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The Algebra of Complex Numbers Is Frobenius In Many Ways

Linear operators over a 2-dimensional vector space representing the algebra of complex numbers

Ref:

We need the Axiom LinearOperator library.

fricas
)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 2-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

fricas
dim:=2

\label{eq1}2(1)
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], ℚ)

\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, i ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()

\label{eq3}\left[{|_{\  1}}, \:{|_{\  i}}\right](3)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)))
fricas
𝐝:ℒ 𝐋      := basisIn()

\label{eq4}\left[{|^{\  1}}, \:{|^{\  i}}\right](4)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)))
fricas
I:𝐋:=[1]   -- identity for composition

\label{eq5}{|_{\  1}^{\  1}}+{|_{\  i}^{\  i}}(5)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
X:𝐋:=[2,1] -- twist

\label{eq6}{|_{\  1 \  1}^{\  1 \  1}}+{|_{\  i \  1}^{\  1 \  i}}+{|_{\  1 \  i}^{\  i \  1}}+{|_{\  i \  i}^{\  i \  i}}(6)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
V:𝐋:=ev(1) -- evaluation

\label{eq7}{|^{\  1 \  1}}+{|^{\  i \  i}}(7)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
Λ:𝐋:=co(1) -- co-evaluation

\label{eq8}{|_{\  1 \  1}}+{|_{\  i \  i}}(8)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
Type: Void

Now generate structure constants for Complex Algebra

The basis consists of the real and imaginary units. We use complex multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Split-complex can be specified by Caley-Dickson parameter (i2)

fricas
i2:=sp('i,[2])

\label{eq9}i^{2}(9)
Type: Symbol
fricas
--i2:= -1  -- complex
QQ := ℂ(ℚ,'i,-i2);
Type: Type

Basis: Each B.i is a complex number

fricas
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)

\label{eq10}\left[ 1, \: i \right](10)
Type: List(CaleyDickson(Expression(Integer),i,-(i[;2])))
fricas
-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

\label{eq11}\left[ 
\begin{array}{cc}
1 & i 
\
i &{i^{2}}
(11)
Type: Matrix(CaleyDickson(Expression(Integer),i,-(i[;2])))
fricas
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real(x/y),M)
Type: Void
fricas
-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
fricas
Compiling function S with type CaleyDickson(Expression(Integer),i,-(
      i[;2])) -> Matrix(Expression(Integer))

\label{eq12}\left[{\left[{\left[ 1, \: 0 \right]}, \:{\left[ 0, \:{i^{2}}\right]}\right]}, \:{\left[{\left[ 0, \: 1 \right]}, \:{\left[ 1, \: 0 \right]}\right]}\right](12)
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{eq13}{|_{\  1}^{\  1 \  1}}+{|_{\  i}^{\  1 \  i}}+{|_{\  i}^{\  i \  1}}+{{i^{2}}\ {|_{\  1}^{\  i \  i}}}(13)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
arity Y

\label{eq14}{{+}^{2}}\over +(14)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)))
fricas
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

\label{eq15}\left[ 
\begin{array}{cc}
{|_{\  1}}&{|_{\  i}}
\
{|_{\  i}}&{{i^{2}}\ {|_{\  1}}}
(15)
Type: Matrix(ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)))

Units

fricas
e:=𝐞.1; i:=𝐞.2;
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Multiplication of arbitrary ccomplex numbers a and b

fricas
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

\label{eq16}{{a_{1}}\ {|_{\  1}}}+{{a_{2}}\ {|_{\  i}}}(16)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

\label{eq17}{{b_{1}}\ {|_{\  1}}}+{{b_{2}}\ {|_{\  i}}}(17)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
(a*b)/Y

\label{eq18}{{\left({{i^{2}}\ {a_{2}}\ {b_{2}}}+{{a_{1}}\ {b_{1}}}\right)}\ {|_{\  1}}}+{{\left({{a_{1}}\ {b_{2}}}+{{a_{2}}\ {b_{1}}}\right)}\ {|_{\  i}}}(18)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Multiplication is Associative

