_{\ j \ j}}+{|_{\ k \ k}}+{|_{\ ij \ ij}}+{|_{\ ik \ ik}}+{|_{\ jk \ jk}}+{|_{\ ijk \ ijk}}$

(8)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);

Type: Void

Now generate structure constants for Pauli 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.

The Pauli Algebra as Cl(3)

Basis: Each B.i is a Clifford number

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q0:=sp('i,[2])

$\label{eq9}i^{2}$

(9)

Type: Symbol

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q1:=sp('j,[2])

$\label{eq10}j^{2}$

(10)

Type: Symbol

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q2:=sp('k,[2])

$\label{eq11}k^{2}$

(11)

Type: Symbol

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QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])

$\label{eq12}\hbox{\axiomType{CliffordAlgebra}\ } \left({3, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\left[ \begin{array}{ccc} {i^{2}}& 0 & 0 \ 0 &{j^{2}}& 0 \ 0 & 0 &{k^{2}}$

(12)

Type: Type

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B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[3]),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])]

$\label{eq13}\left[ 1, \:{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{{e_{1}}\ {e_{2}}}, \:{{e_{1}}\ {e_{3}}}, \:{{e_{2}}\ {e_{3}}}, \:{{e_{1}}\ {e_{2}}\ {e_{3}}}\right]$

(13)

Type: List(CliffordAlgebra?(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))

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

$\label{eq14}\left[ \begin{array}{cccccccc} 1 &{e_{1}}&{e_{2}}&{e_{3}}&{{e_{1}}\ {e_{2}}}&{{e_{1}}\ {e_{3}}}&{{e_{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}} \ {e_{1}}&{i^{2}}& -{{e_{1}}\ {e_{2}}}& -{{e_{1}}\ {e_{3}}}& -{{i^{2}}\ {e_{2}}}& -{{i^{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}} \ {e_{2}}&{{e_{1}}\ {e_{2}}}&{j^{2}}& -{{e_{2}}\ {e_{3}}}&{{j^{2}}\ {e_{1}}}& -{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}} \ {e_{3}}&{{e_{1}}\ {e_{3}}}&{{e_{2}}\ {e_{3}}}&{k^{2}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{k^{2}}\ {e_{1}}}&{{k^{2}}\ {e_{2}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}} \ {{e_{1}}\ {e_{2}}}&{{i^{2}}\ {e_{2}}}& -{{j^{2}}\ {e_{1}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{i^{2}}\ {j^{2}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}& -{{i^{2}}\ {j^{2}}\ {e_{3}}} \ {{e_{1}}\ {e_{3}}}&{{i^{2}}\ {e_{3}}}& -{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{k^{2}}\ {e_{1}}}& -{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{i^{2}}\ {k^{2}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}&{{i^{2}}\ {k^{2}}\ {e_{2}}} \ {{e_{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{j^{2}}\ {e_{3}}}& -{{k^{2}}\ {e_{2}}}&{{j^{2}}\ {e_{1}}\ {e_{3}}}& -{{k^{2}}\ {e_{1}}\ {e_{2}}}& -{{j^{2}}\ {k^{2}}}& -{{j^{2}}\ {k^{2}}\ {e_{1}}} \ {{e_{1}}\ {e_{2}}\ {e_{3}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}& -{{i^{2}}\ {j^{2}}\ {e_{3}}}&{{i^{2}}\ {k^{2}}\ {e_{2}}}& -{{j^{2}}\ {k^{2}}\ {e_{1}}}& -{{i^{2}}\ {j^{2}}\ {k^{2}}}$

(14)

Type: Matrix(CliffordAlgebra?(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))

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S(y) == map(x +-> coefficient(recip(y)*x,[]),M)

Type: Void

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ѕ :=map(S,B)::ℒ ℒ ℒ ℚ

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Compiling function S with type CliffordAlgebra(3,Expression(Integer)
      ,[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]) -> Matrix(Expression(
      Integer))

$\label{eq15}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \:{i^{2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \:{j^{2}}, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \:{k^{2}}, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {j^{2}}}, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {k^{2}}}, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j^{2}}\ {k^{2}}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {j^{2}}\ {k^{2}}}\right]}\right]$

(15)

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

fricas

-- structure constants form a tensor operator
Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim)

$\label{eq16}\begin{array}{@{}l} \displaystyle {|_{\ 1}^{\ 1 \ 1}}+{|_{\ i}^{\ 1 \ i}}+{|_{\ j}^{\ 1 \ j}}+{|_{\ k}^{\ 1 \ k}}+{|_{\ ij}^{\ 1 \ ij}}+{|_{\ ik}^{\ 1 \ ik}}+ \ \ \displaystyle {|_{\ jk}^{\ 1 \ jk}}+{|_{\ ijk}^{\ 1 \ ijk}}+{|_{\ i}^{\ i \ 1}}+{{i^{2}}\ {|_{\ 1}^{\ i \ i}}}+{|_{\ ij}^{\ i \ j}}+ \ \ \displaystyle {|_{\ ik}^{\ i \ k}}+{{i^{2}}\ {|_{\ j}^{\ i \ ij}}}+{{i^{2}}\ {|_{\ k}^{\ i \ ik}}}+{|_{\ ijk}^{\ i \ jk}}+{{i^{2}}\ {|_{\ jk}^{\ i \ ijk}}}+ \ \ \displaystyle {|_{\ j}^{\ j \ 1}}-{|_{\ ij}^{\ j \ i}}+{{j^{2}}\ {|_{\ 1}^{\ j \ j}}}+{|_{\ jk}^{\ j \ k}}-{{j^{2}}\ {|_{\ i}^{\ j \ ij}}}- \ \ \displaystyle {|_{\ ijk}^{\ j \ ik}}+{{j^{2}}\ {|_{\ k}^{\ j \ jk}}}-{{j^{2}}\ {|_{\ ik}^{\ j \ ijk}}}+{|_{\ k}^{\ k \ 1}}-{|_{\ ik}^{\ k \ i}}- \ \ \displaystyle {|_{\ jk}^{\ k \ j}}+{{k^{2}}\ {|_{\ 1}^{\ k \ k}}}+{|_{\ ijk}^{\ k \ ij}}-{{k^{2}}\ {|_{\ i}^{\ k \ ik}}}-{{k^{2}}\ {|_{\ j}^{\ k \ jk}}}+ \ \ \displaystyle {{k^{2}}\ {|_{\ ij}^{\ k \ ijk}}}+{|_{\ ij}^{\ ij \ 1}}-{{i^{2}}\ {|_{\ j}^{\ ij \ i}}}+{{j^{2}}\ {|_{\ i}^{\ ij \ j}}}+{|_{\ ijk}^{\ ij \ k}}- \ \ \displaystyle {{i^{2}}\ {j^{2}}\ {|_{\ 1}^{\ ij \ ij}}}-{{i^{2}}\ {|_{\ jk}^{\ ij \ ik}}}+{{j^{2}}\ {|_{\ ik}^{\ ij \ jk}}}- \ \ \displaystyle {{i^{2}}\ {j^{2}}\ {|_{\ k}^{\ ij \ ijk}}}+{|_{\ ik}^{\ ik \ 1}}-{{i^{2}}\ {|_{\ k}^{\ ik \ i}}}-{|_{\ ijk}^{\ ik \ j}}+ \ \ \displaystyle {{k^{2}}\ {|_{\ i}^{\ ik \ k}}}+{{i^{2}}\ {|_{\ jk}^{\ ik \ ij}}}-{{i^{2}}\ {k^{2}}\ {|_{\ 1}^{\ ik \ ik}}}-{{k^{2}}\ {|_{\ ij}^{\ ik \ jk}}}+ \ \ \displaystyle {{i^{2}}\ {k^{2}}\ {|_{\ j}^{\ ik \ ijk}}}+{|_{\ jk}^{\ jk \ 1}}+{|_{\ ijk}^{\ jk \ i}}-{{j^{2}}\ {|_{\ k}^{\ jk \ j}}}+ \ \ \displaystyle {{k^{2}}\ {|_{\ j}^{\ jk \ k}}}-{{j^{2}}\ {|_{\ ik}^{\ jk \ ij}}}+{{k^{2}}\ {|_{\ ij}^{\ jk \ ik}}}-{{j^{2}}\ {k^{2}}\ {|_{\ 1}^{\ jk \ jk}}}- \ \ \displaystyle {{j^{2}}\ {k^{2}}\ {|_{\ i}^{\ jk \ ijk}}}+{|_{\ ijk}^{\ ijk \ 1}}+{{i^{2}}\ {|_{\ jk}^{\ ijk \ i}}}-{{j^{2}}\ {|_{\ ik}^{\ ijk \ j}}}+ \ \ \displaystyle {{k^{2}}\ {|_{\ ij}^{\ ijk \ k}}}-{{i^{2}}\ {j^{2}}\ {|_{\ k}^{\ ijk \ ij}}}+{{i^{2}}\ {k^{2}}\ {|_{\ j}^{\ ijk \ ik}}}- \ \ \displaystyle {{j^{2}}\ {k^{2}}\ {|_{\ i}^{\ ijk \ jk}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {|_{\ 1}^{\ ijk \ ijk}}}$

