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: - http://arxiv.org/abs/1103.5113
-permuted Frobenius Algebras *Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)* - http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories *Joachim Kock* - http://en.wikipedia.org/wiki/Frobenius_algebra
We need the Axiom LinearOperator library. fricas (1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY Use the following macros for convenient notation fricas -- summation macro Σ(x, Type: Voidfricas -- list macro Ξ(f, Type: Voidfricas -- subscript and superscripts macro sb == subscript Type: Voidfricas macro sp == superscript Type: Void𝐋 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
Type: PositiveInteger?fricas macro ℒ == List Type: Voidfricas macro ℂ == CaleyDickson Type: Voidfricas macro ℚ == Expression Integer Type: Voidfricas 𝐋 := ClosedLinearOperator(OVAR ['1,
Type: Typefricas 𝐞:ℒ 𝐋 := basisOut()
fricas 𝐝:ℒ 𝐋 := basisIn()
fricas I:𝐋:=[1] -- identity for composition
fricas X:𝐋:=[2,
fricas V:𝐋:=ev(1) -- evaluation
fricas Λ:𝐋:=co(1) -- co-evaluation
fricas equate(eq)==map((x, Type: VoidNow 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,
Type: Integerfricas QQ := ℂ(ℚ, Type: TypeBasis: Each B.i is a complex number fricas B:ℒ QQ := map(x +-> hyper x, Units fricas e:=𝐞.1; i:=𝐞.2; Multiplication of arbitrary ccomplex numbers and fricas a:=Σ(sb('a,
fricas b:=Σ(sb('b,
fricas (a*b)/Y
Multiplication is Associative fricas test( ( I Y ) / _ ( Y ) = _ ( Y I ) / _ ( Y ) ) A scalar product is denoted by the (2,0)-tensor fricas U:=Σ(Σ(script('u,
## Definition 1 We say that the scalar product is Y = Y U U In other words, if the (3,0)-tensor:
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 ## Definition 2An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?. The Cartan-Killing Trace fricas Ú:= ( Y Λ ) / _ ( Y I ) / _ V forms a non-degenerate associative scalar product for Y fricas Ũ := Ù
Type: Variable(Ù)fricas test ( Y I ) / Ũ = ( I Y ) / Ũ 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 Type: Matrix(Expression(Integer))fricas nrows(J),
Type: Tuple(PositiveInteger?)The matrix If the null space of the fricas Ñ:=nullSpace(J)
Type: List(Vector(Expression(Integer)))fricas ℰ:=map((x,
Type: List(Equation(Expression(Integer)))This defines a family of pre-Frobenius algebras: fricas zero? eval(ω,
Type: Booleanfricas Ų:𝐋 := eval(U,
Frobenius Form (co-unit) fricas d:=ε1*𝐝.1+εi*𝐝.2
fricas 𝔇:=equate(d= ( e I ) / _ Ų ) fricas Compiling function equate with type Equation(ClosedLinearOperator( OrderedVariableList([1,
Type: List(Equation(Expression(Integer)))Express scalar product in terms of Frobenius form fricas 𝔓:=solve(𝔇, In general the pairing is not symmetric! fricas u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų),
Type: Matrix(Expression(Integer))The scalar product must be non-degenerate: fricas Ů:=determinant u1
Type: Expression(Integer)fricas factor(numer Ů)/factor(denom Ů)
Cartan-Killing is a special case fricas ck:=solve(equate(eval(Ũ, Frobenius scalar product of complex numbers and fricas a:=sb('a,
fricas b:=sb('b,
fricas (a,
fricas (b,
fricas ab:=(a,
fricas solve(equate(ab=0*ab), ## Definition 3Co-scalar product Solve the Snake Relation as a system of linear equations. fricas Ω:𝐋:=Σ(Σ(script('u,
fricas ΩX:=Ω/X; fricas UXΩ:=(I*ΩX)/(Ų*I); fricas ΩXU:=(ΩX*I)/(I*Ų); fricas eq1:=equate(UXΩ=I); Type: List(Equation(Expression(Integer)))fricas eq2:=equate(ΩXU=I); Type: List(Equation(Expression(Integer)))fricas snake:=solve(concat(eq1, Type: List(List(Equation(Expression(Integer))))fricas if #snake ~= 1 then error "no solution" Type: Voidfricas Ω:=eval(Ω,
fricas ΩX:=Ω/X; The common demoninator is fricas squareFreePart((factor denom Ů)::ℚ) / squareFreePart((factor numer Ů)::ℚ)
Type: Expression(Integer)fricas matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)),
Check "dimension" and the snake relations. fricas O:𝐋:= Ω / Ų
fricas test ( I ΩX ) / ( Ų I ) = I
Type: Booleanfricas test ( ΩX I ) / ( I Ų ) = I
Type: BooleanCartan-Killing co-scalar fricas eval(Ω, ## Definition 4Co-algebra Compute the "three-point" function and use it to define co-multiplication. fricas W:= (Y I) / Ų Cartan-Killing co-multiplication fricas eval(λ, fricas test ( I ΩX ) / ( Y I ) = λ Co-associativity fricas test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) ) fricas test e / λ = ΩX
Type: BooleanFrobenius Condition (fork) fricas H := Y / λ
Type: Fraction(Polynomial(Integer))fricas test ( λ I ) / ( I Y ) = H The Cartan-Killing form makes H of the Frobenius condition idempotent fricas test( eval(H, And it is unique. fricas h1:=map(numer, Handle fricas Φ := λ / Y
Type: Fraction(Polynomial(Integer))The Cartan-Killing form makes Φ of the identity fricas test( eval(Φ, and it can be the identity in only this one way. fricas solve(equate(Φ=I), If handle is identity then fork is idempotent but the converse is not true fricas Φ1:=map(numer, Figure 12 fricas φφ:= _ ( Ω Ω ) / _ ( X I I ) / _ ( I X I ) / _ ( I I X ) / _ ( Y Y ); For Cartan-Killing this is just the co-scalar fricas test(eval(φφ, Bi-algebra conditions fricas ΦΦ:= _ ( λ λ ) / _ ( I I X ) / _ ( I X I ) / _ ( I I X ) / _ ( Y Y ) ; The Cartan Killing form is a bi-algebra fricas bi1:=map(numer, |