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

\label{eq19} \mbox{\rm true} (19)
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{eq20}{{u^{1, \: 1}}\ {|^{\  1 \  1}}}+{{u^{1, \: 2}}\ {|^{\  1 \  i}}}+{{u^{2, \: 1}}\ {|^{\  i \  1}}}+{{u^{2, \: 2}}\ {|^{\  i \  i}}}(20)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),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:


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\label{eq21}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (21)
(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

\label{eq22}\begin{array}{@{}l}
\displaystyle
{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|^{\  1 \  i \  1}}}+{{\left({u^{2, \: 2}}-{{i^{2}}\ {u^{1, \: 1}}}\right)}\ {|^{\  1 \  i \  i}}}+ 
\
\
\displaystyle
{{\left(-{u^{2, \: 2}}+{{i^{2}}\ {u^{1, \: 1}}}\right)}\ {|^{\  i \  i \  1}}}+{{\left(-{{i^{2}}\ {u^{2, \: 1}}}+{{i^{2}}\ {u^{1, \: 2}}}\right)}\ {|^{\  i \  i \  i}}}
(22)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),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{eq23}{2 \ {|^{\  1 \  1}}}+{2 \ {i^{2}}\ {|^{\  i \  i}}}(23)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

\label{eq24}{2 \ {|^{\  1 \  1}}}+{2 \ {i^{2}}\ {|^{\  i \  i}}}(24)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test(Ù=Ú)

\label{eq25} \mbox{\rm true} (25)
Type: Boolean

forms a non-degenerate associative scalar product for Y

fricas
Ũ := Ù

\label{eq26}{2 \ {|^{\  1 \  1}}}+{2 \ {i^{2}}\ {|^{\  i \  i}}}(26)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

\label{eq27} \mbox{\rm true} (27)
Type: Boolean
fricas
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)

\label{eq28}4 \ {i^{2}}(28)
Type: Expression(Integer)

General Solution

We may consider the problem where multiplication Y is given, and look for all associative scalar products U = U(Y)

This problem can be solved using linear algebra.

fricas
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);
Type: Matrix(Expression(Integer))
fricas
nrows(J),ncols(J)

\label{eq29}\left[ 8, \: 4 \right](29)
Type: Tuple(PositiveInteger)

The matrix J transforms the coefficients of the tensor U into coefficients of the tensor \Phi. We are looking for the general linear family of tensors U=U(Y,p_i) such that J transforms U into \Phi=0 for any such U.

If the null space of the J matrix is not empty we can use the basis to find all non-trivial solutions for U:

fricas
Ñ:=nullSpace(J)

\label{eq30}\left[{\left[ 0, \: 1, \: 1, \: 0 \right]}, \:{\left[{1 \over{i^{2}}}, \: 0, \: 0, \: 1 \right]}\right](30)
Type: List(Vector(Expression(Integer)))
fricas
ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )

\label{eq31}\left[{{u^{1, \: 1}}={{p_{2}}\over{i^{2}}}}, \:{{u^{1, \: 2}}={p_{1}}}, \:{{u^{2, \: 1}}={p_{1}}}, \:{{u^{2, \: 2}}={p_{2}}}\right](31)
Type: List(Equation(Expression(Integer)))

This defines a family of pre-Frobenius algebras:

fricas
zero? eval(ω,ℰ)

\label{eq32} \mbox{\rm true} (32)
Type: Boolean
fricas
Ų:𝐋 := eval(U,ℰ)

\label{eq33}{{{p_{2}}\over{i^{2}}}\ {|^{\  1 \  1}}}+{{p_{1}}\ {|^{\  1 \  i}}}+{{p_{1}}\ {|^{\  i \  1}}}+{{p_{2}}\ {|^{\  i \  i}}}(33)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Frobenius Form (co-unit)

fricas
d:=ε1*𝐝.1+εi*𝐝.2

\label{eq34}{�� 1 \ {|^{\  1}}}+{�� i \ {|^{\  i}}}(34)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
𝔇:=equate(d=
    (    e I   ) / _
          Ų    )
fricas
Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i]),Expression(Integer))) -> List(Equation
      (Expression(Integer)))