(16)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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

$\label{eq17}\left[ \begin{array}{cccccccc} {|_{\ 1}}&{|_{\ i}}&{|_{\ j}}&{|_{\ k}}&{|_{\ ij}}&{|_{\ ik}}&{|_{\ jk}}&{|_{\ ijk}} \ {|_{\ i}}&{{i^{2}}\ {|_{\ 1}}}& -{|_{\ ij}}& -{|_{\ ik}}& -{{i^{2}}\ {|_{\ j}}}& -{{i^{2}}\ {|_{\ k}}}&{|_{\ ijk}}&{{i^{2}}\ {|_{\ jk}}} \ {|_{\ j}}&{|_{\ ij}}&{{j^{2}}\ {|_{\ 1}}}& -{|_{\ jk}}&{{j^{2}}\ {|_{\ i}}}& -{|_{\ ijk}}& -{{j^{2}}\ {|_{\ k}}}& -{{j^{2}}\ {|_{\ ik}}} \ {|_{\ k}}&{|_{\ ik}}&{|_{\ jk}}&{{k^{2}}\ {|_{\ 1}}}&{|_{\ ijk}}&{{k^{2}}\ {|_{\ i}}}&{{k^{2}}\ {|_{\ j}}}&{{k^{2}}\ {|_{\ ij}}} \ {|_{\ ij}}&{{i^{2}}\ {|_{\ j}}}& -{{j^{2}}\ {|_{\ i}}}&{|_{\ ijk}}& -{{i^{2}}\ {j^{2}}\ {|_{\ 1}}}&{{i^{2}}\ {|_{\ jk}}}& -{{j^{2}}\ {|_{\ ik}}}& -{{i^{2}}\ {j^{2}}\ {|_{\ k}}} \ {|_{\ ik}}&{{i^{2}}\ {|_{\ k}}}& -{|_{\ ijk}}& -{{k^{2}}\ {|_{\ i}}}& -{{i^{2}}\ {|_{\ jk}}}& -{{i^{2}}\ {k^{2}}\ {|_{\ 1}}}&{{k^{2}}\ {|_{\ ij}}}&{{i^{2}}\ {k^{2}}\ {|_{\ j}}} \ {|_{\ jk}}&{|_{\ ijk}}&{{j^{2}}\ {|_{\ k}}}& -{{k^{2}}\ {|_{\ j}}}&{{j^{2}}\ {|_{\ ik}}}& -{{k^{2}}\ {|_{\ ij}}}& -{{j^{2}}\ {k^{2}}\ {|_{\ 1}}}& -{{j^{2}}\ {k^{2}}\ {|_{\ i}}} \ {|_{\ ijk}}&{{i^{2}}\ {|_{\ jk}}}& -{{j^{2}}\ {|_{\ ik}}}&{{k^{2}}\ {|_{\ ij}}}& -{{i^{2}}\ {j^{2}}\ {|_{\ k}}}&{{i^{2}}\ {k^{2}}\ {|_{\ j}}}& -{{j^{2}}\ {k^{2}}\ {|_{\ i}}}& -{{i^{2}}\ {j^{2}}\ {k^{2}}\ {|_{\ 1}}}$

(17)

Type: Matrix(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))

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XY := X/Y;

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Units

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e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication of arbitrary quaternions and

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a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

$\label{eq18}\begin{array}{@{}l} \displaystyle {{a_{1}}\ {|_{\ 1}}}+{{a_{2}}\ {|_{\ i}}}+{{a_{3}}\ {|_{\ j}}}+{{a_{4}}\ {|_{\ k}}}+{{a_{5}}\ {|_{\ ij}}}+{{a_{6}}\ {|_{\ ik}}}+{{a_{7}}\ {|_{\ jk}}}+ \ \ \displaystyle {{a_{8}}\ {|_{\ ijk}}}$

(18)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

$\label{eq19}\begin{array}{@{}l} \displaystyle {{b_{1}}\ {|_{\ 1}}}+{{b_{2}}\ {|_{\ i}}}+{{b_{3}}\ {|_{\ j}}}+{{b_{4}}\ {|_{\ k}}}+{{b_{5}}\ {|_{\ ij}}}+{{b_{6}}\ {|_{\ ik}}}+{{b_{7}}\ {|_{\ jk}}}+ \ \ \displaystyle {{b_{8}}\ {|_{\ ijk}}}$

(19)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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(a*b)/Y

$\label{eq20}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{i^{2}}\ {j^{2}}\ {k^{2}}\ {a_{8}}\ {b_{8}}}-{{j^{2}}\ {k^{2}}\ {a_{7}}\ {b_{7}}}-{{i^{2}}\ {k^{2}}\ {a_{6}}\ {b_{6}}}- \ \ \displaystyle {{i^{2}}\ {j^{2}}\ {a_{5}}\ {b_{5}}}+{{k^{2}}\ {a_{4}}\ {b_{4}}}+{{j^{2}}\ {a_{3}}\ {b_{3}}}+{{i^{2}}\ {a_{2}}\ {b_{2}}}+ \ \ \displaystyle {{a_{1}}\ {b_{1}}}$

