References - Frobenius algebras and 2D topological quantum field theories
by Joachim Kock
Especially "Tensor calculus (linear algebra in coordinates)" in section 2.3.31, page 123.
See also: - http://ncatlab.org/nlab/show/Frobenius+algebra by John Baez, et al.
- Ideals, Varieties, and Algorithms by David A. Cox, et al.
## 2-d ExampleUse LinearOperator (LOP) fricas (1) -> )lib CARTEN MONAL PROP LOP
Type: Typefricas -- basis dx:=basisIn()$L
fricas Dx:=basisOut()$L
fricas -- summation macro Σ(f, Type: Voidfricas -- identity I:L:=[1]
fricas -- twist X:L:=[2,
fricas -- co-evaluation Λ:L:=co(1)
fricas -- evaluation V:L:=ev(1)
## AlgebraAn n-dimensional algebra is represented by a (2,1)-tensor viewed as a linear operator with two inputs and one output . For example in 2 dimensions fricas Y:=Σ(Σ(Σ(script(y,
## MultiplicationGiven two vectors and fricas P:=Σ(script(p,
fricas Q:=Σ(script(q,
the tensor operates on their tensor product to yield a vector fricas R:=(P,
Pictorially: P Q Y R or more explicitly Pi Qj \/ \ Rk ## Associator An algebra is said to be Y = Y Y Y i j k i j k i j k \ | / \/ / \ \/ \ | / \ / \ / \|/ = e k - i e | \/ \/ | \ / l l l This requires that the following (3,1)-tensor
fricas YY := (Y,
## Commutator The algebra is Y = Y i j i j j i \ / = \/ - \/ | \ / k k k This requires that the following (2,1)-tensor
fricas YC:=Y-(X/Y)
A basis for the ideal defined by the coefficients of the commutator is given by: fricas groebner(ravel(YC)) ## Anti-commutator The algebra is Y = -Y i j i j j i \ / = \/ = \/ | \ / k k k This requires that the following (2,1)-tensor
fricas YA:=Y+(X/Y)
A basis for the ideal defined by the coefficients of the commutator is given by: fricas groebner(ravel(YA)) ## Jacobi The X Y = Y + Y Y Y Y i j k i j k i j k i j k \ | / \ / / \ \ / \ \ / \ | / \ / / \ \ / \ 0 \ | / \/ / \ \/ \/ \ \ | / \ / \ / \ \ \|/ = e k - i e - e j | \/ \/ \/ | \ / / l l l l An algebra satisfies the Jacobi identity if and only if the following (3,1)-tensor
fricas YX := YY - (I,
## Scalar ProductA 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: i j k i j k i j k \ | / \/ / \ \/ \|/ = \ / - \ / 0 0 0
fricas YU := (Y,
## Definition 2 An algebra with a non-degenerate associative scalar product
is called We may consider the problem where multiplication Y is given, and look for all associative scalar products or we may consider an scalar product U as given, and look for all algebras such that the scalar product is associative. This problem can be solved using linear algebra. fricas )expose MCALCFN Type: Matrix(Expression(Integer))fricas yy := transpose matrix [concat(map(variables, Type: Matrix(Polynomial(Integer))fricas K::OutputForm * yy::OutputForm = 0
Type: Equation(OutputForm?)The matrix A necessary condition for the equation to have a non-trivial
solution is that the matrix ## Theorem 1All 2-dimensional pre-Frobenius algebras are symmetric. Proof: Consider the determinant of the matrix fricas Kd := factor(determinant(K)::DMP(concat map(variables,
Type: Factored(DistributedMultivariatePolynomial?([u[;1,The scalar product must also be non-degenerate fricas Ud:DMP(concat map(variables,
therefore U must be symmetric. fricas nthFactor(Kd, ## Theorem 2All 2-dimensional algebras with associative scalar product are commutative. Proof: The basis of the null space of the symmetric
