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# Edit detail for The Algebra of Complex Numbers Is Frobenius In Many Ways revision 9 of 9

 1 2 3 4 5 6 7 8 9 Editor: Bill Page Time: 2011/06/04 08:32:42 GMT-7 Note: new notation

changed:
-𝐋 := ClosedLinearOperator(OVAR ['1,'2], ℚ)
𝐋 := ClosedLinearOperator(OVAR ['1,'i], ℚ)

changed:
-Split-complex can be specified by Caley-Dickson parameter (q0 = -1)
-\begin{axiom}
---q:=1  -- split-complex
-q:=sp('i,)
-QQ := ℂ(ℚ,'i,q);
-\end{axiom}
-
-Basis: Each B.i is a quaternion number
Split-complex can be specified by Caley-Dickson parameter (i2)
\begin{axiom}
i2:=sp('i,)
--i2:= -1  -- complex
QQ := ℂ(ℚ,'i,-i2);
\end{axiom}

Basis: Each B.i is a complex number

changed:
-S(y) == map(x +-> real real(x/y),M)
S(y) == map(x +-> real(x/y),M)

changed:
-Multiplication of arbitrary quaternions $a$ and $b$
Units
\begin{axiom}
e:=𝐞.1; i:=𝐞.2;
\end{axiom}

Multiplication of arbitrary ccomplex numbers $a$ and $b$

removed:
-u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
-J::OutputForm * u::OutputForm = 0

removed:
-\end{axiom}
-
-In general the pairing is symmetric!
-\begin{axiom}

changed:
-matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim)
-\end{axiom}
-
-This is the most general form of the "dot product" of two quaternions
-\begin{axiom}
-(a*b)/Ų
-(a*a)/Ų
-\end{axiom}
\end{axiom}

Frobenius Form (co-unit)
\begin{axiom}
d:=ε1*𝐝.1+εi*𝐝.2
𝔇:=equate(d=
(    e I   ) / _
Ų    )
\end{axiom}

Express scalar product in terms of Frobenius form
\begin{axiom}
𝔓:=solve(𝔇,Ξ(sb('p,[i]), i,1..#Ñ)).1
Ų:=eval(Ų,𝔓)
test
Y     /
d     =  Ų

\end{axiom}

In general the pairing is not symmetric!
\begin{axiom}
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
--eigenvectors(u1::Matrix FRAC POLY INT)
\end{axiom}

The scalar product must be non-degenerate:
\begin{axiom}
Ů:=determinant u1
factor(numer Ů)/factor(denom Ů)
\end{axiom}

changed:
-ck:=solve(equate(Ũ=Ų),Ξ(sb('p,[i]), i,1..#Ñ)).1
-\end{axiom}
-
-The scalar product must be non-degenerate:
-\begin{axiom}
-Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim)
-factor(numer Ů)/factor(denom Ů)
-\end{axiom}
ck:=solve(equate(eval(Ũ,𝔓)=Ų),[ε1,εi]).1
\end{axiom}

Frobenius scalar product of complex numbers $a$ and $b$
\begin{axiom}
a:=sb('a,)*e+sb('a,)*i
b:=sb('b,)*e+sb('b,)*i
(a,a)/Ų
(b,b)/Ų
ab:=(a,b)/Ų
solve(equate(ab=0*ab),[sb('b,),sb('b,),i2])
\end{axiom}

changed:
-  Co-pairing
Co-scalar product

changed:
-d1:=(I*Ω)/(Ų*I);
-d2:=(Ω*I)/(I*Ų);
-eq1:=equate(d1=I);
-eq2:=equate(d2=I);
ΩX:=Ω/X;
UXΩ:=(I*ΩX)/(Ų*I);
ΩXU:=(ΩX*I)/(I*Ų);
eq1:=equate(UXΩ=I);
eq2:=equate(ΩXU=I);

changed:
-matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)
-\end{axiom}
ΩX:=Ω/X;
\end{axiom}

The common demoninator is $1/\sqrt{\mathring{U}}$
\begin{axiom}
squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
\end{axiom}

changed:
-d:𝐋:=
O:𝐋:=

removed:
-       X    /

changed:
-    (    I Ω     )  /
-    (     Ų I    )  =  I
-
-test
-    (     Ω I    )  /
-    (    I Ų     )  =  I
-
-\end{axiom}
(    I ΩX     )  /
(     Ų I     )  =  I

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

\end{axiom}

Cartan-Killing co-scalar
\begin{axiom}
eval(Ω,ck)
\end{axiom}

changed:
-W:=(Y,I)/Ų
-λ:=(Ω,I,Ω)/(I,W,I)
-\end{axiom}
-
-\begin{axiom}
-
-test
-     (    I Ω     )  /
-     (     Y I    )  =  λ
-
-test
-     (     Ω I    )  /
-     (    I Y     )  =  λ
-
-\end{axiom}

W:=
(Y I) /
Ų

λ:=
(  ΩX I ΩX  ) /
(  I  W  I  )