(20)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication is Associative

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test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )

$\label{eq21} \mbox{\rm true}$

(21)

Type: Boolean

A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$

fricas

U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

$\label{eq22}\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^{1, \: 5}}\ {|^{\ 1 \ ij}}}+{{u^{1, \: 6}}\ {|^{\ 1 \ ik}}}+{{u^{1, \: 7}}\ {|^{\ 1 \ jk}}}+{{u^{1, \: 8}}\ {|^{\ 1 \ ijk}}}+ \ \ \displaystyle {{u^{2, \: 1}}\ {|^{\ i \ 1}}}+{{u^{2, \: 2}}\ {|^{\ i \ i}}}+{{u^{2, \: 3}}\ {|^{\ i \ j}}}+{{u^{2, \: 4}}\ {|^{\ i \ k}}}+{{u^{2, \: 5}}\ {|^{\ i \ ij}}}+ \ \ \displaystyle {{u^{2, \: 6}}\ {|^{\ i \ ik}}}+{{u^{2, \: 7}}\ {|^{\ i \ jk}}}+{{u^{2, \: 8}}\ {|^{\ i \ ijk}}}+{{u^{3, \: 1}}\ {|^{\ j \ 1}}}+ \ \ \displaystyle {{u^{3, \: 2}}\ {|^{\ j \ i}}}+{{u^{3, \: 3}}\ {|^{\ j \ j}}}+{{u^{3, \: 4}}\ {|^{\ j \ k}}}+{{u^{3, \: 5}}\ {|^{\ j \ ij}}}+ \ \ \displaystyle {{u^{3, \: 6}}\ {|^{\ j \ ik}}}+{{u^{3, \: 7}}\ {|^{\ j \ jk}}}+{{u^{3, \: 8}}\ {|^{\ j \ ijk}}}+{{u^{4, \: 1}}\ {|^{\ k \ 1}}}+ \ \ \displaystyle {{u^{4, \: 2}}\ {|^{\ k \ i}}}+{{u^{4, \: 3}}\ {|^{\ k \ j}}}+{{u^{4, \: 4}}\ {|^{\ k \ k}}}+{{u^{4, \: 5}}\ {|^{\ k \ ij}}}+ \ \ \displaystyle {{u^{4, \: 6}}\ {|^{\ k \ ik}}}+{{u^{4, \: 7}}\ {|^{\ k \ jk}}}+{{u^{4, \: 8}}\ {|^{\ k \ ijk}}}+{{u^{5, \: 1}}\ {|^{\ ij \ 1}}}+ \ \ \displaystyle {{u^{5, \: 2}}\ {|^{\ ij \ i}}}+{{u^{5, \: 3}}\ {|^{\ ij \ j}}}+{{u^{5, \: 4}}\ {|^{\ ij \ k}}}+{{u^{5, \: 5}}\ {|^{\ ij \ ij}}}+ \ \ \displaystyle {{u^{5, \: 6}}\ {|^{\ ij \ ik}}}+{{u^{5, \: 7}}\ {|^{\ ij \ jk}}}+{{u^{5, \: 8}}\ {|^{\ ij \ ijk}}}+{{u^{6, \: 1}}\ {|^{\ ik \ 1}}}+ \ \ \displaystyle {{u^{6, \: 2}}\ {|^{\ ik \ i}}}+{{u^{6, \: 3}}\ {|^{\ ik \ j}}}+{{u^{6, \: 4}}\ {|^{\ ik \ k}}}+{{u^{6, \: 5}}\ {|^{\ ik \ ij}}}+ \ \ \displaystyle {{u^{6, \: 6}}\ {|^{\ ik \ ik}}}+{{u^{6, \: 7}}\ {|^{\ ik \ jk}}}+{{u^{6, \: 8}}\ {|^{\ ik \ ijk}}}+{{u^{7, \: 1}}\ {|^{\ jk \ 1}}}+ \ \ \displaystyle {{u^{7, \: 2}}\ {|^{\ jk \ i}}}+{{u^{7, \: 3}}\ {|^{\ jk \ j}}}+{{u^{7, \: 4}}\ {|^{\ jk \ k}}}+{{u^{7, \: 5}}\ {|^{\ jk \ ij}}}+ \ \ \displaystyle {{u^{7, \: 6}}\ {|^{\ jk \ ik}}}+{{u^{7, \: 7}}\ {|^{\ jk \ jk}}}+{{u^{7, \: 8}}\ {|^{\ jk \ ijk}}}+{{u^{8, \: 1}}\ {|^{\ ijk \ 1}}}+ \ \ \displaystyle {{u^{8, \: 2}}\ {|^{\ ijk \ i}}}+{{u^{8, \: 3}}\ {|^{\ ijk \ j}}}+{{u^{8, \: 4}}\ {|^{\ ijk \ k}}}+{{u^{8, \: 5}}\ {|^{\ ijk \ ij}}}+ \ \ \displaystyle {{u^{8, \: 6}}\ {|^{\ ijk \ ik}}}+{{u^{8, \: 7}}\ {|^{\ ijk \ jk}}}+{{u^{8, \: 8}}\ {|^{\ ijk \ ijk}}}$

(22)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 1

We say that the scalar product is associative if the tensor equation holds:

    Y   =   Y
     U     U

In other words, if the (3,0)-tensor:

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

$\label{eq23} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(23)

(three-point function) is zero.

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

fricas

ω:𝐋 :=                 _
     (    Y I    )  /  _
           U        -  _
     (    I Y    )  /  _
           U;

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 2

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

The Cartan-Killing Trace

fricas

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

$\label{eq24}\begin{array}{@{}l} \displaystyle {8 \ {|^{\ 1 \ 1}}}+{8 \ {i^{2}}\ {|^{\ i \ i}}}+{8 \ {j^{2}}\ {|^{\ j \ j}}}+{8 \ {k^{2}}\ {|^{\ k \ k}}}-{8 \ {i^{2}}\ {j^{2}}\ {|^{\ ij \ ij}}}- \ \ \displaystyle {8 \ {i^{2}}\ {k^{2}}\ {|^{\ ik \ ik}}}-{8 \ {j^{2}}\ {k^{2}}\ {|^{\ jk \ jk}}}-{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ {|^{\ ijk \ ijk}}}$

(24)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

$\label{eq25}\begin{array}{@{}l} \displaystyle {8 \ {|^{\ 1 \ 1}}}+{8 \ {i^{2}}\ {|^{\ i \ i}}}+{8 \ {j^{2}}\ {|^{\ j \ j}}}+{8 \ {k^{2}}\ {|^{\ k \ k}}}-{8 \ {i^{2}}\ {j^{2}}\ {|^{\ ij \ ij}}}- \ \ \displaystyle {8 \ {i^{2}}\ {k^{2}}\ {|^{\ ik \ ik}}}-{8 \ {j^{2}}\ {k^{2}}\ {|^{\ jk \ jk}}}-{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ {|^{\ ijk \ ijk}}}$

