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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)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
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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)
Type: Type
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ⅇ:List ℒ := basisOut()
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
Now generate structure constants for Octonion Algebra
The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.
Split-complex, co-quaternions and split-octonions can be specified by Caley-Dickson parameters
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--q0:=sb('q,[0])
q0:=1 -- not split-complex
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--q1:=sb('q,[1])
q1:=1 -- not co-quaternion
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q2:=sb('q,[2])
Type: Symbol
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--q2:=1 -- split-octonion
QQ := ℂ(ℂ(ℂ(ℚ,'i,q0),'j,q1),'k,q2);
Type: Type
Basis: Each B.i is a octonion number
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B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::List List ℚ)
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-- Multiplication table:
M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]
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-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real real(x/y),M)
Type: Void
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-- The result is a nested list
ѕ :=map(S,B)::List List List ℚ;
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Compiling function S with type CaleyDickson(CaleyDickson(
CaleyDickson(Expression(Integer),i,1),j,1),k,q[2]) -> Matrix(
Expression(Integer))
Type: List(List(List(Expression(Integer))))
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-- 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)
There are no library operations named ⅆ
Use HyperDoc Browse or issue
)what op ⅆ
to learn if there is any operation containing " ⅆ " in its name.
Cannot find a definition or applicable library operation named ⅆ
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
FriCAS will attempt to step through and interpret the code.
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
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
Use HyperDoc Browse or issue
)what op ⅆ
to learn if there is any operation containing " ⅆ " in its name.
Cannot find a definition or applicable library operation named ⅆ
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
FriCAS will attempt to step through and interpret the code.
There are no library operations named ⅆ
Use HyperDoc Browse or issue
)what op ⅆ
to learn if there is any operation containing " ⅆ " in its name.
Cannot find a definition or applicable library operation named ⅆ
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
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;
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)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
forms a non-degenerate associative scalar product for Y
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Ũ := Ù
Type: Variable(Ù)
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test
( Y I ) /
Ũ =
( I Y ) /
Ũ
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
to learn more about the available operations.
Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(I)
Perhaps you should use "@" to indicate the required return type,
or "$" 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 2 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(U)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
The matrix J
transforms the coefficients of the tensor
into coefficients of the tensor . We are looking for
the general linear family of tensors such that
J
transforms into for any such .
If the null space of the J
matrix is not empty we can use
the basis to find all non-trivial solutions for U:
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Ñ:=nullSpace(J);
There are 3 exposed and 3 unexposed library operations named
nullSpace having 1 argument(s) but none was determined to be
applicable. Use HyperDoc Browse, or issue
)display op nullSpace
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
nullSpace with argument type(s)
Variable(J)
Perhaps you should use "@" to indicate the required return type,
or "$" 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 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([0,1,2,3,4,5,6,7]),Expression(Integer))
Variable(ℰ)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
The pairing is necessarily diagonal!
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Ų:ℒ := eval(U,ℰ)
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matrix [[(ⅇ.i ⅇ.j)/Ų for i in 1..dim] for j in 1..dim]
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
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]
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
Definition 3
Co-pairing
Solve the Snake Relation as a system of linear equations.
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Um:=matrix [[(ⅇ.i*ⅇ.j)/Ų for i in 1..dim] for j in 1..dim]
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
Check "dimension" and the snake relations.
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d:ℒ:=
Ω /
X /
Ų
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test
( I Ω ) /
( Ų I ) = 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
to learn more about the available operation.
Cannot find a definition or applicable library operation named I
with argument type(s)
Variable(Ω)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
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W:=(Y,I)/Ų
There are 16 exposed and 12 unexposed library operations named /
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op /
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named /
with argument type(s)
Tuple(OrderedVariableList([Y,I]))
ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
Perhaps you should use "@" to indicate the required return type,
or "$" 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)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Frobenius Condition
Octonion algebra fails the Frobenius Condition!
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H :=
Y /
λ
Type: Fraction(Polynomial(Integer))
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Hr := (λ,I)/(I,Y)
There are 16 exposed and 12 unexposed library operations named /
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op /
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named /
with argument type(s)
Tuple(OrderedVariableList([λ,I]))
Tuple(OrderedVariableList([I,Y]))
Perhaps you should use "@" to indicate the required return type,
or "$" 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
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
Handle and handle element
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Φ:ℒ :=
λ /
X /
Y
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Φ1:= i /
Φ
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
Definition 5
Co-unit i
U
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ι:ℒ:=
( i I ) /
( Ų )
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer)
Internal Error
The function setelt with signature (Integer)$(Integer)(Integer) is missing
from domain Vector(Integer)
Y=U
ι
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test
Y /
ι = Ų
Type: Boolean
For example:
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ex1:=[q[2]=1,p[1]=1]
Type: List(Equation(Polynomial(Integer)))
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Ų0:ℒ :=eval(Ų,ex1)
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Ω0:ℒ :=eval(Ω,ex1)$ℒ
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λ0:ℒ :=eval(λ,ex1)$ℒ
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H0:ℒ :=eval(H,ex1)$ℒ