\label{eq35}\left[{�� 1 ={{p_{2}}\over{i^{2}}}}, \:{�� i ={p_{1}}}\right](35)
Type: List(Equation(Expression(Integer)))

Express scalar product in terms of Frobenius form

fricas
𝔓:=solve(𝔇,Ξ(sb('p,[i]), i,1..#Ñ)).1

\label{eq36}\left[{{p_{1}}= �� i}, \:{{p_{2}}={{i^{2}}\  �� 1}}\right](36)
Type: List(Equation(Expression(Integer)))
fricas
Ų:=eval(Ų,𝔓)

\label{eq37}{�� 1 \ {|^{\  1 \  1}}}+{�� i \ {|^{\  1 \  i}}}+{�� i \ {|^{\  i \  1}}}+{{i^{2}}\  �� 1 \ {|^{\  i \  i}}}(37)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test
        Y     /
        d     =  Ų

\label{eq38} \mbox{\rm true} (38)
Type: Boolean

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{eq39}\left[ 
\begin{array}{cc}
�� 1 & �� i 
\
�� i &{{i^{2}}\  �� 1}
(39)
Type: Matrix(Expression(Integer))

The scalar product must be non-degenerate:

fricas
Ů:=determinant u1

\label{eq40}-{{�� i}^{2}}+{{i^{2}}\ {{�� 1}^{2}}}(40)
Type: Expression(Integer)
fricas
factor(numer Ů)/factor(denom Ů)

\label{eq41}-{\left({{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}\right)}(41)
Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))))

Cartan-Killing is a special case

fricas
ck:=solve(equate(eval(Ũ,𝔓)=Ų),[ε1,εi]).1

\label{eq42}\left[{�� 1 = 2}, \:{�� i = 0}\right](42)
Type: List(Equation(Expression(Integer)))

Frobenius scalar product of complex numbers a and b

fricas
a:=sb('a,[1])*e+sb('a,[2])*i

\label{eq43}{{a_{1}}\ {|_{\  1}}}+{{a_{2}}\ {|_{\  i}}}(43)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
b:=sb('b,[1])*e+sb('b,[2])*i

\label{eq44}{{b_{1}}\ {|_{\  1}}}+{{b_{2}}\ {|_{\  i}}}(44)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
(a,a)/Ų

\label{eq45}{2 \ {a_{1}}\ {a_{2}}\  �� i}+{{\left({{i^{2}}\ {{a_{2}}^{2}}}+{{a_{1}}^{2}}\right)}\  �� 1}(45)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
(b,b)/Ų

\label{eq46}{2 \ {b_{1}}\ {b_{2}}\  �� i}+{{\left({{i^{2}}\ {{b_{2}}^{2}}}+{{b_{1}}^{2}}\right)}\  �� 1}(46)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
ab:=(a,b)/Ų

\label{eq47}{{\left({{a_{1}}\ {b_{2}}}+{{a_{2}}\ {b_{1}}}\right)}\  �� i}+{{\left({{i^{2}}\ {a_{2}}\ {b_{2}}}+{{a_{1}}\ {b_{1}}}\right)}\  �� 1}(47)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
solve(equate(ab=0*ab),[sb('b,[1]),sb('b,[2]),i2])

\label{eq48}\left[{\left[{{b_{1}}={{{{\left(-{{a_{1}}\ {b_{2}}}-{{a_{2}}\ {b_{1}}}\right)}\  �� i}-{{a_{1}}\ {b_{1}}\  �� 1}}\over{{a_{2}}\ {b_{2}}\  �� 1}}}\right]}\right](48)
Type: List(List(Equation(Expression(Integer))))

Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

fricas
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)

