Quaternion Algebra Is Frobenius In Many Ways
Linear operators over a 4-dimensional vector space representing quaternion algebra
Ref:
We need the Axiom LinearOperator library.
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(1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY
>> System error:
The value
15684
is not of type
LIST
Use the following macros for convenient notation
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-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
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-- subscript and superscripts
macro sb == subscript
Type: Void
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macro sp == superscript
Type: Void
ℒ is the domain of 4-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.
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dim:=4
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macro ℂ == CaleyDickson
Type: Void
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macro ℚ == Expression Integer
Type: Void
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ℒ := ClosedLinearOperator(OVAR ['1,'i,'j,'k], ℚ)
There are no library operations named ClosedLinearOperator
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)what op ClosedLinearOperator
to learn if there is any operation containing "
ClosedLinearOperator " in its name.
Cannot find a definition or applicable library operation named
ClosedLinearOperator with argument type(s)
Type
Type
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or "$" to specify which version of the function you need.
Now generate structure constants for Quaternion Algebra
The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.
Split-complex and co-quaternions can be specified by Caley-Dickson parameters (i2, j2)
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i2:=sp('i,[2])
Type: Symbol
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--i2:= -1 -- complex
j2:=sp('j,[2])
Type: Symbol
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--j2:= -1 -- quaternion
k2:=-i2*j2;
Type: Polynomial(Integer)
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QQ := ℂ(ℂ(ℚ,'i,-i2),'j,-j2);
There are no library operations named CaleyDickson
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)what op CaleyDickson
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its name.
Cannot find a definition or applicable library operation named
CaleyDickson with argument type(s)
Type
Variable(i)
Polynomial(Integer)
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or "$" to specify which version of the function you need.
Basis: Each B.i is a quaternion number
fricas
B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::List List ℚ)
QQ is not a valid type.
Units
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q:=ⅇ.1; i:=ⅇ.2; j:=ⅇ.3; k:=ⅇ.4;
There are no library operations named ⅇ
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)what op ⅇ
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Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
Multiplication of arbitrary quaternions
and
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a:=Σ(sb('a,[i])*ⅇ.i, i,1..dim)
There are no library operations named ⅇ
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)what op ⅇ
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Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
FriCAS will attempt to step through and interpret the code.
There are no library operations named ⅇ
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)what op ⅇ
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Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
Multiplication is Associative
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test(
( I Y ) / _
( Y ) = _
( Y I ) / _
( Y ) )
There are no exposed library operations named I but there is one
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)display op I
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with argument type(s)
Variable(Y)
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A scalar product is denoted by the (2,0)-tensor
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U:=Σ(Σ(script('u,[[],[i,j]])*ⅆ.i*ⅆ.j, i,1..dim), j,1..dim)
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)what op ⅆ
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Cannot find a definition or applicable library operation named ⅆ
with argument type(s)
PositiveInteger
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FriCAS will attempt to step through and interpret the code.
There are no library operations named ⅆ
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)what op ⅆ
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Cannot find a definition or applicable library operation named ⅆ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
Definition 1
We say that the scalar product is associative if the tensor
equation holds:
Y = Y
U U
In other words, if the (3,0)-tensor:
(three-point function) is zero.
Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.
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ω:ℒ :=
( Y I ) /
U -
( I Y ) /
U
ℒ is not a valid type.
Definition 2
An algebra with a non-degenerate associative scalar product
is called a [Frobenius Algebra]?.
The Cartan-Killing Trace
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Ú:=
( Y Λ ) / _
( Y I ) / _
V
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
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Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(Λ)
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or "$" to specify which version of the function you need.
forms a non-degenerate associative scalar product for Y
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Ũ := Ù
Type: Variable(Ù)
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test
( Y I ) /
Ũ =
( I Y ) /
Ũ
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
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Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(I)
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or "$" to specify which version of the function you need.
General Solution
We may consider the problem where multiplication Y is given,
and look for all associative scalar products
This problem can be solved using linear algebra.
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)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame
initial
J := jacobian(ravel ω,concat map(variables,ravel U)::List Symbol);
There are 1 exposed and 0 unexposed library operations named ravel
having 1 argument(s) but none was determined to be applicable.
