Edit detail for SandBoxFrobeniusAlgebra revision 20 of 26

added:
References

- "Frobenius algebras and 2D topological quantum field theories":http://mat.uab.es/~kock/TQFT.html
  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":http://www.cs.amherst.edu/~dac/iva.html#MR5
  by David A. Cox, et al.


changed:
-$Y=\{ {y_k}^{ji} \ i,j,k =1,2, ... n \}$
-viewed as an operator with two inputs 'i,j' and one
-output 'k'. For example in 2 dimensions
$Y=\{ {y^k}_{ij} \ i,j,k =1,2, ... n \}$
viewed as an operator with two inputs $i,j$ and one
output $k$. For example in 2 dimensions

changed:
-  [[[script(y,[[k],[j,i]])
  [[[script(y,[[i,j],[k]])

changed:
-Given two vectors $P=\{ p_i \}$ and $Q=\{ q_j \}$
-\begin{axiom}
-P:T := unravel([script(p,[[i]]) for i in 1..n])
-Q:T := unravel([script(q,[[i]]) for i in 1..n])
Given two vectors $P=\{ p^i \}$ and $Q=\{ q^j \}$
\begin{axiom}
P:T := unravel([script(p,[[],[i]]) for i in 1..n])
Q:T := unravel([script(q,[[],[i]]) for i in 1..n])

changed:
-yield a vector $R=\{ r_k = {y_k}^{ji} p_i q_j \}$
yield a vector $R=\{ r_k = {y^k}_{ij} p^i q^j \}$

changed:
-  Q   P
-  2j  3i
-   \ /
-    |
-    1k
-    R
  P Q
   Y
   R

  or more explicitly

  Pi Qj
   \/
    \
     Rk

changed:
-An algebra $Y$ is commutative if the tensor
-$\Pi = \{ {\pi_k}^{ji} = {y_k}^{ji}-{y^k}^{ij} \}$
-(the commutator) is zero::
-
-   Y  -  X
-         Y
-
-\begin{axiom}
-XY:=Y-reindex(Y,[1,3,2])
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
\begin{equation}
\Psi  = \{ {\psi_l}^{ijk} =  {y^e}_{ij} {y^l}_{ek} - {y^l}_{ie} {y^e}_{jk} \}
\end{equation}
(associator) is zero.
\begin{axiom}
YY := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(YY)
\end{axiom}

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
\begin{equation}
\mathcal{C} = \{ {c^k}_{ij} = {y^k}_{ij} - {y^k}_{ji} \}
\end{equation}
(commutator) is zero.
\begin{axiom}
YC:=Y-reindex(Y,[1,3,2])

changed:
-groebner(ravel(XY))
-\end{axiom}
-The algebra 'Y' is anti-commutative if the tensor
-$\Xi = \{ {\xi_k}^{ji} = {y_k}^{ji}+{y^k}^{ij} \}$
-(the anti-commutator) is zero::
-
-   Y  +  X
-         Y
-
-\begin{axiom}
-XX:=Y+reindex(Y,[1,3,2])
groebner(ravel(YC))
\end{axiom}

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
\begin{equation}
\mathcal{A} = \{ {a^k}_{ij} = {y^k}_{ij} + {y^k}_{ji} \}
\end{equation}
(anti-commutator) is zero.
\begin{axiom}
YA:=Y+reindex(Y,[1,3,2])

changed:
-anti-commutator is given by:
-\begin{axiom}
-groebner(ravel(XX))
-\end{axiom}
-An algebra is associative if::
-
-  Y    =    Y
-   Y       Y
-
-  Note: right figure is mirror image of left!
-
-  2  3 6   2 5  6      2  3  4
-   \/ /     \ \/        \ | /
-    \/   =   \/    =     \|/
-     \       /            |
-      4     1             1
-
-In other words an algebra is associative if and only
-if the following (3,1)-tensor
-$\Psi  = \{ {\psi_s}^{kji} =  {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} \}$
commutator is given by:
\begin{axiom}
groebner(ravel(YA))
\end{axiom}

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
\begin{equation}
\Theta = \{ {\theta^l}_{ijk} =  {y^l}_{ek} {y^e}_{ij} - {y^l}_{ie} {y^e}_{jk} - {y^l}_{ej} {y^e}_{ik} \}
\end{equation}

changed:
-\begin{axiom}
-YY := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(YY)
-\end{axiom}
-
-The Jacobi identity requires the following (3,1)-tensor
-$\Phi = \{ {\phi_s}^{kji} =  {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} - {y_s}^{ri} {y_r}^{jk} \}$
-to be zero::
-
-  Y    -    Y  -   X
-   Y       Y       Y
-                  Y
-
-  2    3 6   2 5    6   2 6  3   
-   \  / /     \ \  /     \ \/   
-    \/ /       \ \/       \/\   
-     \/    -    \/    -    \/   
-      \         /           \   
-       4       1             4  
-
-\begin{axiom}
-YYX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(YYX)

\begin{axiom}
YX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(YX)

changed:
-$U = \{ u^{ij} \}$
$U = \{ u_{ij} \}$

changed:
-  We say that the scalar product is "associative" if the tensor
  We say that the scalar product is *associative* if the tensor

changed:
-  In other words, if the following tensor is zero
-  $ \Omega = \{ \omega^{kji} =  {Y_r}^{kj} U^{ri} - U^{kr} {Y_r}^{ji} \} $
  In other words, if the (3,0)-tensor::

