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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.

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
-- 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.

fricas
dim:=8
 (1)
Type: PositiveInteger?
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
ℒ := ClosedLinearOperator(OVAR ['0,'1,'2,'3,'4,'5,'6,'7], ℚ)
 (2)
Type: Type
fricas
ⅇ:List ℒ      := basisOut()
 (3)
Type: List(ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)))
fricas
ⅆ:List ℒ      := basisIn()
 (4)
Type: List(ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)))
fricas
I:ℒ:=[1];   -- identity for composition
Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
X:ℒ:=[2,1]; -- twist
Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
V:ℒ:=ev(1) -- evaluation
 (5)
Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
Λ:ℒ:=co(1) -- co-evaluation
 (6)
Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(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

fricas
--q0:=sb('q,[0])
q0:=1  -- not split-complex
 (7)
Type: PositiveInteger?
fricas
--q1:=sb('q,[1])
q1:=1  -- not co-quaternion
 (8)
Type: PositiveInteger?
fricas
q2:=sb('q,[2])
 (9)
Type: Symbol
fricas
--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

fricas
B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::List List ℚ)  (10) Type: List(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,q[2])) fricas -- Multiplication table: M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]  (11) Type: Matrix(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),i,1),j,1),k,q[2])) fricas -- Function to divide the matrix entries by a basis element S(y) == map(x +-> real real real(x/y),M) Type: Void fricas -- The result is a nested list ѕ :=map(S,B)::List List List ℚ; fricas 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)))) 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)  (12) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas arity Y  (13) Type: ClosedProp?(ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))) fricas matrix [[(ⅇ.i*ⅇ.j)/Y for i in 1..dim] for j in 1..dim]  (14) Type: Matrix(ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))) 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)  (15) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),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:  (16) (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; Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),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  (17) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas Ù:= ( Λ Y ) / _ ( I Y ) / _ V  (18) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas test(Ù=Ú)  (19) Type: Boolean forms a non-degenerate associative scalar product for Y fricas Ũ := Ù  (20) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas test ( Y I ) / Ũ = ( I Y ) / Ũ  (21) Type: Boolean fricas determinant [[retract((ⅇ.i * ⅇ.j)/Ũ) for j in 1..dim] for i in 1..dim]  (22) 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)::List Symbol); Type: Matrix(Expression(Integer)) fricas u := transpose matrix [concat map(variables,ravel U)::List Symbol]; Type: Matrix(Polynomial(Integer)) fricas J::OutputForm * u::OutputForm = 0  (23) Type: Equation(OutputForm?) fricas nrows(J),ncols(J)  (24) 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); 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) ) >> Error detected within library code: reducing over an empty list needs the 3 argument form This defines a family of Frobenius algebras: fricas 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!

fricas
Ų:ℒ := eval(U,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
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,

## Definition 4

Co-algebra

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

fricas
W:=(Y,I)/Ų
3
+  0
-- -
2 0
+
arity warning: ----
2
+  0
-- -
2 0
+
 (28)
Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
λ:=(Ω,I,Ω)/(I,W,I)
4
+ +
- --
+  4
+
arity warning: ----
5
0 +
- --
0  4
+
>> System error:
Heap exhausted (no more space for allocation).
There are still 807796736 bytes available; the request was for 1073741840 bytes.
PROCEED WITH CAUTION.

fricas
test
(    I Ω     )  /
(     Y I    )  =  λ
2
+ +
- --
+  2
+
arity warning: ----
3
0 +
- --
0  2
+
 (29)
Type: Boolean
fricas
test
(     Ω I    )  /
(    I Y     )  =  λ
There are no library operations named Ω
Use HyperDoc Browse or issue
)what op Ω
to learn if there is any operation containing " Ω " in its name.
Cannot find a definition or applicable library operation named Ω
with argument type(s)
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. ## Frobenius Condition Octonion algebra fails the Frobenius Condition! fricas H := Y / λ 2 + 0 -- - + 0 arity warning: ---- + 0 - - + 0  (30) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas Hr := (λ,I)/(I,Y) 2 + + - -- + 2 + arity warning: ---- 3 0 + - -- 0 2 +  (31) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas test(Hr = H )  (32) Type: Boolean fricas Hl := (I,λ)/(Y,I); 2 + + - -- + 2 + arity warning: ---- 3 0 + - -- 0 2 + Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas test( Hl = H )  (33) Type: Boolean fricas test( Hr = Hl )  (34) Type: Boolean 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 fricas i:=ⅇ.1  (35) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas test i / λ = Ω 0 0 - - + 0 arity warning: --- + 0 - - + 0  (36) Type: Boolean Handle and handle element fricas Φ:ℒ := λ / X / Y 2 0 + - -- 0 2 + arity warning: ---- 2 0 + - -- 0 2 +  (37) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas Φ1:= i / Φ 0 + - - + + arity warning: ---- 2 0 + - -- 0 +  (38) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) ## Definition 5 Co-unit  i U  fricas ι:ℒ:= ( i I ) / ( Ų ) + 0 -- - 2 0 + arity warning: ---- 2 + 0 -- - 2 0 +  (39) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) Y=U ι  fricas test Y / ι = Ų  (40) Type: Boolean For example: fricas ex1:=[q[2]=1,p[1]=1]  (41) Type: List(Equation(Polynomial(Integer))) fricas Ų0:ℒ :=eval(Ų,ex1)  (42) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas Ω0:ℒ :=eval(Ω,ex1)$ℒ
 (43)
Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))
fricas
λ0:ℒ  :=eval(λ,ex1)$ℒ  (44) Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer)) fricas H0:ℒ :=eval(H,ex1)$ℒ
 (45)
Type: ClosedLinearOperator(OrderedVariableList([0,1,2,3,4,5,6,7]),Expression(Integer))

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