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SandBox Quaternion Algebra is Frobenius in Many Ways
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SandBoxPauliAlgebra
  
last edited 15 years ago by Bill Page

Edit detail for SandBoxPauliAlgebra revision 4 of 7
1 2 3 4 5 6 7
Editor: Bill Page
Time: 2011/05/31 19:37:32 GMT-7
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:

We need the Axiom LinearOperator? library.

fricas
(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

fricas
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
fricas
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
fricas
-- subscript and superscripts
macro sb == subscript
Type: Void
fricas
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.

fricas
dim:=8
\label{eq1}8(1)
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
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
fricas
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)))
fricas
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)))
fricas
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))
fricas
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))
fricas
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))
fricas
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))
fricas
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

fricas
q0:=sp('i,[2])
\label{eq9}i^{2}(9)
Type: Symbol
fricas
q1:=sp('j,[2])
\label{eq10}j^{2}(10)
Type: Symbol
fricas
q2:=sp('k,[2])
\label{eq11}k^{2}(11)
Type: Symbol
fricas
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
fricas
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]]]))
fricas
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]]]))
fricas
S(y) == map(x +-> coefficient(recip(y)*x,[]),M)
Type: Void
fricas
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
fricas
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))
fricas
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)))
fricas
XY := X/Y;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication is Associative

fricas
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )
\label{eq18} \mbox{\rm true} (18)
Type: Boolean

A scalar product is denoted by the (2,0)-tensor

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

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{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)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
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}}} (21)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test(Ù=Ú)
\label{eq22} \mbox{\rm true} (22)
Type: Boolean

forms a non-degenerate associative scalar product for Y

fricas
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}}} (23)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
\label{eq24} \mbox{\rm true} (24)
Type: Boolean

The scalar product must be non-degenerate:

fricas
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
\label{eq25}{16777216}\ {{i^{2}}^{4}}\ {{j^{2}}^{4}}\ {{k^{2}}^{4}}(25)
Type: Expression(Integer)
fricas
factor(numer Ů)/factor(denom Ů)
\label{eq26}{16777216}\ {{i^{2}}^{4}}\ {{j^{2}}^{4}}\ {{k^{2}}^{4}}(26)
Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))))

Definition 3

Co-scalar product (pairing)

Solve the [Snake Relation]? as a system of linear equations.

fricas
mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ũ), i,1..dim), j,1..dim)
\label{eq27}\left[ \begin{array}{cccccccc} {\frac{1}{8}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 &{\frac{1}{8 \ {i^{2}}}}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{\frac{1}{8 \ {j^{2}}}}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 &{\frac{1}{8 \ {k^{2}}}}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & -{\frac{1}{8 \ {i^{2}}\ {j^{2}}}}& 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & -{\frac{1}{8 \ {i^{2}}\ {k^{2}}}}& 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{1}{8 \ {j^{2}}\ {k^{2}}}}& 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{1}{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}}} (27)
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))

Check "dimension" and the snake relations.

fricas
d:𝐋:=
       Ω    /
       Ũ
\label{eq28}8(28)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
    (    I ΩX     )  /
    (     Ũ I     )  =  I
\label{eq29} \mbox{\rm true} (29)
Type: Boolean
fricas
test
    (     ΩX I    )  /
    (    I Ũ      )  =  I
\label{eq30} \mbox{\rm true} (30)
Type: Boolean

Definition 4

Co-algebra

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

fricas
W:=(Y I) / Ũ;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Cartan-Killing co-multiplication

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{eq31} \mbox{\rm true} (31)
Type: Boolean

Frobenius Condition (fork)

fricas
H := Y / λ;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
     (   λ I   )  /
     (  I Y    )  =  H
\label{eq32} \mbox{\rm true} (32)
Type: Boolean
fricas
test
     (   I λ   )  /
     (    Y I  )  =  H
\label{eq33} \mbox{\rm true} (33)
Type: Boolean

The Cartan-Killing form makes H of the Frobenius condition idempotent

fricas
test( H=H/H )
\label{eq34} \mbox{\rm true} (34)
Type: Boolean

Handle

fricas
Φ := λ / Y;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

The Cartan-Killing form makes Φ of the identity

fricas
test( Φ=I )
\label{eq35} \mbox{\rm true} (35)
Type: Boolean

Definition 5

Unit

fricas
e:=𝐞.1
\label{eq36}|_{\ 1}(36)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
         e     /
         λ     =    ΩX
\label{eq37} \mbox{\rm true} (37)
Type: Boolean

Co-unit

fricas
d:=
    (    e I   ) /
          Ũ
\label{eq38}8 \ {|^{\ 1}}(38)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
        Y     /
        d     =  Ũ
\label{eq39} \mbox{\rm true} (39)
Type: Boolean

Figure 12

fricas
ΩXΩ:= ΩX * Ω;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
YXY:= Y * XY;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
arity(ΩXΩ)
\label{eq40}\frac{0}{{+}^{4}}(40)
Type: ClosedProp?(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
fricas
φφ := ΩXΩ / (I X I ) / YXY;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
φφ1:=map((x:ℚ):ℚ+->numer x,φφ)
\label{eq41}{|_{\ 1 \ 1}}+{|_{\ i \ i}}+{|_{\ j \ j}}+{|_{\ k \ k}}-{|_{\ ij \ ij}}-{|_{\ ik \ ik}}-{|_{\ jk \ jk}}-{|_{\ ijk \ ijk}}(41)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
φφ2:=denom(ravel(φφ).1)
\label{eq42}8(42)
Type: SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))
fricas
test(φφ=(1/φφ2)*φφ1)
\label{eq43} \mbox{\rm false} (43)
Type: Boolean

For Cartan-Killing this is just the co-scalar

fricas
test(φφ=Ω)
\label{eq44} \mbox{\rm true} (44)
Type: Boolean
fricas
test((e,e)/H=Ω)
\label{eq45} \mbox{\rm true} (45)
Type: Boolean

Bi-algebra conditions

fricas
ΦΦ:=          _
  (  λ λ  ) / _
  ( I I X ) / _
  ( I X I ) / _
  (  YXY  ) ;

   >> System error:
   Heap exhausted (no more space for allocation).
13434880 bytes available, 134217744 requested.

PROCEED WITH CAUTION.