Octonion Algebra Is Frobenius In Just One Way
Linear operators over a 8-dimensional vector space representing octonnion 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
macro sb == subscript
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.
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dim:=8
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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 ['0,'1,'2,'3,'4,'5,'6,'7], 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 "$" to specify which version of the function you need.
Now generate structure constants for Octonion Algebra
The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.
Split-complex, co-quaternions and split-octonions can be specified by Caley-Dickson parameters
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--q0:=sb('q,[0])
q0:=1 -- not split-complex
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--q1:=sb('q,[1])
q1:=1 -- not co-quaternion
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q2:=sb('q,[2])
Type: Symbol
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--q2:=1 -- split-octonion
QQ := ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2);
There are no library operations named CaleyDickson
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)what op CaleyDickson
to learn if there is any operation containing " CaleyDickson " in
its name.
Cannot find a definition or applicable library operation named
CaleyDickson with argument type(s)
Type
Variable(i)
PositiveInteger
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or "$" to specify which version of the function you need.
Basis: Each B.i is a octonion number
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B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::List List ℚ)
QQ is not a valid type.
A scalar product is denoted by the (2,0)-tensor
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U:=Σ(Σ(script('u,[[],[i,j]])*ⅆ.i*ⅆ.j, i,1..dim), j,1..dim)
There are no library operations named ⅆ
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)what op ⅆ
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Cannot find a definition or applicable library operation named ⅆ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
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 "$" to specify which version of the function you need.
Definition 1
We say that the scalar product is associative if the tensor
equation holds:
Y = Y
U U
In other words, if the (3,0)-tensor:
(three-point function) is zero.
Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.
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ω:ℒ :=(Y*I)/U - (I*Y)/U;
ℒ is not a valid type.
Definition 2
An algebra with a non-degenerate associative scalar product
is called a [Frobenius Algebra]?.
The Cartan-Killing Trace
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Ú:=
( Y Λ ) / _
( Y I ) / _
V
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
to learn more about the available operations.
Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(Λ)
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or "$" 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
to learn more about the available operations.
Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(I)
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or "$" to specify which version of the function you need.
General Solution
We may consider the problem where multiplication Y is given,
and look for all associative scalar products
This problem can be solved using linear algebra.
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)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame
initial
J := jacobian(ravel ω,concat map(variables,ravel U)::List Symbol);
There are 1 exposed and 0 unexposed library operations named ravel
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op ravel
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel
with argument type(s)
Variable(ω)
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or "$" 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
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
nullSpace with argument type(s)
Variable(J)
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or "$" to specify which version of the function you need.
This defines a family of Frobenius algebras:
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zero? eval(ω,ℰ)
There are 10 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
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)display op eval
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package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval
with argument type(s)
Variable(ω)
Variable(ℰ)
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or "$" to specify which version of the function you need.
The pairing is necessarily diagonal!
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Ų:ℒ := eval(U,ℰ)
ℒ is not a valid type.
The scalar product must be non-degenerate:
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Ů:=determinant [[retract((ⅇ.i * ⅇ.j)/Ų) for j in 1..dim] for i in 1..dim]
There are no library operations named ⅇ
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)what op ⅇ
to learn if there is any operation containing " ⅇ " in its name.
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.
Definition 3
Co-pairing
Solve the Snake Relation as a system of linear equations.
fricas
Um:=matrix [[(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
to learn if there is any operation containing " ⅇ " in its name.
Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
FriCAS will attempt to step through and interpret the code.
There are no library operations named ⅇ
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)what op ⅇ
to learn if there is any operation containing " ⅇ " in its name.
Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
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or "$" to specify which version of the function you need.
Check "dimension" and the snake relations.
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d:ℒ:=
Ω /
X /
Ų
ℒ is not a valid type.
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
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W:=(Y,I)/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 /
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 /
with argument type(s)
Tuple(OrderedVariableList([Y,I]))
Variable(Ų)
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or "$" to specify which version of the function you need.
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test
( I Ω ) /
( Y I ) = λ
There are no exposed library operations named I but there is one
unexposed operation with that name. Use HyperDoc Browse or issue
)display op I
to learn more about the available operation.
Cannot find a definition or applicable library operation named I
with argument type(s)
Variable(Ω)
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or "$" to specify which version of the function you need.
Frobenius Condition
Octonion algebra fails the Frobenius Condition!
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H :=
Y /
λ
Type: Fraction(Polynomial(Integer))
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Hr := (λ,I)/(I,Y)
There are 11 exposed and 15 unexposed library operations named /
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op /
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 /
with argument type(s)
Tuple(OrderedVariableList([λ,I]))
Tuple(OrderedVariableList([I,Y]))
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Perhaps this is not too surprising since Octonion algebra is
non-associative. Nevertheless Octonions are "Frobenius" in a more
general sense because there is a non-degenerate associative
pairing.
i = Unit of the algebra
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i:=ⅇ.1
There are no library operations named ⅇ
Use HyperDoc Browse or issue
)what op ⅇ
to learn if there is any operation containing " ⅇ " in its name.
Cannot find a definition or applicable library operation named ⅇ
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Handle and handle element
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Φ:ℒ :=
λ /
X /
Y
ℒ is not a valid type.
Definition 5
Co-unit
i
U
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ι:ℒ:=
( i I ) /
( Ų )
ℒ is not a valid type.
Y=U
ι
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test
Y /
ι = Ų
Type: Boolean
For example:
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ex1:=[q[2]=1,p[1]=1]
Type: List(Equation(Polynomial(Integer)))
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Ų0:ℒ :=eval(Ų,ex1)
ℒ is not a valid type.