\end{axiom}

Cartan-Killing co-multiplication
\begin{axiom}
eval(λ,ck)
\end{axiom}

\begin{axiom}

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

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

\end{axiom}

\begin{axiom}

test
e     /
λ     =    ΩX

\end{axiom}

changed:
-ck4:=solve(h2,[p,p])
-test( eval(H,ck4.2)=eval(H/H,ck4.2) )
-determinant Ξ(Ξ(retract( (𝐞.i * 𝐞.j)/eval(Ų,ck4.2) ), j,1..dim), i,1..dim)
-\end{axiom}
ck4:=solve(h2,[ε1,εi])
test( eval(H,ck4.2)=eval(H/H,ck4.1) )
determinant Ξ(Ξ(retract( (𝐞.i * 𝐞.j)/eval(Ų,ck4.1) ), j,1..dim), i,1..dim)
\end{axiom}

changed:
-Φ:𝐋 :=
Φ :=

changed:
-solve(equate(Φ=I),[p,p])
-\end{axiom}
solve(equate(Φ=I),[ε1,εi])
\end{axiom}

changed:
-Bi-algebra conditions
-\begin{axiom}
-ΦΦ:=         _
-  (  λ λ  ) / _
Figure 12
\begin{axiom}

φφ:=          _
( Ω  Ω  ) / _
( X I I ) / _

changed:
-  (  Y Y  );
-
-test( H/H = ΦΦ )
-\end{axiom}
-The Cartan Killing form is a bi-algebra
-\begin{axiom}
-bi1:=map(numer,ravel(ΦΦ-H)::List FRAC POLY INT);
-bi2:=groebner bi1
-b:=solve( equate(ΦΦ=H), [sb('p,[i]) for i in 1..#Ñ] )
-test(eval(Ų, b.1)=Ũ)
-\end{axiom}
-
-Scalars for figure 12 (only 1 non-zero for Cartan-Killing form)
-
-\begin{axiom}
-
-φφ:=          _
-  (  Ω Ω  ) / _
-  ( I X I ) / _
-  (  Y Y  );
( I I X ) / _
(  Y  Y );

changed:
-eval(φφ,ck)
-\end{axiom}
-
-Definition 5
-
-  i = Unit of the algebra
-
-\begin{axiom}
-
-i:=𝐞.1
-test
-         i     /
-         λ     =    Ω
-
-\end{axiom}
-
-Co-unit
-
-<center><pre>
-i
-U
-</pre></center>
-
-\begin{axiom}
-
-ι:𝐋:=
-    (    i I   ) /
-          Ų
-
-\end{axiom}
-<center><pre>
-Y=U
-ι
-</pre></center>
-\begin{axiom}
-test
-        Y     /
-        ι     =  Ų
-
-\end{axiom}
-
-For example:
-\begin{axiom}
-ex1:=[q=1,p=0,p=1]
-Ų0:𝐋  :=eval(Ų,ex1)
-Ω0:𝐋  :=eval(Ω,ex1)$𝐋 -λ0:𝐋 :=eval(λ,ex1)$𝐋
-Φ0:𝐋 :=eval(Φ,ex1)$𝐋 -\end{axiom} \end{axiom} For Cartan-Killing this is just the co-scalar \begin{axiom} test(eval(φφ,ck)=eval(Ω,ck)) test(eval((e,e)/H,ck)=eval(Ω,ck)) \end{axiom} Bi-algebra conditions \begin{axiom} ΦΦ:= _ ( λ λ ) / _ ( I I X ) / _ ( I X I ) / _ ( I I X ) / _ ( Y Y ) ; test((e,e)/ΦΦ=φφ) test(eval(ΦΦ,ck)=eval(H,ck)) test(eval(ΦΦ/(d,d),ck)=Ũ) test(eval(H/(d,d),ck)=Ũ) \end{axiom} The Cartan Killing form is a bi-algebra \begin{axiom} bi1:=map(numer,ravel(ΦΦ-H)::List FRAC POLY INT); bi2:=groebner bi1 b:=solve( equate(ΦΦ=H), [ε1,εi] ) test(eval(Ų, b.1)=Ũ) \end{axiom}  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 (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], ℚ) (2) Type: Type fricas 𝐞:ℒ 𝐋 := basisOut() (3) Type: List(ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))) fricas 𝐝:ℒ 𝐋 := basisIn() (4) Type: List(ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))) fricas I:𝐋:= -- identity for composition (5) Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) fricas X:𝐋:=[2,1] -- twist (6) Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) fricas V:𝐋:=ev(1) -- evaluation (7) Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) fricas Λ:𝐋:=co(1) -- co-evaluation (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,) (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) (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) (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)) (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) (13)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
arity Y (14)
Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)))
fricas
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim) (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 and fricas
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim) (16)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim) (17)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
(a*b)/Y (18)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Multiplication is Associative

fricas
test(
( I Y ) / _
(  Y  ) = _
( Y I ) / _
(  Y  ) ) (19)
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) (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:  (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 (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 (23)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
Ù:=
(  Λ Y  ) / _
( I Y   ) / _
V (24)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test(Ù=Ú) (25)
Type: Boolean

forms a non-degenerate associative scalar product for Y

fricas
Ũ := Ù (26)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
test
(    Y I    )  /
Ũ        =
(    I Y    )  /
Ũ (27)
Type: Boolean
fricas
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim) (28)
Type: Expression(Integer)

General Solution

We may consider the problem where multiplication Y is given, and look for all associative scalar products 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) (29)
Type: Tuple(PositiveInteger?)

