Topics

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  • FriCASContributions
    • Frobenius Algebra, Vector Spaces and Polynomial Ideals
      • SandBoxFrobeniusAlgebra <-- You are here.

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 Example

Use LinearOperator (LOP)

(1) -> )lib CARTEN MONAL PROP LOP

 >> System error:
 The value
 15684
is not of type
 LIST

Algebra

An n-dimensional algebra is represented by a (2,1)-tensor Y=\{ {y^k}_{ij} \ i,j,k =1,2, ... n \} viewed as a linear operator with two inputs i,j and one output k. For example in 2 dimensions

Y:=Σ(Σ(Σ(script(y,[[i,j],[k]]),dx,i),dx,j),Dx,k)

 There are no library operations named Σ 
 Use HyperDoc Browse or issue
 )what op Σ
 to learn if there is any operation containing " Σ " in its name.

 Cannot find a definition or applicable library operation named Σ 
 with argument type(s) 
 Symbol
 Variable(dx)
 Variable(i)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Multiplication

Given two vectors P=\{ p^i \} and Q=\{ q^j \}

P:=Σ(script(p,[[],[i]]),Dx,i)

 There are no library operations named Σ 
 Use HyperDoc Browse or issue
 )what op Σ
 to learn if there is any operation containing " Σ " in its name.

 Cannot find a definition or applicable library operation named Σ 
 with argument type(s) 
 Symbol
 Variable(Dx)
 Variable(i)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

the tensor Y operates on their tensor product to yield a vector R=\{ r_k = {y^k}_{ij} p^i q^j \}

R:=(P,Q)/Y

 There are 11 exposed and 15 unexposed library operations named / 
 having 2 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op /
 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 / 
 with argument type(s) 
 Tuple(OrderedVariableList([P,Q]))
 Variable(Y)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Pictorially:

 P Q
 Y
 R
 or more explicitly
 Pi Qj
 \/
 \
 Rk

Associator

An algebra is said to be associative if:

 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

(associator) is zero.

YY := (Y,I)/Y - (I,Y)/Y

 There are 11 exposed and 15 unexposed library operations named / 
 having 2 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op /
 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 / 
 with argument type(s) 
 Tuple(OrderedVariableList([Y,I]))
 Variable(Y)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Commutator

The algebra Y is commutative if:

 Y = Y
 i j i j j i
 \ / = \/ - \/
 | \ /
 k k k

This requires that the following (2,1)-tensor

(commutator) is zero.

YC:=Y-(X/Y)

A basis for the ideal defined by the coefficients of the commutator is given by:

groebner(ravel(YC))

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Fraction(Polynomial(Integer))

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Anti-commutator

The algebra Y is anti-commutative if:

 Y = -Y
 i j i j j i
 \ / = \/ = \/
 | \ /
 k k k

This requires that the following (2,1)-tensor

(anti-commutator) is zero.

YA:=Y+(X/Y)

A basis for the ideal defined by the coefficients of the commutator is given by:

groebner(ravel(YA))

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Fraction(Polynomial(Integer))

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Jacobi

The Jacobi identity is:

 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

is zero.

YX := YY - (I,X)/(Y,I)/Y

 There are 11 exposed and 15 unexposed library operations named / 
 having 2 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op /
 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 / 
 with argument type(s) 
 Tuple(OrderedVariableList([I,X]))
 Tuple(OrderedVariableList([Y,I]))

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Scalar Product

A scalar product is denoted by the (2,0)-tensor U = \{ u_{ij} \}

U:=Σ(Σ(script(u,[[],[i,j]]),dx,i),dx,j)

 There are no library operations named Σ 
 Use HyperDoc Browse or issue
 )what op Σ
 to learn if there is any operation containing " Σ " in its name.

 Cannot find a definition or applicable library operation named Σ 
 with argument type(s) 
 Symbol
 Variable(dx)
 Variable(i)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

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:

 i j k i j k i j k
 \ | / \/ / \ \/
 \|/ = \ / - \ /
 0 0 0

(three-point function) is zero.

YU := (Y,I)/U - (I,Y)/U

 There are 11 exposed and 15 unexposed library operations named / 
 having 2 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op /
 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 / 
 with argument type(s) 
 Tuple(OrderedVariableList([Y,I]))
 Variable(U)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Definition 2

An algebra with a non-degenerate associative scalar product is called pre-Frobenius.

We may consider the problem where multiplication Y is given, and look for all associative scalar products U = U(Y) or we may consider an scalar product U as given, and look for all algebras Y=Y(U) such that the scalar product is associative.

This problem can be solved using linear algebra.

)expose MCALCFN

 MultiVariableCalculusFunctions is now explicitly exposed in frame 
 initial 
K := jacobian(ravel(YU),concat(map(variables,ravel(Y)))::List Symbol);

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Variable(YU)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

The matrix K transforms the coefficients of the tensor Y into coefficients of the tensor \Phi. We are looking for coefficients of the tensor U such that K transforms the tensor Y into \Phi=0 for any Y.

A necessary condition for the equation to have a non-trivial solution is that the matrix K be degenerate.

Theorem 1

All 2-dimensional pre-Frobenius algebras are symmetric.

Proof: Consider the determinant of the matrix K above.