(25)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

test(Ù=Ú)

$\label{eq26} \mbox{\rm true}$

(26)

Type: Boolean

forms a non-degenerate associative scalar product for Y

fricas

Ũ := Ù

$\label{eq27}\begin{array}{@{}l} \displaystyle {8 \ {|^{\ 1 \ 1}}}+{8 \ {i^{2}}\ {|^{\ i \ i}}}+{8 \ {j^{2}}\ {|^{\ j \ j}}}+{8 \ {k^{2}}\ {|^{\ k \ k}}}-{8 \ {i^{2}}\ {j^{2}}\ {|^{\ ij \ ij}}}- \ \ \displaystyle {8 \ {i^{2}}\ {k^{2}}\ {|^{\ ik \ ik}}}-{8 \ {j^{2}}\ {k^{2}}\ {|^{\ jk \ jk}}}-{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ {|^{\ ijk \ ijk}}}$

(27)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

$\label{eq28} \mbox{\rm true}$

(28)

Type: Boolean

fricas

determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)

$\label{eq29}{16777216}\ {{i^{2}}^{4}}\ {{j^{2}}^{4}}\ {{k^{2}}^{4}}$

(29)

Type: Expression(Integer)

General Solution

Frobenius Form (co-unit)

fricas

d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8

$\label{eq30}{�� 1 \ {|^{\ 1}}}+{�� i \ {|^{\ i}}}+{�� j \ {|^{\ j}}}+{�� k \ {|^{\ k}}}+{�� ij \ {|^{\ ij}}}+{�� ik \ {|^{\ ik}}}+{�� jk \ {|^{\ jk}}}+{�� ijk \ {|^{\ ijk}}}$

(30)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

Ų:= Y/d

$\label{eq31}\begin{array}{@{}l} \displaystyle {�� 1 \ {|^{\ 1 \ 1}}}+{�� i \ {|^{\ 1 \ i}}}+{�� j \ {|^{\ 1 \ j}}}+{�� k \ {|^{\ 1 \ k}}}+{�� ij \ {|^{\ 1 \ ij}}}+{�� ik \ {|^{\ 1 \ ik}}}+ \ \ \displaystyle {�� jk \ {|^{\ 1 \ jk}}}+{�� ijk \ {|^{\ 1 \ ijk}}}+{�� i \ {|^{\ i \ 1}}}+{{i^{2}}\ �� 1 \ {|^{\ i \ i}}}+{�� ij \ {|^{\ i \ j}}}+ \ \ \displaystyle {�� ik \ {|^{\ i \ k}}}+{{i^{2}}\ �� j \ {|^{\ i \ ij}}}+{{i^{2}}\ �� k \ {|^{\ i \ ik}}}+{�� ijk \ {|^{\ i \ jk}}}+{{i^{2}}\ �� jk \ {|^{\ i \ ijk}}}+ \ \ \displaystyle {�� j \ {|^{\ j \ 1}}}-{�� ij \ {|^{\ j \ i}}}+{{j^{2}}\ �� 1 \ {|^{\ j \ j}}}+{�� jk \ {|^{\ j \ k}}}-{{j^{2}}\ �� i \ {|^{\ j \ ij}}}- \ \ \displaystyle {�� ijk \ {|^{\ j \ ik}}}+{{j^{2}}\ �� k \ {|^{\ j \ jk}}}-{{j^{2}}\ �� ik \ {|^{\ j \ ijk}}}+{�� k \ {|^{\ k \ 1}}}-{�� ik \ {|^{\ k \ i}}}- \ \ \displaystyle {�� jk \ {|^{\ k \ j}}}+{{k^{2}}\ �� 1 \ {|^{\ k \ k}}}+{�� ijk \ {|^{\ k \ ij}}}-{{k^{2}}\ �� i \ {|^{\ k \ ik}}}-{{k^{2}}\ �� j \ {|^{\ k \ jk}}}+ \ \ \displaystyle {{k^{2}}\ �� ij \ {|^{\ k \ ijk}}}+{�� ij \ {|^{\ ij \ 1}}}-{{i^{2}}\ �� j \ {|^{\ ij \ i}}}+{{j^{2}}\ �� i \ {|^{\ ij \ j}}}+{�� ijk \ {|^{\ ij \ k}}}- \ \ \displaystyle {{i^{2}}\ {j^{2}}\ �� 1 \ {|^{\ ij \ ij}}}-{{i^{2}}\ �� jk \ {|^{\ ij \ ik}}}+{{j^{2}}\ �� ik \ {|^{\ ij \ jk}}}- \ \ \displaystyle {{i^{2}}\ {j^{2}}\ �� k \ {|^{\ ij \ ijk}}}+{�� ik \ {|^{\ ik \ 1}}}-{{i^{2}}\ �� k \ {|^{\ ik \ i}}}-{�� ijk \ {|^{\ ik \ j}}}+ \ \ \displaystyle {{k^{2}}\ �� i \ {|^{\ ik \ k}}}+{{i^{2}}\ �� jk \ {|^{\ ik \ ij}}}-{{i^{2}}\ {k^{2}}\ �� 1 \ {|^{\ ik \ ik}}}-{{k^{2}}\ �� ij \ {|^{\ ik \ jk}}}+ \ \ \displaystyle {{i^{2}}\ {k^{2}}\ �� j \ {|^{\ ik \ ijk}}}+{�� jk \ {|^{\ jk \ 1}}}+{�� ijk \ {|^{\ jk \ i}}}-{{j^{2}}\ �� k \ {|^{\ jk \ j}}}+ \ \ \displaystyle {{k^{2}}\ �� j \ {|^{\ jk \ k}}}-{{j^{2}}\ �� ik \ {|^{\ jk \ ij}}}+{{k^{2}}\ �� ij \ {|^{\ jk \ ik}}}-{{j^{2}}\ {k^{2}}\ �� 1 \ {|^{\ jk \ jk}}}- \ \ \displaystyle {{j^{2}}\ {k^{2}}\ �� i \ {|^{\ jk \ ijk}}}+{�� ijk \ {|^{\ ijk \ 1}}}+{{i^{2}}\ �� jk \ {|^{\ ijk \ i}}}-{{j^{2}}\ �� ik \ {|^{\ ijk \ j}}}+ \ \ \displaystyle {{k^{2}}\ �� ij \ {|^{\ ijk \ k}}}-{{i^{2}}\ {j^{2}}\ �� k \ {|^{\ ijk \ ij}}}+{{i^{2}}\ {k^{2}}\ �� j \ {|^{\ ijk \ ik}}}- \ \ \displaystyle {{j^{2}}\ {k^{2}}\ �� i \ {|^{\ ijk \ jk}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {|^{\ ijk \ ijk}}}$

(31)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

In general the pairing is not symmetric!

fricas

u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)