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)display op ravel
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package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel
with argument type(s)
Variable(ω)
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or "$" to specify which version of the function you need.
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:
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Ñ:=nullSpace(J)
There are 3 exposed and 3 unexposed library operations named
nullSpace having 1 argument(s) but none was determined to be
applicable. Use HyperDoc Browse, or issue
)display op nullSpace
to learn more about the available operations. Perhaps
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will allow you to apply the operation.
Cannot find a definition or applicable library operation named
nullSpace with argument type(s)
Variable(J)
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or "$" to specify which version of the function you need.
This defines a family of pre-Frobenius algebras:
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zero? eval(ω,ℰ)
There are 10 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
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)display op eval
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Cannot find a definition or applicable library operation named eval
with argument type(s)
Variable(ω)
Variable(ℰ)
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or "$" to specify which version of the function you need.
Frobenius Form (co-unit)
fricas
d:=ε1*ⅆ.1+εi*ⅆ.2+εj*ⅆ.3+εk*ⅆ.4
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Cannot find a definition or applicable library operation named ⅆ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
Express scalar product in terms of Frobenius form
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Ξ:=solve(%,[sb('p,[i]) for i in 1..#Ñ]).1
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having 1 argument(s) but none was determined to be applicable.
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)display op #
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Cannot find a definition or applicable library operation named #
with argument type(s)
Variable(Ñ)
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In general the pairing is not symmetric!
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u1:=matrix [[retract((ⅇ.i ⅇ.j)/Ų) for i in 1..dim] for j in 1..dim]
There are no library operations named ⅇ
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)what op ⅇ
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Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
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FriCAS will attempt to step through and interpret the code.
There are no library operations named ⅇ
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)what op ⅇ
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Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
The scalar product must be non-degenerate:
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Ů:=determinant u1
There are 3 exposed and 1 unexposed library operations named
determinant having 1 argument(s) but none was determined to be
applicable. Use HyperDoc Browse, or issue
)display op determinant
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determinant with argument type(s)
Variable(u1)
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or "$" to specify which version of the function you need.
Cartan-Killing is a special case
fricas
ck:=solve(equate(eval(Ũ,Ξ)=Ų),[ε1,εi,εj,εk]).1
There are 10 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
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)display op eval
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Cannot find a definition or applicable library operation named eval
with argument type(s)
Variable(Ù)
Variable(Ξ)
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or "$" to specify which version of the function you need.
Frobenius scalar product of "vector" quaternions
and
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a:=sb('a,[1])*i+sb('a,[2])*j
Type: Polynomial(Integer)
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b:=sb('b,[1])*i+sb('b,[2])*j
Type: Polynomial(Integer)
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(a,a)/Ų
There are 11 exposed and 15 unexposed library operations named /
having 2 argument(s) but none was determined to be applicable.
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)display op /
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Cannot find a definition or applicable library operation named /
with argument type(s)
Tuple(Polynomial(Integer))
Variable(Ų)
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or "$" to specify which version of the function you need.
Definition 3
Co-scalar product
Solve the Snake Relation as a system of linear equations.
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Ω:ℒ:=Σ(Σ(script('u,[[i,j]])*ⅇ.i*ⅇ.j, i,1..dim), j,1..dim)
ℒ is not a valid type.
The common demoninator is
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squareFreePart factor denom Ů / squareFreePart factor numer Ů
There are 3 exposed and 3 unexposed library operations named denom
having 1 argument(s) but none was determined to be applicable.
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)display op denom
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Cannot find a definition or applicable library operation named denom
with argument type(s)
Variable(Ů)
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or "$" to specify which version of the function you need.
Check "dimension" and the snake relations.
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O:ℒ:=
Ω /
Ų
ℒ is not a valid type.
Cartan-Killing co-scalar
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eval(Ω,ck)
There are 10 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
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)display op eval
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Cannot find a definition or applicable library operation named eval
with argument type(s)
Variable(Ω)
Variable(ck)
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or "$" to specify which version of the function you need.
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
fricas
W:=
(Y I) /
Ų
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
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Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(I)
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or "$" to specify which version of the function you need.
