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last edited 12 years ago by Bill Page |
Edit detail for SandBoxPauliAlgebra revision 4 of 7
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Editor: Bill Page
Time: 2011/05/31 19:37:32 GMT-7 |
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| Note: Clifford algebras | ||
changed: -QQ:=CliffordAlgebra(3,ℚ,matrix [[-q0,0,0],[0,-q1,0],[0,0,-q2]]) QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]]) removed: -arity Y changed: -\end{axiom} - -Multiplication of arbitrary Clifford numbers $a$ and $b$ -\begin{axiom} -a:=Σ(sb('a,[i])*𝐞.i, i,1..dim) -b:=Σ(sb('b,[i])*𝐞.i, i,1..dim) -(a*b)/Y -\end{axiom} XY := X/Y; \end{axiom} removed: -$U = \{ u_{ij} \}$ changed: -determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim) -\end{axiom} - -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. - -\begin{axiom} -)expose MCALCFN -J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol); -nrows(J),ncols(J) -\end{axiom} - -The matrix 'J' transforms the coefficients of the tensor $U$ -into coefficients of the tensor $\Phi$. We are looking for -the general linear family of tensors $U=U(Y,p_i)$ such that -'J' transforms $U$ into $\Phi=0$ for any such $U$. - -If the null space of the 'J' matrix is not empty we can use -the basis to find all non-trivial solutions for U: - -\begin{axiom} -Ñ:=nullSpace(J); -ℰ:=map((x,y)+->x=y, concat map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) ); -\end{axiom} - -This defines a family of pre-Frobenius algebras: -\begin{axiom} -zero? eval(ω,ℰ) -\end{axiom} - -In general the pairing is not symmetric! -\begin{axiom} -Ų:𝐋 := eval(U,ℰ) -matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim) -\end{axiom} - -Cartan-Killing is a special case -\begin{axiom} -ck:=solve(equate(Ũ=Ų),Ξ(sb('p,[i]), i,1..#Ñ)).1 -\end{axiom} \end{axiom} changed: -Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim) Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim) changed: -mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim) mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ũ), i,1..dim), j,1..dim) changed: - Ų Ũ changed: - ( Ų I ) = I ( Ũ I ) = I changed: - ( I Ų ) = I - -\end{axiom} - -Cartan-Killing co-scalar -\begin{axiom} -eval(Ω,ck) -\end{axiom} ( I Ũ ) = I \end{axiom} changed: -W:=(Y I) / Ų; - -\end{axiom} W:=(Y I) / Ũ; \end{axiom} changed: -eval(λ,ck) -\end{axiom} - -\begin{axiom} - ---test changed: -test( eval(H,ck)=eval(H/H,ck) ) -\end{axiom} test( H=H/H ) \end{axiom} changed: -test( eval(Φ,ck)=I ) -\end{axiom} test( Φ=I ) \end{axiom} changed: - Ų Ũ changed: - d = Ų - -\end{axiom} d = Ũ \end{axiom} changed: -φφ:= _ - ( Ω Ω ) / _ - ( X I I ) / _ - ( I X I ) / _ - ( I I X ) / _ - ( Y Y ); ΩXΩ:= ΩX * Ω; YXY:= Y * XY; arity(ΩXΩ) φφ := ΩXΩ / (I X I ) / YXY; changed: -test(eval(φφ,ck)=eval(Ω,ck)) -test(eval((e,e)/H,ck)=eval(Ω,ck)) -\end{axiom} test(φφ=Ω) test((e,e)/H=Ω) \end{axiom} changed: - ( I I X ) / _ - ( Y Y ) ; ( YXY ) ; changed: -test(eval(ΦΦ,ck)=eval(H/H,ck)) -test(eval(ΦΦ/(d,d),ck)=Ũ) -test(eval(H/(d,d),ck)=Ũ) -\end{axiom} - test(ΦΦ=H/H) test(ΦΦ/(d,d)=Ũ) test(H/(d,d)=Ũ) \end{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 AlgebrasZbigniew 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.
axiom
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/zope2/var/LatexWiki/ARITY.NRLIB/ARITY ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/zope2/var/LatexWiki/CMONAL.NRLIB/CMONAL ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/zope2/var/LatexWiki/CPROP.NRLIB/CPROP ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/zope2/var/LatexWiki/CLOP.NRLIB/CLOP CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY
Use the following macros for convenient notation
axiom
-- summation macro Σ(x,i, n)==reduce(+, [x for i in n])
Type: Void
axiom
-- list macro Ξ(f,i, n)==[f for i in n]
Type: Void
axiom
-- subscript and superscripts macro sb == subscript
Type: Void
axiom
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.
axiom
dim:=8
 | (1) |
Type: PositiveInteger?