Kd := factor(determinant(K)::DMP(concat map(variables,ravel(U)),FRAC INT))

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Variable(U)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

The scalar product must also be non-degenerate

Ud:DMP(concat map(variables,ravel(U)),FRAC INT) := determinant [[retract((Dx.i,Dx.j)/U) for j in 1..#Dx] for i in 1..#Dx]

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Variable(U)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

therefore U must be symmetric.

nthFactor(Kd,1)

 There are no exposed library operations named nthFactor but there 
 are 3 unexposed operations with that name. Use HyperDoc Browse or
 issue
 )display op nthFactor
 to learn more about the available operations.

 Cannot find a definition or applicable library operation named 
 nthFactor with argument type(s) 
 Variable(Kd)
 PositiveInteger

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Theorem 2

All 2-dimensional algebras with associative scalar product are commutative.

Proof: The basis of the null space of the symmetric K matrix are all symmetric

YUS := (I,Y)/US - (Y,I)/US

 There are 11 exposed and 15 unexposed library operations named / 
 having 2 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op /
 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 / 
 with argument type(s) 
 Tuple(OrderedVariableList([I,Y]))
 Variable(US)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

This defines a 4-parameter family of 2-d pre-Frobenius algebras

test(eval(YUS,SS)=0*YUS)

 There are 10 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) 
 Variable(YUS)
 Variable(SS)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Alternatively we may consider

J := jacobian(ravel(YU),concat(map(variables,ravel(U)))::List Symbol);

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Variable(YU)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

The matrix J transforms the coefficients of the tensor U into coefficients of the tensor \Phi. We are looking for coefficients of the tensor Y such that J transforms the tensor U into \Phi=0 for any U.

A necessary condition for the equation to have a non-trivial solution is that all 70 of the 4x4 sub-matrices of J are degenerate. To this end we can form the polynomial ideal of the determinants of these sub-matrices.

JP:=ideal concat concat concat
 [[[[ determinant(
 matrix([row(J,i1),row(J,i2),row(J,i3),row(J,i4)]))::FRAC POLY INT
 for i4 in (i3+1)..maxRowIndex(J) ] 
 for i3 in (i2+1)..(maxRowIndex(J)-1) ]
 for i2 in (i1+1)..(maxRowIndex(J)-2) ]
 for i1 in minRowIndex(J)..(maxRowIndex(J)-3) ];

 There are 2 exposed and 0 unexposed library operations named 
 minRowIndex having 1 argument(s) but none was determined to be 
 applicable. Use HyperDoc Browse, or issue
 )display op minRowIndex
 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 
 minRowIndex with argument type(s) 
 Variable(J)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Theorem 3

If 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

YYI:=ideal(ravel(YY)::List FRAC POLY INT);

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Variable(YY)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Y-forms

Three 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
 \/ \/

LS:=
 ( Y Λ )/ _
 ( Y I )/ _
 V

 There are no exposed library operations named Y but there are 2 
 unexposed operations with that name. Use HyperDoc Browse or issue
 )display op Y
 to learn more about the available operations.

 Cannot find a definition or applicable library operation named Y 
 with argument type(s) 
 Variable(Λ)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Left and right deer:

 RD LD
 \ /\/ \/\ /
 Y /\ /\ Y
 Y ) ( Y
 \/ \/
 i j i j
 \ /\ / \ /\ /
 \ f \ / \ / f /
 \/ \/ \/ \/
 \ /\ /\ /
 e / \ / \ e
 \/ \ / \/
 \ / \ /
 f / \ f
 \/ \/

Left and right deer forms are identical but different from snails.

RD:=
 ( I Λ I ) / _
 ( Y X ) / _
 ( Y I ) / _
 V

 There are no library operations named Λ 
 Use HyperDoc Browse or issue
 )what op Λ
 to learn if there is any operation containing " Λ " in its name.

 Cannot find a definition or applicable library operation named Λ 
 with argument type(s) 
 Variable(I)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Left and right turtles:

 RT LT
 /\ / / \ \ /\
 ( Y / \ Y )
 \ Y Y /
 \/ \/
 i j i j
 /\ / / \ \ /\
 / f / / \ \ f \
 / \/ / \ \/ \
 \ \ / \ / /
 \ e / \ e /
 \ \/ \/ /
 \ / \ /
 \ f f /
 \/ \/

RT:=
 ( Λ I I ) / _
 ( I Y I ) / _
 ( I Y ) / _
 V

 There are no exposed library operations named I but there is one 
 unexposed operation with that name. Use HyperDoc Browse or issue
 )display op I
 to learn more about the available operation.

 Cannot find a definition or applicable library operation named I 
 with argument type(s) 
 Variable(I)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

The turles are symmetric

test(RT = X/RT)

test(LT = X/LT)

Five of the six forms are independent.

test(RT=RS)

test(RT=LS)

test(RT=RD)

test(LT=RS)

test(LT=LS)

test(LT=RD)

Associativity implies right turtle equals right snail and left turtle equals left snail.

in?(ideal(ravel(RT-RS)::List FRAC POLY INT),YYI)

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Polynomial(Integer)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

If the Jacobi identity holds then both snails are zero

in?(ideal(ravel(RS)::List FRAC POLY INT),YXI)

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Variable(RS)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

and right turtle and deer have opposite signs

in?(ideal(ravel(RT+RD)::List FRAC POLY INT),YXI)

 There are 1 exposed and 0 unexposed library operations named ravel 
 having 1 argument(s) but none was determined to be applicable. 
 Use HyperDoc Browse, or issue
 )display op ravel
 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 ravel
 with argument type(s) 
 Polynomial(Integer)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.