$\label{eq32}\left[ \begin{array}{cccccccc} �� 1 & �� i & �� j & �� k & �� ij & �� ik & �� jk & �� ijk \ �� i &{{i^{2}}\ �� 1}& - �� ij & - �� ik & -{{i^{2}}\ �� j}& -{{i^{2}}\ �� k}& �� ijk &{{i^{2}}\ �� jk} \ �� j & �� ij &{{j^{2}}\ �� 1}& - �� jk &{{j^{2}}\ �� i}& - �� ijk & -{{j^{2}}\ �� k}& -{{j^{2}}\ �� ik} \ �� k & �� ik & �� jk &{{k^{2}}\ �� 1}& �� ijk &{{k^{2}}\ �� i}&{{k^{2}}\ �� j}&{{k^{2}}\ �� ij} \ �� ij &{{i^{2}}\ �� j}& -{{j^{2}}\ �� i}& �� ijk & -{{i^{2}}\ {j^{2}}\ �� 1}&{{i^{2}}\ �� jk}& -{{j^{2}}\ �� ik}& -{{i^{2}}\ {j^{2}}\ �� k} \ �� ik &{{i^{2}}\ �� k}& - �� ijk & -{{k^{2}}\ �� i}& -{{i^{2}}\ �� jk}& -{{i^{2}}\ {k^{2}}\ �� 1}&{{k^{2}}\ �� ij}&{{i^{2}}\ {k^{2}}\ �� j} \ �� jk & �� ijk &{{j^{2}}\ �� k}& -{{k^{2}}\ �� j}&{{j^{2}}\ �� ik}& -{{k^{2}}\ �� ij}& -{{j^{2}}\ {k^{2}}\ �� 1}& -{{j^{2}}\ {k^{2}}\ �� i} \ �� ijk &{{i^{2}}\ �� jk}& -{{j^{2}}\ �� ik}&{{k^{2}}\ �� ij}& -{{i^{2}}\ {j^{2}}\ �� k}&{{i^{2}}\ {k^{2}}\ �� j}& -{{j^{2}}\ {k^{2}}\ �� i}& -{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1}$

(32)

Type: Matrix(Expression(Integer))

The scalar product must be non-degenerate:

fricas

--Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1

$\label{eq33}1$

(33)

Type: PositiveInteger?

Cartan-Killing is a special case

fricas

ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1

fricas

Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
       -> List(Equation(Expression(Integer)))

$\label{eq34}\left[{�� 1 = 8}, \:{�� i = 0}, \:{�� j = 0}, \:{�� k = 0}, \:{�� ij = 0}, \:{�� ik = 0}, \:{�� jk = 0}, \:{�� ijk = 0}\right]$

(34)

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

Frobenius scalar product of "vector" quaternions and

fricas

a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k

$\label{eq35}{{a_{1}}\ {|_{\ i}}}+{{a_{2}}\ {|_{\ j}}}+{{a_{3}}\ {|_{\ k}}}$

(35)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

b:=sb('b,[1])*i+sb('b,[2])*j+sb('b,[3])*k

$\label{eq36}{{b_{1}}\ {|_{\ i}}}+{{b_{2}}\ {|_{\ j}}}+{{b_{3}}\ {|_{\ k}}}$

(36)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

(a,a)/Ų

$\label{eq37}{\left({{k^{2}}\ {{a_{3}}^{2}}}+{{j^{2}}\ {{a_{2}}^{2}}}+{{i^{2}}\ {{a_{1}}^{2}}}\right)}\ �� 1$

(37)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

(b,b)/Ų

$\label{eq38}{\left({{k^{2}}\ {{b_{3}}^{2}}}+{{j^{2}}\ {{b_{2}}^{2}}}+{{i^{2}}\ {{b_{1}}^{2}}}\right)}\ �� 1$

(38)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

(a,b)/Ų

$\label{eq39}\begin{array}{@{}l} \displaystyle {{\left({{a_{2}}\ {b_{3}}}-{{a_{3}}\ {b_{2}}}\right)}\ �� jk}+{{\left({{a_{1}}\ {b_{3}}}-{{a_{3}}\ {b_{1}}}\right)}\ �� ik}+{{\left({{a_{1}}\ {b_{2}}}-{{a_{2}}\ {b_{1}}}\right)}\ �� ij}+ \ \ \displaystyle {{\left({{k^{2}}\ {a_{3}}\ {b_{3}}}+{{j^{2}}\ {a_{2}}\ {b_{2}}}+{{i^{2}}\ {a_{1}}\ {b_{1}}}\right)}\ �� 1}$

(39)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

fricas

mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);

Type: Union(Matrix(Expression(Integer)),...)

fricas

Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim);

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

ΩX:=Ω/X;

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

The common demoninator is $1/\sqrt{\mathring{U}}$

fricas

--squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)