axiom
macro ℒ == List
Type: Void
axiom
macro ℂ == CaleyDickson
Type: Void
axiom
macro ℚ == Expression Integer
Type: Void
axiom
𝐋 := ClosedLinearOperator(OVAR ['1,'i, 'j, 'k, 'ij, 'ik, 'jk, 'ijk], ℚ)
![\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, i , j , k , ij , ik , jk , ijk ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))](images/645834808841087549-16.0px.png "\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, i , j , k , ij , ik , jk , ijk ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))") | (2) |
Type: Type
axiom
𝐞:ℒ 𝐋 := basisOut()
![\label{eq3}\left[{|_{\ 1}}, \:{|_{\ i}}, \:{|_{\ j}}, \:{|_{\ k}}, \:{|_{\ ij}}, \:{|_{\ ik}}, \:{|_{\ jk}}, \:{|_{\ ijk}}\right]](images/3628059914917409431-16.0px.png "\label{eq3}\left[{|_{\ 1}}, \:{|_{\ i}}, \:{|_{\ j}}, \:{|_{\ k}}, \:{|_{\ ij}}, \:{|_{\ ik}}, \:{|_{\ jk}}, \:{|_{\ ijk}}\right]") | (3) |
axiom
𝐝:ℒ 𝐋 := basisIn()
![\label{eq4}\left[{|^{\ 1}}, \:{|^{\ i}}, \:{|^{\ j}}, \:{|^{\ k}}, \:{|^{\ ij}}, \:{|^{\ ik}}, \:{|^{\ jk}}, \:{|^{\ ijk}}\right]](images/5005601258201691986-16.0px.png "\label{eq4}\left[{|^{\ 1}}, \:{|^{\ i}}, \:{|^{\ j}}, \:{|^{\ k}}, \:{|^{\ ij}}, \:{|^{\ ik}}, \:{|^{\ jk}}, \:{|^{\ ijk}}\right]") | (4) |
axiom
I:𝐋:=[1] -- identity for composition
 | (5) |
axiom
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}}](images/4501158356527718379-16.0px.png "\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) |
axiom
V:𝐋:=ev(1) -- evaluation
 | (7) |
axiom
Λ:𝐋:=co(1) -- co-evaluation
 | (8) |
axiom
equate(eq)==map((x,y)+->(x=y), ravel lhs eq, ravel rhs eq);
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.
The Pauli Algebra as Cl(3)
Basis: Each B.i is a Clifford number
axiom
q0:=sp('i,
 | (9) |
Type: Symbol
axiom
q1:=sp('j,
 | (10) |
Type: Symbol
axiom
q2:=sp('k,
 | (11) |
Type: Symbol
axiom
QQ:=CliffordAlgebra(3,ℚ, matrix [[q0, 0, 0], [0, q1, 0], [0, 0, q2]])
![\label{eq12}\hbox{\axiomType{CliffordAlgebra}\ } (3, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , [ [ * 001 i (2) , 0, 0 ] , [ 0, * 001 j (2) , 0 ] , [ 0, 0, * 001 k (2) ] ])](images/7284212448618751061-16.0px.png "\label{eq12}\hbox{\axiomType{CliffordAlgebra}\ } (3, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , [ [ * 001 i (2) , 0, 0 ] , [ 0, * 001 j (2) , 0 ] , [ 0, 0, * 001 k (2) ] ])") | (12) |
Type: Type
axiom
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]](images/4070421524196703516-16.0px.png "\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) |
axiom
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}}}](images/5852572173596026448-16.0px.png "\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,
axiom
S(y) == map(x +-> coefficient(recip(y)*x,[]), M)
Type: Void
axiom
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
axiom
Compiling function S with type CliffordAlgebra(3,Expression(Integer) , [[*001i(2), 0, 0], [0, *001j(2), 0], [0, 0, *001k(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]](images/6869713486503648537-16.0px.png "\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))))
axiom
-- 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}}}](images/7488340274450820960-16.0px.png "\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) |
axiom
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}}}](images/7627477422961147532-16.0px.png "\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,
axiom
XY := X/Y;
Multiplication is Associative
axiom
test( ( I Y ) / _ ( Y ) = _ ( Y I ) / _ ( Y ) )
 | (18) |
Type: Boolean
A scalar product is denoted by the (2,0)-tensor
axiom
U:=Σ(Σ(script('u,
![\label{eq19}\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}}}](images/15940054754840550-16.0px.png "\label{eq19}\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}}}") | (19) |
Definition 1
We say that the scalar product is associative if the tensor equation holds:
Y = Y
U U
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.
axiom
ω:𝐋 := _
( Y I ) / _
U - _
( I Y ) / _
U;
Definition 2
An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.