\label{eq49}{{u_{1, \: 1}}\ {|_{\  1 \  1}}}+{{u_{1, \: 2}}\ {|_{\  1 \  i}}}+{{u_{2, \: 1}}\ {|_{\  i \  1}}}+{{u_{2, \: 2}}\ {|_{\  i \  i}}}(49)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
UXΩ:=(I*ΩX)/(Ų*I);
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
ΩXU:=(ΩX*I)/(I*Ų);
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
eq1:=equate(UXΩ=I);
Type: List(Equation(Expression(Integer)))
fricas
eq2:=equate(ΩXU=I);
Type: List(Equation(Expression(Integer)))
fricas
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));
Type: List(List(Equation(Expression(Integer))))
fricas
if #snake ~= 1 then error "no solution"
Type: Void
fricas
Ω:=eval(Ω,snake(1))

\label{eq50}\begin{array}{@{}l}
\displaystyle
-{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  1}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  i}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  1}}}- 
\
\
\displaystyle
{{�� 1 \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  i}}}
(50)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

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

fricas
squareFreePart factor denom Ů / squareFreePart factor numer Ů
Function: squareFree : % -> Factored(%) is missing from domain: Factored(SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))) Internal Error The function squareFree with signature (Factored $)$ is missing from domain Factored (SparseMultivariatePolynomial (Integer) (Kernel (Expression (Integer))))

Check "dimension" and the snake relations.

fricas
O:𝐋:=
       Ω    /
       Ų

\label{eq51}2(51)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test
    (    I ΩX     )  /
    (     Ų I     )  =  I

\label{eq52} \mbox{\rm true} (52)
Type: Boolean
fricas
test
    (     ΩX I    )  /
    (    I Ų      )  =  I

\label{eq53} \mbox{\rm true} (53)
Type: Boolean

Cartan-Killing co-scalar

fricas
eval(Ω,ck)

\label{eq54}{{1 \over 2}\ {|_{\  1 \  1}}}+{{1 \over{2 \ {i^{2}}}}\ {|_{\  i \  i}}}(54)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Definition 4

Co-algebra

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

fricas
W:=
  (Y I) /
    Ų

\label{eq55}\begin{array}{@{}l}
\displaystyle
{�� 1 \ {|^{\  1 \  1 \  1}}}+{�� i \ {|^{\  1 \  1 \  i}}}+{�� i \ {|^{\  1 \  i \  1}}}+{{i^{2}}\  �� 1 \ {|^{\  1 \  i \  i}}}+{�� i \ {|^{\  i \  1 \  1}}}+ 
\
\
\displaystyle
{{i^{2}}\  �� 1 \ {|^{\  i \  1 \  i}}}+{{i^{2}}\  �� 1 \ {|^{\  i \  i \  1}}}+{{i^{2}}\  �� i \ {|^{\  i \  i \  i}}}
(55)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
λ:=
  (  ΩX I ΩX  ) /
  (  I  W  I  )

\label{eq56}\begin{array}{@{}l}
\displaystyle
-{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  1}^{\  1}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  i}^{\  1}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  1}^{\  1}}}- 
\
\
\displaystyle
{{�� 1 \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  i}^{\  1}}}+{{{{i^{2}}\  �� i}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  1}^{\  i}}}-{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  i}^{\  i}}}- 
\
\
\displaystyle
{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  1}^{\  i}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  i}^{\  i}}}
(56)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Cartan-Killing co-multiplication

fricas
eval(λ,ck)

\label{eq57}{{1 \over 2}\ {|_{\  1 \  1}^{\  1}}}+{{1 \over{2 \ {i^{2}}}}\ {|_{\  i \  i}^{\  1}}}+{{1 \over 2}\ {|_{\  1 \  i}^{\  i}}}+{{1 \over 2}\ {|_{\  i \  1}^{\  i}}}(57)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

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

\label{eq58} \mbox{\rm true} (58)
Type: Boolean
fricas
test
     (     ΩX I    )  /
     (    I  Y     )  =  λ

\label{eq59} \mbox{\rm true} (59)
Type: Boolean

Co-associativity

fricas
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )

\label{eq60} \mbox{\rm true} (60)
Type: Boolean

fricas
test
         e     /
         λ     =    ΩX

\label{eq61} \mbox{\rm true} (61)
Type: Boolean

Frobenius Condition (fork)

fricas
H :=
         Y    /
         λ

\label{eq62}\begin{array}{@{}l}
\displaystyle
-{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  1}^{\  1 \  1}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  i}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  1}^{\  1 \  1}}}-{{�� 1 \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  i}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{i^{2}}\  �� i}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  1}^{\  1 \  i}}}-{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  i}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  1}^{\  1 \  i}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  i}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{i^{2}}\  �� i}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  1}^{\  i \  1}}}-{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  i}^{\  i \  1}}}- 
\
\
\displaystyle
{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  1}^{\  i \  1}}}+{{�� i \over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  i}^{\  i \  1}}}- 
\
\
\displaystyle
{{{{{i^{2}}^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  1}^{\  i \  i}}}+{{{{i^{2}}\  �� i}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1 \  i}^{\  i \  i}}}+ 
\
\
\displaystyle
{{{{i^{2}}\  �� i}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  1}^{\  i \  i}}}-{{{{i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i \  i}^{\  i \  i}}}
(62)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test
     (   λ I   )  /
     (  I Y    )  =  H

\label{eq63} \mbox{\rm true} (63)
Type: Boolean
fricas
test
     (   I λ   )  /
     (    Y I  )  =  H

\label{eq64} \mbox{\rm true} (64)
Type: Boolean

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

fricas
test( eval(H,ck)=eval(H/H,ck) )

\label{eq65} \mbox{\rm true} (65)
Type: Boolean

And it is unique.

fricas
h1:=map(numer,ravel(H-H/H)::List FRAC POLY INT);
Type: List(Polynomial(Integer))
fricas
h2:=groebner h1

\label{eq66}\begin{array}{@{}l}
\displaystyle
\left[{{�� i}^{3}}, \:{{{\left(�� 1 + 2 \right)}\ {{�� i}^{2}}}-{{i^{2}}\ {{�� 1}^{3}}}+{2 \ {i^{2}}\ {{�� 1}^{2}}}}, \:{{2 \ {i^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {{�� 1}^{3}}}+{2 \ {{i^{2}}^{2}}\ {{�� 1}^{2}}}}, \: \right.
\
\
\displaystyle
\left.{{i^{2}}\  �� 1 \  �� i}, \:{{{{i^{2}}^{2}}\ {{�� 1}^{4}}}-{2 \ {{i^{2}}^{2}}\ {{�� 1}^{3}}}}\right] 
(66)
Type: List(Polynomial(Integer))
fricas
ck4:=solve(h2,[ε1,εi])

\label{eq67}\left[{\left[{�� 1 = 2}, \:{�� i = 0}\right]}, \:{\left[{�� 1 = 0}, \:{�� i = 0}\right]}\right](67)
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
fricas
test( eval(H,ck4.2)=eval(H/H,ck4.1) )
>> Error detected within library code: catdef: division by zero

Handle

fricas
Φ :=
         λ     /
         Y

\label{eq68}\begin{array}{@{}l}
\displaystyle
-{{{2 \ {i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1}^{\  1}}}+{{{2 \  �� i}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i}^{\  1}}}+{{{2 \ {i^{2}}\  �� i}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  1}^{\  i}}}- 
\
\
\displaystyle
{{{2 \ {i^{2}}\  �� 1}\over{{{�� i}^{2}}-{{i^{2}}\ {{�� 1}^{2}}}}}\ {|_{\  i}^{\  i}}}
(68)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

The Cartan-Killing form makes Φ of the identity

fricas
test( eval(Φ,ck)=I )

\label{eq69} \mbox{\rm true} (69)
Type: Boolean

and it can be the identity in only this one way.

fricas
solve(equate(Φ=I),[ε1,εi])

\label{eq70}\left[{\left[{�� 1 = 2}, \:{�� i = 0}\right]}\right](70)
Type: List(List(Equation(Expression(Integer))))

If handle is identity then fork is idempotent but the converse is not true

fricas
Φ1:=map(numer,ravel(Φ-I)::List FRAC POLY INT);
Type: List(Polynomial(Integer))
fricas
Φ2:=groebner Φ1

