Edit detail for Frobenius Algebra, Vector Spaces and Polynomial Ideals revision 2 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.

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

(1) -> n:=2

(1)

T:=CartesianTensor(1,n,FRAC POLY INT)

(2)

Y:T := unravel(concat concat
  [[[script(y,[[i,j],[k]])
    for i in 1..n]
      for j in 1..n]
        for k in 1..n]
          )

(3)

Given two vectors and

P:T := unravel([script(p,[[],[i]]) for i in 1..n])

(4)

Q:T := unravel([script(q,[[],[i]]) for i in 1..n])

(5)

the tensor operates on their tensor product to yield a vector

R:=contract(contract(Y,3,product(P,Q),1),2,3)

(6)

Pictorially:

  P Q
   Y
   R

  or more explicitly

  Pi Qj
   \/
    \
     Rk

In Axiom we may use the more convenient tensor inner product denoted by * that combines tensor product with a contraction on the last index of the first tensor and the first index of the second tensor.

R:=(Y*P)*Q

(7)

An algebra is said to be associative if:

  Y    =    Y
   Y       Y

Note: the right hand side of the equation above is implicitly the mirror image of the left hand side:

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

This requires that the following (3,1)-tensor

(8)

(associator) is zero.

YY := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(YY)

(9)

The algebra is commutative if:

  Y = Y

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

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

(10)

(commutator) is zero.

YC:=Y-reindex(Y,[1,3,2])

(11)

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

groebner(ravel(YC))

(12)

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

(13)

(anti-commutator) is zero.

YA:=Y+reindex(Y,[1,3,2])

(14)

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

groebner(ravel(YA))

(15)

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

(16)

is zero.

YX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(YX)

(17)

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

U:T := unravel(concat
  [[script(u,[[],[j,i]])
    for i in 1..n]
      for j in 1..n]
        )

(18)

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

(19)

(three-point function) is zero.

YU := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y

(20)

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

(21)

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))

(22)

The scalar product must also be non-degenerate

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

(23)

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:T :=  reindex(reindex(US,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-US*Y

   >> Error detected within library code:
   The list is not a permutation.

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

test(unravel(map(x+->subst(x,SS),ravel YUS))$T=0*YU)

   There are 3 exposed and 0 unexposed library operations named subst 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op subst
      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 subst
      with argument type(s) 
                        Fraction(Polynomial(Integer))
                                Variable(SS)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.

   Anonymous user functions created with +-> that are processed in 
      interpret-code mode must have result target information 
      available. This information is not present so FriCAS cannot 
      proceed any further. This may be remedied by declaring the 
      function.

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

(24)

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)]))
      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(%)

(25)

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;

in?(JP,YYI)  -- associative

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YCI:=ideal ravel YC;

in?(JP,YCI)  -- commutative

(27)

YAI:=ideal ravel YA;

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

(28)

YXI:=ideal ravel YX;

in?(JP,YXI) -- Jacobi identity

(29)

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
         \/              \/

(30)

LS:=contract(Y*Y,1,2)

(31)

RS:=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,2),[2,1])

(32)

test(LS=RS)

(33)

Left and right deer:

   RD                 LD

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

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

(34)

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

RD:=contract(Y*Y,1,3)

(35)

LD:=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,3),[2,1])

(36)

test(LD=RD)

(37)

test(RD=RS)

(38)

test(RD=LS)

(39)

Left and right turtles:

  RT                   LT

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

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

(40)

RT:=contract(Y*Y,1,4)

(41)

LT:=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,4),[2,1])

(42)

test(LT=RT)

(43)

The turles are symmetric

test(RT=reindex(RT,[2,1]))

(44)

test(LT=reindex(LT,[2,1]))

(45)

Five of the six forms are independent.

test(RT=RS)

(46)

test(RT=LS)

(47)

test(RT=RD)

(48)

test(LT=RS)

(49)

test(LT=LS)

(50)

test(LT=RD)

(51)

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

in?(ideal ravel(RT-RS),YYI)

(52)

in?(ideal ravel(LT-LS),YYI)

(53)

If the Jacobi identity holds then both snails are zero

in?(ideal ravel(RS),YXI)

(54)

in?(ideal ravel(LS),YXI)

(55)

and right turtle and deer have opposite signs

in?(ideal ravel(RT+RD),YXI)

(56)

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