fricas YUS := (I, This defines a 4-parameter family of 2-d pre-Frobenius algebras fricas test(eval(YUS, Alternatively we may consider fricas J := jacobian(ravel(YU), Type: Matrix(Expression(Integer))fricas uu := transpose matrix [concat(map(variables, Type: Matrix(Polynomial(Integer))fricas J::OutputForm * uu::OutputForm = 0
Type: Equation(OutputForm?)The matrix A necessary condition for the equation to have a non-trivial
solution is that all 70 of the 4x4 sub-matrices of fricas JP:=ideal concat concat concat [[[[ determinant( matrix([row(J, Type: PolynomialIdeal?(Fraction(Integer),fricas #generators(%)
Type: PositiveInteger?## Theorem 3If a 2-d algebra is associative, commutative, anti-commutative or if it satisfies the Jacobi identity then it is a pre-Frobenius algebra. Proof Consider the ideals of the associator, commutator, anti-commutator and Jacobi identity fricas YYI:=ideal(ravel(YY)::List FRAC POLY INT); Type: PolynomialIdeal?(Fraction(Integer),fricas in?(JP,
Type: Booleanfricas YCI:=ideal(ravel(YC)::List FRAC POLY INT); Type: PolynomialIdeal?(Fraction(Integer),fricas in?(JP,
Type: Booleanfricas YAI:=ideal(ravel(YA)::List FRAC POLY INT); Type: PolynomialIdeal?(Fraction(Integer),fricas in?(JP,
Type: Booleanfricas YXI:=ideal(ravel(YX)::List FRAC POLY INT); Type: PolynomialIdeal?(Fraction(Integer),fricas in?(JP,
Type: Boolean## Y-formsThree traces of two graftings of an algebra gives six (2,0)-forms. Left snail and right snail: LS RS Y /\ /\ Y Y ) ( Y \/ \/ i j j i \/ \/ \ /\ /\ / e f \ / f e \/ \ / \/ \ / \ / f / \ f \/ \/
fricas LS:= ( Y Λ )/ _ ( Y I )/ _ V
fricas RS:= ( Λ Y )/ _ ( I Y )/ _ V
fricas test(LS=RS)
Type: BooleanLeft and right deer: RD LD \ /\/ \/\ / Y /\ /\ Y Y ) ( Y \/ \/ i j i j \ /\ / \ /\ / \ f \ / \ / f / \/ \/ \/ \/ \ /\ /\ / e / \ / \ e \/ \ / \/ \ / \ / f / \ f \/ \/
fricas RD:= ( I Λ I ) / _ ( Y X ) / _ ( Y I ) / _ V
fricas LD:= ( I Λ I ) / _ ( X Y ) / _ ( I Y ) / _ V
fricas test(LD=RD)
Type: Booleanfricas test(RD=RS)
Type: Booleanfricas test(RD=LS)
Type: BooleanLeft and right turtles: RT LT /\ / / \ \ /\ ( Y / \ Y ) \ Y Y / \/ \/ i j i j /\ / / \ \ /\ / f / / \ \ f \ / \/ / \ \/ \ \ \ / \ / / \ e / \ e / \ \/ \/ / \ / \ / \ f f / \/ \/
fricas RT:= ( Λ I I ) / _ ( I Y I ) / _ ( I Y ) / _ V
fricas LT:= ( I I Λ ) / _ ( I Y I ) / _ ( Y I ) / _ V
fricas test(LT=RT)
Type: BooleanThe turles are symmetric fricas test(RT = X/RT)
Type: Booleanfricas test(LT = X/LT)
Type: BooleanFive of the six forms are independent. fricas test(RT=RS)
Type: Booleanfricas test(RT=LS)
Type: Booleanfricas test(RT=RD)
Type: Booleanfricas test(LT=RS)
Type: Booleanfricas test(LT=LS)
Type: Booleanfricas test(LT=RD)
Type: BooleanAssociativity implies right turtle equals right snail and left turtle equals left snail. fricas in?(ideal(ravel(RT-RS)::List FRAC POLY INT),
Type: Booleanfricas in?(ideal(ravel(LT-LS)::List FRAC POLY INT),
Type: BooleanIf the Jacobi identity holds then both snails are zero fricas in?(ideal(ravel(RS)::List FRAC POLY INT),
Type: Booleanfricas in?(ideal(ravel(LS)::List FRAC POLY INT),
Type: Booleanand right turtle and deer have opposite signs fricas in?(ideal(ravel(RT+RD)::List FRAC POLY INT),
Type: Boolean |