\label{eq71}\left[ �� i , \:{{{i^{2}}\ {{�� 1}^{2}}}-{2 \ {i^{2}}\  �� 1}}\right](71)
Type: List(Polynomial(Integer))
fricas
in?(ideal h2, ideal Φ2)

\label{eq72} \mbox{\rm true} (72)
Type: Boolean
fricas
in?(ideal Φ2, ideal h2)

\label{eq73} \mbox{\rm false} (73)
Type: Boolean

Figure 12

fricas
φφ:=          _
  ( Ω  Ω  ) / _
  ( X I I ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y );
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
φφ1:=map((x:ℚ):ℚ+->numer x,φφ)

\label{eq74}\begin{array}{@{}l}
\displaystyle
{{\left({2 \ {i^{2}}\ {{�� i}^{2}}}+{2 \ {{i^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\ {|_{\  1 \  1}}}-{4 \ {i^{2}}\  �� 1 \  �� i \ {|_{\  1 \  i}}}-{4 \ {i^{2}}\  �� 1 \  �� i \ {|_{\  i \  1}}}+ 
\
\
\displaystyle
{{\left({2 \ {{�� i}^{2}}}+{2 \ {i^{2}}\ {{�� 1}^{2}}}\right)}\ {|_{\  i \  i}}}
(74)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
φφ2:=denom(ravel(φφ).1)

\label{eq75}{{�� i}^{4}}-{2 \ {i^{2}}\ {{�� 1}^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {{�� 1}^{4}}}(75)
Type: SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))
fricas
test(φφ=(1/φφ2)*φφ1)

\label{eq76} \mbox{\rm true} (76)
Type: Boolean

For Cartan-Killing this is just the co-scalar

fricas
test(eval(φφ,ck)=eval(Ω,ck))

\label{eq77} \mbox{\rm true} (77)
Type: Boolean
fricas
test(eval((e,e)/H,ck)=eval(Ω,ck))

\label{eq78} \mbox{\rm true} (78)
Type: Boolean

Bi-algebra conditions

fricas
ΦΦ:=          _
  (  λ λ  ) / _
  ( I I X ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y ) ;
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test((e,e)/ΦΦ=φφ)

\label{eq79} \mbox{\rm true} (79)
Type: Boolean
fricas
test(eval(ΦΦ,ck)=eval(H,ck))

\label{eq80} \mbox{\rm true} (80)
Type: Boolean
fricas
test(eval(ΦΦ/(d,d),ck)=Ũ)

\label{eq81} \mbox{\rm true} (81)
Type: Boolean
fricas
test(eval(H/(d,d),ck)=Ũ)

\label{eq82} \mbox{\rm true} (82)
Type: Boolean

The Cartan Killing form is a bi-algebra

fricas
bi1:=map(numer,ravel(ΦΦ-H)::List FRAC POLY INT);
Type: List(Polynomial(Integer))
fricas
bi2:=groebner bi1

\label{eq83}\begin{array}{@{}l}
\displaystyle
\left[{{�� i}^{3}}, \:{{{\left(�� 1 + 2 \right)}\ {{�� i}^{2}}}-{{i^{2}}\ {{�� 1}^{3}}}+{2 \ {i^{2}}\ {{�� 1}^{2}}}}, \:{{2 \ {i^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {{�� 1}^{3}}}+{2 \ {{i^{2}}^{2}}\ {{�� 1}^{2}}}}, \: \right.
\
\
\displaystyle
\left.{{i^{2}}\  �� 1 \  �� i}, \:{{{{i^{2}}^{2}}\ {{�� 1}^{4}}}-{2 \ {{i^{2}}^{2}}\ {{�� 1}^{3}}}}\right] 
(83)
Type: List(Polynomial(Integer))
fricas
b:=solve( equate(ΦΦ=H), [ε1,εi] )

\label{eq84}\left[{\left[{�� 1 = 2}, \:{�� i = 0}\right]}\right](84)
Type: List(List(Equation(Expression(Integer))))
fricas
test(eval(Ų, b.1)=Ũ)

\label{eq85} \mbox{\rm true} (85)
Type: Boolean




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