The matrix J transforms the coefficients of the tensor into coefficients of the tensor . We are looking for the general linear family of tensors such that J transforms into for any such .

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) (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) ) (31)
Type: List(Equation(Expression(Integer)))

This defines a family of pre-Frobenius algebras:

fricas
zero? eval(ω,ℰ) (32)
Type: Boolean
fricas
Ų:𝐋 := eval(U,ℰ) (33)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Frobenius Form (co-unit)

fricas
d:=ε1*𝐝.1+εi*𝐝.2 (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))) (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
>> Error detected within library code:
index out of range

In general the pairing is not symmetric!

fricas
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim) (36)
Type: Matrix(Expression(Integer))

The scalar product must be non-degenerate:

fricas
Ů:=determinant u1 (37)
Type: Expression(Integer)
fricas
factor(numer Ů)/factor(denom Ů) (38)
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
There are 12 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op eval
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval
with argument type(s)
ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
Variable(𝔓)
Perhaps you should use "@" to indicate the required return type,

## Definition 4

Co-algebra

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

fricas
W:=
(Y I) /
Ų (50)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
fricas
λ:=
(  ΩX I ΩX  ) /
(  I  W  I  ) (51)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

Cartan-Killing co-multiplication

fricas
eval(λ,ck)
There are 12 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op eval
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval
with argument type(s)
ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
Variable(ck)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. fricas test ( I ΩX ) / ( Y I ) = λ (52) Type: Boolean fricas test ( ΩX I ) / ( I Y ) = λ (53) Type: Boolean Co-associativity fricas test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) ) (54) Type: Boolean fricas test e / λ = ΩX (55) Type: Boolean Frobenius Condition (fork) fricas H := Y / λ (56) Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) fricas test ( λ I ) / ( I Y ) = H (57) Type: Boolean fricas test ( I λ ) / ( Y I ) = H (58) Type: Boolean The Cartan-Killing form makes H of the Frobenius condition idempotent fricas test( eval(H,ck)=eval(H/H,ck) ) There are 12 exposed and 6 unexposed library operations named eval having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op eval to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named eval with argument type(s) ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) Variable(ck) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

And it is unique.

fricas
h1:=map(numer,ravel(H-H/H)::List FRAC POLY INT);
Cannot convert the value from type List(Expression(Integer)) to List
(Fraction(Polynomial(Integer))) .

Handle

fricas
Φ :=
λ     /
Y (59)
Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))

The Cartan-Killing form makes Φ of the identity

fricas
test( eval(Φ,ck)=I )
There are 12 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op eval
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval
with argument type(s)
ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
Variable(ck)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. and it can be the identity in only this one way. fricas solve(equate(Φ=I),[ε1,εi]) (60) Type: List(List(Equation(Expression(Integer)))) If handle is identity then fork is idempotent but the converse is not true fricas Φ1:=map(numer,ravel(Φ-I)::List FRAC POLY INT); Cannot convert the value from type List(Expression(Integer)) to List (Fraction(Polynomial(Integer))) . Figure 12 fricas φφ:= _ ( Ω Ω ) / _ ( X I I ) / _ ( I X I ) / _ ( I I X ) / _ ( Y Y ); Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) fricas φφ1:=map((x:ℚ):ℚ+->numer x,φφ) (61) Type: ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) fricas φφ2:=denom(ravel(φφ).1) (62) Type: SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))) fricas test(φφ=(1/φφ2)*φφ1) (63) Type: Boolean For Cartan-Killing this is just the co-scalar fricas test(eval(φφ,ck)=eval(Ω,ck)) There are 12 exposed and 6 unexposed library operations named eval having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op eval to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named eval with argument type(s) ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer)) Variable(ck) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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) (64)
Type: Boolean
fricas
test(eval(ΦΦ,ck)=eval(H,ck))
There are 12 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op eval
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval
with argument type(s)
ClosedLinearOperator(OrderedVariableList([1,i]),Expression(Integer))
Variable(ck)
Perhaps you should use "@" to indicate the required return type,
or "\$" to specify which version of the function you need.

The Cartan Killing form is a bi-algebra

fricas
bi1:=map(numer,ravel(ΦΦ-H)::List FRAC POLY INT);
Cannot convert the value from type List(Expression(Integer)) to List
(Fraction(Polynomial(Integer))) .