\label{eq40}\left[
\begin{array}{cccccccc}
{-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� ijk \ �� k}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ijk \ �� jk}-{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ijk \ �� ik \ �� j}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� ij}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� jk \ �� k}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ �� i \ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ �� i \ {{�� ik}^{2}}}+{{j^{2}}\ {k^{2}}\ �� i \ {{�� ijk}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\ �� i \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\ �� i}}&{{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ �� j \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� k}-{{{i^{2}}^{2}}\ {k^{2}}\ �� j \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� jk}+{{{i^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� ik}}&{{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {j^{2}}\ {{�� jk}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}+{{i^{2}}\ {k^{2}}\ �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\ �� ij \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\ �� ij \ {{�� ik}^{2}}}-{{k^{2}}\ �� ij \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}&{-{{i^{2}}\ {{j^{2}}^{2}}\ �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}+{{i^{2}}\ {j^{2}}\ �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}+{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left(-{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}&{-{{{i^{2}}^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}+{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}-{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}+{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}-{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}
\
{{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� jk \ �� k}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ �� i \ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ �� i \ {{�� ik}^{2}}}+{{j^{2}}\ {k^{2}}\ �� i \ {{�� ijk}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\ �� i \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\ �� i}}&{-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� k}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\ �� ij \ �� ijk \ �� k}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� jk}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ijk \ �� jk}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� j}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\ �� ijk \ �� ik \ �� j}+{{{j^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� ik}^{2}}}+{{j^{2}}\ {k^{2}}\ �� 1 \ {{�� ijk}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{-{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� jk}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}-{{i^{2}}\ {k^{2}}\ �� ij \ {{�� jk}^{2}}}+{{i^{2}}\ {{k^{2}}^{2}}\ �� ij \ {{�� j}^{2}}}-{{j^{2}}\ {k^{2}}\ �� ij \ {{�� ik}^{2}}}+{{k^{2}}\ �� ij \ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left({{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}&{{{i^{2}}\ {{j^{2}}^{2}}\ �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}-{{i^{2}}\ {j^{2}}\ �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}-{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left({{j^{2}}\ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}&{-{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� j \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� k}+{{i^{2}}\ {k^{2}}\ �� j \ {{�� jk}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� jk}-{{i^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}-{{k^{2}}\ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}-{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� ik}}&{-{{i^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {{�� jk}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{{j^{2}}^{2}}\ {{�� ik}^{2}}}-{{j^{2}}\ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}-{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� jk}+{2 \ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� j}+{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}-{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}+{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}-{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}&{-{{i^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}+{{i^{2}}\ {{�� jk}^{3}}}+{{\left(-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}-{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}-{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}
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{{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ �� j \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� k}-{{{i^{2}}^{2}}\ {k^{2}}\ �� j \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� jk}+{{{i^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� ik}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}+{{i^{2}}\ {k^{2}}\ �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\ �� ij \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\ �� ij \ {{�� ik}^{2}}}-{{k^{2}}\ �� ij \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}&{-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� ijk \ �� k}+{{{i^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� i \ �� ijk \ �� jk}-{{{i^{2}}^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� ijk \ �� ik \ �� j}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� ik}^{2}}}+{{i^{2}}\ {k^{2}}\ �� 1 \ {{�� ijk}^{2}}}+{{i^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{{{{i^{2}}^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}-{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left({{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� jk \ �� k}+{{i^{2}}\ {k^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {k^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}+{{i^{2}}\ {{k^{2}}^{2}}\ �� i \ {{�� j}^{2}}}-{{j^{2}}\ {k^{2}}\ �� i \ {{�� ik}^{2}}}+{{k^{2}}\ �� i \ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\ �� i \ {{�� ij}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\ �� i}}&{-{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}+{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}-{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}+{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}-{{�� ijk}^{3}}+{{\left({{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}&{-{{{i^{2}}^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left({{{i^{2}}^{2}}\ {{�� jk}^{2}}}-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}-{2 \ {i^{2}}\ {k^{2}}\ �� i \ �� ij \ �� jk}+{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� j}+{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}-{{i^{2}}\ �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}-{{i^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}-{{j^{2}}\ {{�� ik}^{3}}}+{{\left({{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}
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{{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {j^{2}}\ {{�� jk}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}&{-{{i^{2}}\ {{j^{2}}^{2}}\ �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}+{{i^{2}}\ {j^{2}}\ �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}+{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left(-{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}&{-{{{i^{2}}^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}+{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}&{-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ �� 1 \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ �� ij \ �� ijk \ �� k}+{{{i^{2}}^{2}}\ {j^{2}}\ �� 1 \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ �� i \ �� ijk \ �� jk}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ �� ijk \ �� ik \ �� j}+{{i^{2}}\ {{j^{2}}^{2}}\ �� 1 \ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\ �� 1 \ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� ij}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}&{{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}-{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}+{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}-{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}&{{{i^{2}}\ {{j^{2}}^{2}}\ �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ �� ij \ �� jk \ �� k}+{{i^{2}}\ {j^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ �� i \ {{�� ik}^{2}}}+{{j^{2}}\ �� i \ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\ �� i \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}\ �� i}}&{{{{i^{2}}^{2}}\ {j^{2}}\ �� j \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ �� ij \ �� ik \ �� k}-{{{i^{2}}^{2}}\ �� j \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ �� i \ �� ik \ �� jk}+{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}+{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk \ �� ik}}&{{{i^{2}}\ {j^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ �� i \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}+{{i^{2}}\ �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\ �� ij \ {{�� j}^{2}}}+{{j^{2}}\ �� ij \ {{�� ik}^{2}}}-{�� ij \ {{�� ijk}^{2}}}+{{k^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}