Cartan-Killing co-multiplication
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eval(λ,ck)
There are 10 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
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)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)
Variable(λ)
Variable(ck)
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or "$" to specify which version of the function you need.
fricas
test
( I ΩX ) /
( Y I ) = λ
There are no exposed library operations named I but there is one
unexposed operation with that name. Use HyperDoc Browse or issue
)display op I
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Cannot find a definition or applicable library operation named I
with argument type(s)
Variable(ΩX)
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or "$" to specify which version of the function you need.
Co-associativity
fricas
test(
( λ ) / _
( I λ ) = _
( λ ) / _
( λ I ) )
There are no exposed library operations named I but there is one
unexposed operation with that name. Use HyperDoc Browse or issue
)display op I
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Cannot find a definition or applicable library operation named I
with argument type(s)
Variable(λ)
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or "$" to specify which version of the function you need.
fricas
test
q /
λ = ΩX
Type: Boolean
Frobenius Condition (fork)
fricas
H :=
Y /
λ
Type: Fraction(Polynomial(Integer))
fricas
test
( λ I ) /
( I Y ) = H
There are no library operations named λ
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)what op λ
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Cannot find a definition or applicable library operation named λ
with argument type(s)
Variable(I)
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or "$" to specify which version of the function you need.
The Cartan-Killing form makes H of the Frobenius condition idempotent
fricas
test( eval(H,ck)=eval(H/H,ck) )
There are 10 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)
Fraction(Polynomial(Integer))
Variable(ck)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
But it is not unique. E.g. other idempots
fricas
h1:=map(numer,ravel(H-H/H)::List FRAC POLY INT);
There are 1 exposed and 0 unexposed library operations named ravel
having 1 argument(s) but none was determined to be applicable.
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)display op ravel
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will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel
with argument type(s)
Fraction(Polynomial(Integer))
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or "$" to specify which version of the function you need.
Handle and handle element
fricas
Φ :=
λ /
Y
Type: Fraction(Polynomial(Integer))
fricas
Φ1 := q /
Φ
Type: Fraction(Polynomial(Integer))
The Cartan-Killing form makes Φ into the identity
fricas
test( eval(Φ,ck)=I )
There are 10 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)
Fraction(Polynomial(Integer))
Variable(ck)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
but it can be the identity in many ways. For example,
fricas
solve(equate(eval(Φ,[ε1=1,εi=1,εj=1,j2=-1])=I),[εk])
There are no library operations named equate
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)what op equate
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name.
Cannot find a definition or applicable library operation named
equate with argument type(s)
Equation(Fraction(Polynomial(Integer)))
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or "$" to specify which version of the function you need.
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);
There are 1 exposed and 0 unexposed library operations named ravel
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op ravel
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 ravel
with argument type(s)
Fraction(Polynomial(Integer))
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Figure 12
fricas
φφ:= _
( Ω Ω ) / _
( X I I ) / _
( I X I ) / _
( I I X ) / _
( Y Y );
There are no library operations named Ω
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)what op Ω
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Cannot find a definition or applicable library operation named Ω
with argument type(s)
Variable(Ω)
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or "$" to specify which version of the function you need.
For Cartan-Killing this is just the co-scalar
fricas
test(eval(φφ,ck)=eval(Ω,ck))
There are 10 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)
Variable(φφ)
Variable(ck)
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or "$" to specify which version of the function you need.
Bi-algebra conditions
fricas
ΦΦ:= _
( λ λ ) / _
( I I X ) / _
( I X I ) / _
( I I X ) / _
( Y Y ) ;
There are no library operations named λ
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)what op λ
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Cannot find a definition or applicable library operation named λ
with argument type(s)
Variable(λ)
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or "$" to specify which version of the function you need.
Theorem 8.3
fricas
u2:=map(retract,matrix [ _
[q/d, i/d, j/d, k/d], _
[i/d, i2*q/d, k/d, i2*j/d], _
[j/d, -k/d, j2*q/d, -j2*i/d], _
[k/d, -i2*j/d, j2*i/d, k2*q/d]])
>> Error detected within library code:
Denominator not equal to 1