## Sedenion Algebra is Frobenius in just one way!Linear operators over a 16-dimensional vector space representing Sedenion Algebra 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
- http://en.wikipedia.org/wiki/Sedenion
We need the Axiom LinearOperator library. fricas )library CARTEN MONAL PROP LOP CALEY Use the following macros for convenient notation fricas -- summation macro Σ(x, Type: Voidfricas -- subscript macro sb == subscript Type: Voidℒ is the domain of 16-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients. fricas dim:=16
Type: PositiveIntegerfricas macro ℂ == CaleyDickson Type: Voidfricas macro ℚ == Expression Integer Type: Voidfricas ℒ := LinearOperator(OVAR ['0,
Type: Typefricas ⅇ:List ℒ := basisOut() Now generate structure constants for Sedenion 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, split-octonions and seneions can be specified by Caley-Dickson parameters fricas --q0:=sb('q,
Type: PositiveIntegerfricas --q1:=sb('q,
Type: PositiveIntegerfricas --q2:=sb('q,
Type: PositiveIntegerfricas --q3:=sb('q,
Type: PositiveIntegerfricas QQ := ℂ(ℂ(ℂ(ℂ(ℚ, Type: TypeBasis: Each B.i is a sedennion number fricas B:List QQ := map(x +-> hyper x,
Type: List(CaleyDickson(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),fricas -- Multiplication table: M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]
Type: Matrix(CaleyDickson(CaleyDickson(CaleyDickson(CaleyDickson(Expression(Integer),fricas -- Function to divide the matrix entries by a basis element S(y) == map(x +-> real real real real(x/y), Type: Voidfricas -- The result is a nested list ѕ :=map(S, fricas Compiling function S with type CaleyDickson(CaleyDickson( CaleyDickson(CaleyDickson(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, 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; Type: LinearOperator(OrderedVariableList([0,## Definition 2An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?. 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 The matrix If the null space of the fricas Ñ:=nullSpace(J); This defines a family of Frobenius algebras: fricas zero? eval(ω, The pairing is necessarily diagonal! fricas Ų:ℒ := eval(U,
Type: LinearOperator(OrderedVariableList([0,fricas matrix [[(ⅇ.i ⅇ.j)/Ų for i in 1..dim] for j in 1..dim] The scalar product must be non-degenerate: fricas Ů:=determinant [[retract((ⅇ.i * ⅇ.j)/Ų) for j in 1..dim] for i in 1..dim] ## Definition 3Co-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]; Check "dimension" and the snake relations. fricas d:ℒ:= Ω / X / Ų
Type: LinearOperator(OrderedVariableList([0,fricas test ( I Ω ) / ( Ų I ) = I ## Definition 4Co-algebra Compute the "three-point" function and use it to define co-multiplication. Too slow: \begin{axiom} W:=(Y,I)/Ų; λ:=(Ω,I,Ω)/(I,W,I) \end{axiom} fricas λ:= (I, ## Frobenius ConditionLike Octonion algebra Sedenion algebra also fails the Frobenius Condition! Too slow to complete here: \begin{axiom} Χ := Y / λ ; Χr := (λ,I)/(I,Y) test(Χr = Χ ) Χl := (I,λ)/(Y,I); --test( Χl = Χ ) test( Χr = Χl ) \end{axiom} Perhaps this is not too surprising since like Octonion Seden algebra is non-associative (in fact also non-alternative). Nevertheless Sedenions are "Frobenius" in a more general sense just because there is a non-degenerate associative pairing. i = Unit of the algebra fricas i:=ⅇ.1 Handle fricas H:ℒ := λ / X / Y
Type: LinearOperator(OrderedVariableList([0,## Definition 5
i U fricas ι:ℒ:= ( i I ) / ( Ų ) Y=U ι fricas test Y / ι = Ų
Type: BooleanFor example: fricas ex1:=[q[3]=1,
Type: List(Equation(Polynomial(Integer)))fricas Ų0:ℒ :=eval(Ų,
Type: LinearOperator(OrderedVariableList([0,fricas Ω0:ℒ :=eval(Ω,
Type: LinearOperator(OrderedVariableList([0,fricas λ0:ℒ :=eval(λ,
Type: LinearOperator(OrderedVariableList([0,fricas H0:ℒ :=eval(H,
Type: LinearOperator(OrderedVariableList([0, |