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{{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}+{{i^{2}}\ {k^{2}}\ �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\ �� ij \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\ �� ij \ {{�� ik}^{2}}}-{{k^{2}}\ �� ij \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� j \ {{�� k}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� k}-{{i^{2}}\ {k^{2}}\ �� j \ {{�� jk}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� jk}+{{i^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left({{j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}+{{k^{2}}\ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}+{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� ik}}&{-{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� jk \ �� k}-{{i^{2}}\ {k^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\ {k^{2}}\ �� ik \ �� j}+{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}-{{i^{2}}\ {{k^{2}}^{2}}\ �� i \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\ �� i \ {{�� ik}^{2}}}-{{k^{2}}\ �� i \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ �� i \ {{�� ij}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\ �� i}}&{{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}-{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}+{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}-{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� k}^{2}}}+{2 \ {k^{2}}\ �� ij \ �� ijk \ �� k}-{{i^{2}}\ {k^{2}}\ �� 1 \ {{�� jk}^{2}}}+{2 \ {k^{2}}\ �� i \ �� ijk \ �� jk}+{{i^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� j}^{2}}}-{2 \ {k^{2}}\ �� ijk \ �� ik \ �� j}-{{j^{2}}\ {k^{2}}\ �� 1 \ {{�� ik}^{2}}}-{{k^{2}}\ �� 1 \ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\ �� 1 \ {{�� ij}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\ �� 1 \ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{-{{i^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}+{{i^{2}}\ {{�� jk}^{3}}}+{{\left(-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}-{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}-{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}-{{i^{2}}\ �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}-{{i^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}-{{j^{2}}\ {{�� ik}^{3}}}+{{\left({{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}&{-{{i^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left({{i^{2}}\ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}-{2 \ {k^{2}}\ �� i \ �� ij \ �� jk}+{2 \ {k^{2}}\ �� ij \ �� ik \ �� j}+{2 \ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}
\
{-{{i^{2}}\ {{j^{2}}^{2}}\ �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}+{{i^{2}}\ {j^{2}}\ �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}+{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left(-{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}&{{{i^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left(-{{i^{2}}\ {j^{2}}\ {{�� jk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ {{�� ik}^{2}}}+{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}+{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� jk}-{2 \ {j^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� j}-{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}&{-{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}+{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}-{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}+{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}-{{�� ijk}^{3}}+{{\left({{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}&{-{{i^{2}}\ {{j^{2}}^{2}}\ �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ �� ij \ �� jk \ �� k}-{{i^{2}}\ {j^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ �� ik \ �� j}+{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ {{�� j}^{2}}}+{{{j^{2}}^{2}}\ �� i \ {{�� ik}^{2}}}-{{j^{2}}\ �� i \ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ �� i \ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}\ �� i}}&{{{i^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}-{{i^{2}}\ {{�� jk}^{3}}}+{{\left({{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{j^{2}}\ {{�� ik}^{2}}}+{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}+{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}+{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}&{{{i^{2}}\ {{j^{2}}^{2}}\ �� 1 \ {{�� k}^{2}}}+{2 \ {j^{2}}\ �� ij \ �� ijk \ �� k}-{{i^{2}}\ {j^{2}}\ �� 1 \ {{�� jk}^{2}}}+{2 \ {j^{2}}\ �� i \ �� ijk \ �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� j}^{2}}}-{2 \ {j^{2}}\ �� ijk \ �� ik \ �� j}-{{{j^{2}}^{2}}\ �� 1 \ {{�� ik}^{2}}}-{{j^{2}}\ �� 1 \ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\ �� 1 \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� i}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}&{{{i^{2}}\ {j^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ �� i \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}+{{i^{2}}\ �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\ �� ij \ {{�� j}^{2}}}+{{j^{2}}\ �� ij \ {{�� ik}^{2}}}-{�� ij \ {{�� ijk}^{2}}}+{{k^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}&{{{i^{2}}\ {j^{2}}\ �� j \ {{�� k}^{2}}}-{2 \ {j^{2}}\ �� ij \ �� ik \ �� k}-{{i^{2}}\ �� j \ {{�� jk}^{2}}}-{2 \ {j^{2}}\ �� i \ �� ik \ �� jk}+{{i^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left({{j^{2}}\ {{�� ik}^{2}}}+{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}+{2 \ {j^{2}}\ �� 1 \ �� ijk \ �� ik}}
\
{-{{{i^{2}}^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}+{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}-{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}+{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}-{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}&{{{{i^{2}}^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {{�� jk}^{2}}}+{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}+{2 \ {i^{2}}\ {k^{2}}\ �� i \ �� ij \ �� jk}-{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� ik \ �� j}-{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}&{-{{{i^{2}}^{2}}\ {j^{2}}\ �� j \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ �� ij \ �� ik \ �� k}+{{{i^{2}}^{2}}\ �� j \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ �� i \ �� ik \ �� jk}-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left(-{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}-{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk \ �� ik}}&{-{{i^{2}}\ {j^{2}}\ �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}+{{i^{2}}\ �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}+{{i^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}+{{j^{2}}\ {{�� ik}^{3}}}+{{\left(-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}&{-{{i^{2}}\ {j^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ �� i \ �� jk}+{2 \ {i^{2}}\ {j^{2}}\ �� ik \ �� j}+{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}-{{i^{2}}\ �� ij \ {{�� jk}^{2}}}+{{i^{2}}\ {k^{2}}\ �� ij \ {{�� j}^{2}}}-{{j^{2}}\ �� ij \ {{�� ik}^{2}}}+{�� ij \ {{�� ijk}^{2}}}-{{k^{2}}\ {{�� ij}^{3}}}+{{\left({{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}&{{{{i^{2}}^{2}}\ {j^{2}}\ �� 1 \ {{�� k}^{2}}}+{2 \ {i^{2}}\ �� ij \ �� ijk \ �� k}-{{{i^{2}}^{2}}\ �� 1 \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ �� i \ �� ijk \ �� jk}+{{{i^{2}}^{2}}\ {k^{2}}\ �� 1 \ {{�� j}^{2}}}-{2 \ {i^{2}}\ �� ijk \ �� ik \ �� j}-{{i^{2}}\ {j^{2}}\ �� 1 \ {{�� ik}^{2}}}-{{i^{2}}\ �� 1 \ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\ �� 1 \ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}&{-{{i^{2}}\ {j^{2}}\ �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\ �� ij \ �� jk \ �� k}-{{i^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\ �� ik \ �� j}+{2 \ {i^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}-{{i^{2}}\ {k^{2}}\ �� i \ {{�� j}^{2}}}+{{j^{2}}\ �� i \ {{�� ik}^{2}}}-{�� i \ {{�� ijk}^{2}}}+{{k^{2}}\ �� i \ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}\ �� i}}
\
{{{i^{2}}\ {j^{2}}\ �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ij \ �� k}-{{i^{2}}\ �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� jk}+{{i^{2}}\ {k^{2}}\ �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ �� 1 \ �� ik \ �� j}-{{j^{2}}\ �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ijk}}&{-{{i^{2}}\ {j^{2}}\ �� jk \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ij \ �� k}+{{i^{2}}\ {{�� jk}^{3}}}+{{\left(-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� jk}-{2 \ {j^{2}}\ {k^{2}}\ �� i \ �� ik \ �� j}-{2 \ {j^{2}}\ {k^{2}}\ �� 1 \ �� i \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� ij \ �� j \ �� k}-{{i^{2}}\ �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\ �� i \ �� j \ �� jk}-{{i^{2}}\ {k^{2}}\ �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\ �� 1 \ �� ijk \ �� j}-{{j^{2}}\ {{�� ik}^{3}}}+{{\left({{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ik}}&{{{i^{2}}\ {j^{2}}\ �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ �� i \ �� jk}-{2 \ {i^{2}}\ {j^{2}}\ �� ik \ �� j}-{2 \ {i^{2}}\ {j^{2}}\ �� 1 \ �� ijk}\right)}\ �� k}+{{i^{2}}\ �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\ �� ij \ {{�� j}^{2}}}+{{j^{2}}\ �� ij \ {{�� ik}^{2}}}-{�� ij \ {{�� ijk}^{2}}}+{{k^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� ij}}&{-{{i^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left({{i^{2}}\ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� k}-{2 \ {k^{2}}\ �� i \ �� ij \ �� jk}+{2 \ {k^{2}}\ �� ij \ �� ik \ �� j}+{2 \ {k^{2}}\ �� 1 \ �� ij \ �� ijk}}&{{{i^{2}}\ {j^{2}}\ �� j \ {{�� k}^{2}}}-{2 \ {j^{2}}\ �� ij \ �� ik \ �� k}-{{i^{2}}\ �� j \ {{�� jk}^{2}}}-{2 \ {j^{2}}\ �� i \ �� ik \ �� jk}+{{i^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left({{j^{2}}\ {{�� ik}^{2}}}+{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\ �� j}+{2 \ {j^{2}}\ �� 1 \ �� ijk \ �� ik}}&{-{{i^{2}}\ {j^{2}}\ �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\ �� ij \ �� jk \ �� k}-{{i^{2}}\ �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\ �� ik \ �� j}+{2 \ {i^{2}}\ �� 1 \ �� ijk}\right)}\ �� jk}-{{i^{2}}\ {k^{2}}\ �� i \ {{�� j}^{2}}}+{{j^{2}}\ �� i \ {{�� ik}^{2}}}-{�� i \ {{�� ijk}^{2}}}+{{k^{2}}\ �� i \ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}\ �� i}}&{{{i^{2}}\ {j^{2}}\ �� 1 \ {{�� k}^{2}}}+{2 \ �� ij \ �� ijk \ �� k}-{{i^{2}}\ �� 1 \ {{�� jk}^{2}}}+{2 \ �� i \ �� ijk \ �� jk}+{{i^{2}}\ {k^{2}}\ �� 1 \ {{�� j}^{2}}}-{2 \ �� ijk \ �� ik \ �� j}-{{j^{2}}\ �� 1 \ {{�� ik}^{2}}}-{�� 1 \ {{�� ijk}^{2}}}-{{k^{2}}\ �� 1 \ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ �� 1 \ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}

