Edit detail for Frobenius Algebra, Vector Spaces and Polynomial Ideals revision 3 of 4

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

   CartesianTensor is now explicitly exposed in frame initial 
   CartesianTensor will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN
   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/PROP.NRLIB/PROP
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/LOP.NRLIB/LOP
L:=LOP(OVAR ['1,'2], EXPR INT)

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-- basis
dx:=basisIn()$L

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Dx:=basisOut()$L

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-- summation
macro Σ(f,b,i) == reduce(+,[f*b.i for i in 1..#b])

-- identity
I:L:=[1]

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-- twist
X:L:=[2,1]

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-- co-evaluation
Λ:L:=co(1)

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-- evaluation
V:L:=ev(1)

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Algebra

An 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

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

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Multiplication

Given two vectors and

P:=Σ(script(p,[[],[i]]),Dx,i)

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Q:=Σ(script(q,[[],[i]]),Dx,i)

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the tensor operates on their tensor product to yield a vector

R:=(P,Q)/Y

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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

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(associator) is zero.

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

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Commutator

The algebra is commutative if:

    Y = Y

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

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

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(commutator) is zero.

YC:=Y-(X/Y)

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

groebner(ravel(YC))

   There are 2 exposed and 6 unexposed library operations named 
      groebner having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op groebner
      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 
      groebner with argument type(s) 
                          List(Expression(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 is anti-commutative if:

    Y = -Y

    i   j     i  j     j  i
     \ /   =   \/   =   \/
      |         \       /
      k          k     k

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

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(anti-commutator) is zero.

YA:=Y+(X/Y)

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

groebner(ravel(YA))

   There are 2 exposed and 6 unexposed library operations named 
      groebner having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op groebner
      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 
      groebner with argument type(s) 
                          List(Expression(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

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is zero.

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

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Scalar Product

A scalar product is denoted by the (2,0)-tensor

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

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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

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(three-point function) is zero.

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

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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 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.

)expose MCALCFN

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

yy := transpose matrix [concat(map(variables,ravel(Y)))::List Symbol];

K::OutputForm * yy::OutputForm = 0

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The matrix K transforms the coefficients of the tensor into coefficients of the tensor . We are looking for coefficients of the tensor such that K transforms the tensor into for any .

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))

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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]

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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) 
Factored(DistributedMultivariatePolynomial([u[;1,1],u[;1,2],u[;2,1],u[;2,2]],Fraction(Integer)))
                               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 15 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(LinearOperator(OrderedVariableList([1,2]),Expression(Integer)))
                                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);

uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];

J::OutputForm * uu::OutputForm = 0

(26)

The matrix J transforms the coefficients of the tensor into coefficients of the tensor . We are looking for coefficients of the tensor such that J transforms the tensor into for any .

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) ];

#generators(%)

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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);

in?(JP,YYI)  -- associative

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YCI:=ideal(ravel(YC)::List FRAC POLY INT);

in?(JP,YCI)  -- commutative

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YAI:=ideal(ravel(YA)::List FRAC POLY INT);

in?(JP,YAI)  -- anti-commutative

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YXI:=ideal(ravel(YX)::List FRAC POLY INT);

in?(JP,YXI) -- Jacobi identity

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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
         \/              \/

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

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

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test(LS=RS)

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Left and right deer:

   RD                 LD

   \ /\/              \/\ /
    Y /\              /\ Y
     Y  )            (  Y
      \/              \/

   i            j    i            j
    \    /\    /      \    /\    /
     \  f  \  /        \  /  f  /
      \/    \/          \/    \/
       \    /\          /\    /
        e  /  \        /  \  e
         \/    \      /    \/
          \    /      \    /
           f  /        \  f
            \/          \/

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

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

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

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test(LD=RD)

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test(RD=RS)

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test(RD=LS)

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Left and right turtles:

  RT                   LT

   /\ / /               \ \ /\
  (  Y /                 \ Y  )
   \  Y                   Y  /
    \/                     \/

           i     j      i     j
    /\    /     /        \     \    /\
   /  f  /     /          \     \  f  \
  /    \/     /            \     \/    \
  \     \    /              \    /     /
   \     e  /                \  e     /
    \     \/                  \/     /
     \    /                    \    /
      \  f                      f  /
       \/                        \/

(42)

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

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

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test(LT=RT)

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The turles are symmetric

test(RT = X/RT)

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test(LT = X/LT)

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Five of the six forms are independent.

test(RT=RS)

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test(RT=LS)

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test(RT=RD)

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test(LT=RS)

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test(LT=LS)

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test(LT=RD)

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

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

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in?(ideal(ravel(LT-LS)::List FRAC POLY INT),YYI)

(55)

If the Jacobi identity holds then both snails are zero

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

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

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and right turtle and deer have opposite signs

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

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