Quaternion Algebra Is Frobenius In Many Ways Linear operators over a 4-dimensional vector space representing quaternion 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
We need the Axiom LinearOperator library. fricas )library CARTEN ARITY CMONAL CPROP CLOP CALEY Use the following macros for convenient notation fricas -- summation macro Σ(x, Type: Voidfricas -- subscript and superscripts macro sb == subscript Type: Voidfricas macro sp == superscript Type: Voidℒ is the domain of 4-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients. fricas dim:=4
Type: PositiveIntegerfricas macro ℂ == CaleyDickson Type: Voidfricas macro ℚ == Expression Integer Type: Voidfricas ℒ := ClosedLinearOperator(OVAR ['1,
Type: Typefricas ⅇ:List ℒ := basisOut() Now generate structure constants for Quaternion 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 and co-quaternions can be specified by Caley-Dickson parameters (i2, j2) fricas i2:=sp('i,
Type: Symbolfricas --i2:= -1 -- complex j2:=sp('j,
Type: Symbolfricas --j2:= -1 -- quaternion k2:=-i2*j2; Type: Polynomial(Integer)fricas QQ := ℂ(ℂ(ℚ, Type: TypeBasis: Each B.i is a quaternion number fricas B:List QQ := map(x +-> hyper x,
fricas -- Multiplication table: M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]
fricas -- Function to divide the matrix entries by a basis element S(y) == map(x +-> real real(x/y), Type: Voidfricas -- The result is a nested list ѕ :=map(S, fricas Compiling function S with type 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, Units fricas q:=ⅇ.1; i:=ⅇ.2; j:=ⅇ.3; k:=ⅇ.4; Multiplication of arbitrary quaternions and fricas a:=Σ(sb('a, 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 The matrix If the null space of the fricas Ñ:=nullSpace(J) This defines a family of pre-Frobenius algebras: fricas zero? eval(ω, Frobenius Form (co-unit) fricas d:=ε1*ⅆ.1+εi*ⅆ.2+εj*ⅆ.3+εk*ⅆ.4 Express scalar product in terms of Frobenius form fricas Ξ:=solve(%, In general the pairing is not symmetric! fricas u1:=matrix [[retract((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 u1 Cartan-Killing is a special case fricas ck:=solve(equate(eval(Ũ, Frobenius scalar product of "vector" quaternions and fricas a:=sb('a, ## Definition 3Co-scalar product Solve the Snake Relation as a system of linear equations. fricas Ω:ℒ:=Σ(Σ(script('u, The common demoninator is fricas squareFreePart factor denom Ů / squareFreePart factor numer Ů Check "dimension" and the snake relations. fricas O:ℒ:= Ω / Ų
fricas test ( I ΩX ) / ( Ų I ) = I Cartan-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,
Type: Expression(Integer)fricas test ( I ΩX ) / ( Y I ) = λ Co-associativity fricas test( ( λ ) / _ ( I λ ) = _ ( λ ) / _ ( λ I ) ) fricas test q / λ = ΩX Frobenius 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,
Type: BooleanBut it is not unique. E.g. other idempots fricas h1:=map(numer, Handle and handle element fricas Φ := λ / Y
Type: Fraction(Polynomial(Integer))fricas Φ1 := q / Φ The Cartan-Killing form makes Φ into the identity fricas test( eval(Φ,
Type: Booleanfricas eval(Φ1,
Type: Expression(Integer)but it can be the identity in many ways. For example, fricas solve(equate(eval(Φ, 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 ) ; Theorem 8.3 fricas u2:=map(retract, |