(40)

Type: Matrix(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))

Check "dimension" and the snake relations.

fricas

O:𝐋:= Ω / Ų

$\label{eq41}8$

(41)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

test
    (    I ΩX     )  /
    (     Ų I     )  =  I

$\label{eq42} \mbox{\rm true}$

(42)

Type: Boolean

fricas

test
    (     ΩX I    )  /
    (    I Ų      )  =  I

$\label{eq43} \mbox{\rm true}$

(43)

Type: Boolean

Cartan-Killing co-scalar

fricas

eval(Ω,ck)

$\label{eq44}\begin{array}{@{}l} \displaystyle {{\frac{1}{8}}\ {|_{\ 1 \ 1}}}+{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ i \ i}}}+{{\frac{1}{8 \ {j^{2}}}}\ {|_{\ j \ j}}}+ \ \ \displaystyle {{\frac{1}{8 \ {k^{2}}}}\ {|_{\ k \ k}}}-{{\frac{1}{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\ ij \ ij}}}- \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\ ik \ ik}}}-{{\frac{1}{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\ jk \ jk}}}- \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}}}\ {|_{\ ijk \ ijk}}}$

(44)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 4

Co-algebra

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

fricas

W:= (Y I) / Ų;

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

λ:=                      _
     (    I ΩX     )  /  _
     (     Y I     );

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

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

$\label{eq45} \mbox{\rm true}$

(45)

Type: Boolean

Cartan-Killing co-multiplication

fricas

eval(λ,ck)

$\label{eq46}\begin{array}{@{}l} \displaystyle {{\frac{1}{8}}\ {|_{\ 1 \ 1}^{\ 1}}}+{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ i \ i}^{\ 1}}}+{{\frac{1}{8 \ {j^{2}}}}\ {|_{\ j \ j}^{\ 1}}}+ \ \ \displaystyle {{\frac{1}{8 \ {k^{2}}}}\ {|_{\ k \ k}^{\ 1}}}-{{\frac{1}{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\ ij \ ij}^{\ 1}}}- \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\ ik \ ik}^{\ 1}}}-{{\frac{1}{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\ jk \ jk}^{\ 1}}}- \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}}}\ {|_{\ ijk \ ijk}^{\ 1}}}+{{\frac{1}{8}}\ {|_{\ 1 \ i}^{\ i}}}+{{\frac{1}{8}}\ {|_{\ i \ 1}^{\ i}}}- \ \ \displaystyle {{\frac{1}{8 \ {j^{2}}}}\ {|_{\ j \ ij}^{\ i}}}-{{\frac{1}{8 \ {k^{2}}}}\ {|_{\ k \ ik}^{\ i}}}+{{\frac{1}{8 \ {j^{2}}}}\ {|_{\ ij \ j}^{\ i}}}+ \ \ \displaystyle {{\frac{1}{8 \ {k^{2}}}}\ {|_{\ ik \ k}^{\ i}}}-{{\frac{1}{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\ jk \ ijk}^{\ i}}}- \ \ \displaystyle {{\frac{1}{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\ ijk \ jk}^{\ i}}}+{{\frac{1}{8}}\ {|_{\ 1 \ j}^{\ j}}}+{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ i \ ij}^{\ j}}}+ \ \ \displaystyle {{\frac{1}{8}}\ {|_{\ j \ 1}^{\ j}}}-{{\frac{1}{8 \ {k^{2}}}}\ {|_{\ k \ jk}^{\ j}}}-{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ ij \ i}^{\ j}}}+ \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\ ik \ ijk}^{\ j}}}+{{\frac{1}{8 \ {k^{2}}}}\ {|_{\ jk \ k}^{\ j}}}+ \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\ ijk \ ik}^{\ j}}}+{{\frac{1}{8}}\ {|_{\ 1 \ k}^{\ k}}}+{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ i \ ik}^{\ k}}}+ \ \ \displaystyle {{\frac{1}{8 \ {j^{2}}}}\ {|_{\ j \ jk}^{\ k}}}+{{\frac{1}{8}}\ {|_{\ k \ 1}^{\ k}}}- \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\ ij \ ijk}^{\ k}}}-{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ ik \ i}^{\ k}}}- \ \ \displaystyle {{\frac{1}{8 \ {j^{2}}}}\ {|_{\ jk \ j}^{\ k}}}-{{\frac{1}{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\ ijk \ ij}^{\ k}}}+ \ \ \displaystyle {{\frac{1}{8}}\ {|_{\ 1 \ ij}^{\ ij}}}+{{\frac{1}{8}}\ {|_{\ i \ j}^{\ ij}}}-{{\frac{1}{8}}\ {|_{\ j \ i}^{\ ij}}}+ \ \ \displaystyle {{\frac{1}{8 \ {k^{2}}}}\ {|_{\ k \ ijk}^{\ ij}}}+{{\frac{1}{8}}\ {|_{\ ij \ 1}^{\ ij}}}- \ \ \displaystyle {{\frac{1}{8 \ {k^{2}}}}\ {|_{\ ik \ jk}^{\ ij}}}+{{\frac{1}{8 \ {k^{2}}}}\ {|_{\ jk \ ik}^{\ ij}}}+ \ \ \displaystyle {{\frac{1}{8 \ {k^{2}}}}\ {|_{\ ijk \ k}^{\ ij}}}+{{\frac{1}{8}}\ {|_{\ 1 \ ik}^{\ ik}}}+{{\frac{1}{8}}\ {|_{\ i \ k}^{\ ik}}}- \ \ \displaystyle {{\frac{1}{8 \ {j^{2}}}}\ {|_{\ j \ ijk}^{\ ik}}}-{{\frac{1}{8}}\ {|_{\ k \ i}^{\ ik}}}+{{\frac{1}{8 \ {j^{2}}}}\ {|_{\ ij \ jk}^{\ ik}}}+ \ \ \displaystyle {{\frac{1}{8}}\ {|_{\ ik \ 1}^{\ ik}}}-{{\frac{1}{8 \ {j^{2}}}}\ {|_{\ jk \ ij}^{\ ik}}}- \ \ \displaystyle {{\frac{1}{8 \ {j^{2}}}}\ {|_{\ ijk \ j}^{\ ik}}}+{{\frac{1}{8}}\ {|_{\ 1 \ jk}^{\ jk}}}+ \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}}}\ {|_{\ i \ ijk}^{\ jk}}}+{{\frac{1}{8}}\ {|_{\ j \ k}^{\ jk}}}-{{\frac{1}{8}}\ {|_{\ k \ j}^{\ jk}}}- \ \ \displaystyle {{\frac{1}{8 \ {i^{2}}}}\ {|_{\ ij \ ik}^{\ jk}}}+{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ ik \ ij}^{\ jk}}}+ \ \ \displaystyle {{\frac{1}{8}}\ {|_{\ jk \ 1}^{\ jk}}}+{{\frac{1}{8 \ {i^{2}}}}\ {|_{\ ijk \ i}^{\ jk}}}+{{\frac{1}{8}}\ {|_{\ 1 \ ijk}^{\ ijk}}}+ \ \ \displaystyle {{\frac{1}{8}}\ {|_{\ i \ jk}^{\ ijk}}}-{{\frac{1}{8}}\ {|_{\ j \ ik}^{\ ijk}}}+{{\frac{1}{8}}\ {|_{\ k \ ij}^{\ ijk}}}+ \ \ \displaystyle {{\frac{1}{8}}\ {|_{\ ij \ k}^{\ ijk}}}-{{\frac{1}{8}}\ {|_{\ ik \ j}^{\ ijk}}}+{{\frac{1}{8}}\ {|_{\ jk \ i}^{\ ijk}}}+ \ \ \displaystyle {{\frac{1}{8}}\ {|_{\ ijk \ 1}^{\ ijk}}}$

(46)

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas

test
         e     /
         λ     =    ΩX

$\label{eq47} \mbox{\rm true}$

(47)

Type: Boolean