The Cartan-Killing Trace
axiom
Ú:=
( Y Λ ) / _
( Y I ) / _
V
![\label{eq20}\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}}}](images/2166671089660007256-16.0px.png "\label{eq20}\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}}}") | (20) |
axiom
Ù:=
( Λ Y ) / _
( I Y ) / _
V
![\label{eq21}\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}}}](images/1676023476227563049-16.0px.png "\label{eq21}\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}}}") | (21) |
axiom
test(Ù=Ú)
 | (22) |
Type: Boolean
forms a non-degenerate associative scalar product for Y
axiom
Ũ := Ù
![\label{eq23}\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}}}](images/648383990859280457-16.0px.png "\label{eq23}\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}}}") | (23) |
axiom
test
( Y I ) /
Ũ =
( I Y ) /
Ũ
 | (24) |
Type: Boolean
The scalar product must be non-degenerate:
axiom
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ),j, 1..dim), i, 1..dim)
 | (25) |
Type: Expression(Integer)
axiom
factor(numer Ů)/factor(denom Ů)
 | (26) |
Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,
Definition 3
Co-scalar product (pairing)
Solve the [Snake Relation]? as a system of linear equations.
axiom
mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ũ),i, 1..dim), j, 1..dim)
![\label{eq27}\left[
\begin{array}{cccccccc}
{1 \over 8}& 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 &{1 \over{8 \ {i^{2}}}}& 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 &{1 \over{8 \ {j^{2}}}}& 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 &{1 \over{8 \ {k^{2}}}}& 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & -{1 \over{8 \ {i^{2}}\ {j^{2}}}}& 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & -{1 \over{8 \ {i^{2}}\ {k^{2}}}}& 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & -{1 \over{8 \ {j^{2}}\ {k^{2}}}}& 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -{1 \over{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}}}](images/7394751493717093610-16.0px.png "\label{eq27}\left[
\begin{array}{cccccccc}
{1 \over 8}& 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 &{1 \over{8 \ {i^{2}}}}& 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 &{1 \over{8 \ {j^{2}}}}& 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 &{1 \over{8 \ {k^{2}}}}& 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & -{1 \over{8 \ {i^{2}}\ {j^{2}}}}& 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & -{1 \over{8 \ {i^{2}}\ {k^{2}}}}& 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & -{1 \over{8 \ {j^{2}}\ {k^{2}}}}& 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -{1 \over{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}}}") | (27) |
Type: Union(Matrix(Expression(Integer)),
axiom
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i, 1..dim), j, 1..dim);
axiom
ΩX:=Ω/X;
Check "dimension" and the snake relations.
axiom
d:𝐋:=
Ω /
Ũ
 | (28) |
axiom
test
( I ΩX ) /
( Ũ I ) = I
 | (29) |
Type: Boolean
axiom
test
( ΩX I ) /
( I Ũ ) = I
 | (30) |
Type: Boolean
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
axiom
W:=(Y I) / Ũ;
Cartan-Killing co-multiplication
axiom
λ:= _
( I ΩX ) / _
( Y I ) ;
axiom
test
( ΩX I ) /
( I Y ) = λ
 | (31) |
Type: Boolean
Frobenius Condition (fork)
axiom
H := Y / λ;
axiom
test
( λ I ) /
( I Y ) = H
 | (32) |
Type: Boolean
axiom
test
( I λ ) /
( Y I ) = H
 | (33) |
Type: Boolean
The Cartan-Killing form makes H of the Frobenius condition idempotent
axiom
test( H=H/H )
 | (34) |
Type: Boolean
Handle
axiom
Φ := λ / Y;
The Cartan-Killing form makes Φ of the identity
axiom
test( Φ=I )
 | (35) |
Type: Boolean
Definition 5
Unit
axiom
e:=𝐞.1
 | (36) |
axiom
test
e /
λ = ΩX
 | (37) |
Type: Boolean
Co-unit
axiom
d:=
( e I ) /
Ũ
 | (38) |
axiom
test
Y /
d = Ũ
 | (39) |
Type: Boolean
Figure 12
axiom
ΩXΩ:= ΩX * Ω;
axiom
YXY:= Y * XY;
axiom
arity(ΩXΩ)
 | (40) |
Type: ClosedProp?(ClosedLinearOperator?(OrderedVariableList?([1,
axiom
φφ := ΩXΩ / (I X I ) / YXY;
axiom
φφ1:=map((x:ℚ):ℚ+->numer x,φφ)
 | (41) |
axiom
φφ2:=denom(ravel(φφ).1)
 | (42) |
Type: SparseMultivariatePolynomial?(Integer,
axiom
test(φφ=(1/φφ2)*φφ1)
 | (43) |
Type: Boolean
For Cartan-Killing this is just the co-scalar
axiom
test(φφ=Ω)
 | (44) |
Type: Boolean
axiom
test((e,e)/H=Ω)
 | (45) |
Type: Boolean
Bi-algebra conditions
axiom
ΦΦ:= _ ( λ λ ) / _ ( I I X ) / _ ( I X I ) / _ ( YXY ) ;
axiom
test((e,e)/ΦΦ=φφ)
 | (46) |
Type: Boolean
axiom
test(ΦΦ=H/H)
 | (47) |
Type: Boolean
axiom
test(ΦΦ/(d,d)=Ũ)
 | (48) |
Type: Boolean
axiom
test(H/(d,d)=Ũ)
 | (49) |
Type: Boolean