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

  \begin{equation}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  \end{equation}
  (three-point function) is zero.

changed:
-  An algebra with a non-degenerate associative scalar product is
-  called ''pre-Frobenius''.
  An algebra with a non-degenerate associative scalar product
  is called *pre-Frobenius*.

changed:
-and look for all associative scalar products 'U = U(Y)' or we
and look for all associative scalar products $U = U(Y)$ or we

changed:
-algebras Y=Y(U) such that the scalar product is associative. 
algebras $Y=Y(U)$ such that the scalar product is associative. 

changed:
-into coefficients of the tensor $\Omega$. We are looking for
into coefficients of the tensor $\Phi$. We are looking for

changed:
-tensor $Y$ into $\Omega=0$ for any $Y$.
tensor $Y$ into $\Phi=0$ for any $Y$.

changed:
-  The scalar product of all 2-dimensional pre-Frobenius
-  algebras is symmetric.
  All 2-dimensional pre-Frobenius algebras are symmetric.

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

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

changed:
-YUS:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel YU))
YUS:T :=  reindex(reindex(US,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-US*Y

changed:
-This is a 4-parameter family of 2-d pre-Frobenius algebras with
-a given admissible (i.e. symmetric) scalar product.
-\begin{axiom}
-UASS:T := unravel(map(x+->subst(x,SS),ravel YUS))
-\end{axiom}
-
This defines a 4-parameter family of 2-d pre-Frobenius algebras
\begin{axiom}
test(unravel(map(x+->subst(x,SS),ravel YUS))$T=0*YU)
\end{axiom}

Alternatively we may consider

changed:
-into coefficients of the tensor $\Omega$. We are looking for
into coefficients of the tensor $\Phi$. We are looking for

changed:
-tensor $U$ into $\Omega=0$ for any $U$.
tensor $U$ into $\Phi=0$ for any $U$.

changed:
-  A 2-d algebra is pre-Frobenius if it is associative,
-  commutative, anti-commutative or if it satisfies the
-  Jacobi identity.
  If a 2-d algebra is associative, commutative, anti-commutative
  or if it satisfies the Jacobi identity then it is a
  pre-Frobenius algebra.

changed:
-in?(JP,ideal ravel XY)  -- commutative
-in?(JP,ideal ravel XX)  -- anti-commutative
-in?(JP,ideal ravel YYX) -- Jacobi identity
-\end{axiom}
in?(JP,ideal ravel YC)  -- commutative
in?(JP,ideal ravel YA)  -- anti-commutative
in?(JP,ideal ravel YX) -- Jacobi identity
\end{axiom}

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 an operator with two inputs and one output . For example in 2 dimensions

fricas

(1) -> n:=2

(1)

Type: PositiveInteger?

fricas

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

(2)

Type: Type

fricas

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)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

Given two vectors and

fricas

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

(4)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

fricas

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

(5)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

the tensor operates on their tensor product to yield a vector

fricas

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

(6)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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.

fricas

R:=(Y*P)*Q

(7)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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.

fricas

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

(9)

Type: List(Fraction(Polynomial(Integer)))

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.

fricas

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

(11)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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

fricas

groebner(ravel(YC))

(12)

Type: List(Polynomial(Integer))

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.

fricas

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

(14)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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

fricas

groebner(ravel(YA))

(15)

Type: List(Polynomial(Integer))

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.

fricas

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

(17)

Type: List(Fraction(Polynomial(Integer)))

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

fricas

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

(18)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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.

fricas

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

(20)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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.

fricas

)expose MCALCFN

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

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

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

Type: Matrix(Polynomial(Integer))

fricas

K::OutputForm * yy::OutputForm = 0

(21)

Type: Equation(OutputForm?)

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.

fricas

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

(22)

Type: Factored(DistributedMultivariatePolynomial?([scripts(u, [, 1, 1]), scripts(u, [, 1, 2]), scripts(u, [, 2, 1]), scripts(u, [, 2, 2])],Fraction(Integer)))

The scalar product must also be non-degenerate

fricas

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

(23)

Type: DistributedMultivariatePolynomial?([scripts(u, [, 1, 1]), scripts(u, [, 1, 2]), scripts(u, [, 2, 1]), scripts(u, [, 2, 2])],Fraction(Integer))

therefore U must be symmetric.

fricas

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

fricas

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

fricas

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

fricas

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

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

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

Type: Matrix(Polynomial(Integer))

fricas

J::OutputForm * uu::OutputForm = 0

(24)

Type: Equation(OutputForm?)

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.

fricas

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) ];

Type: PolynomialIdeal?(Fraction(Integer),IndexedExponents?(Symbol),Symbol,Polynomial(Fraction(Integer)))

fricas

#generators(%)

(25)

Type: PositiveInteger?

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

fricas

in?(JP,ideal ravel YY)  -- associative

(26)

Type: Boolean

fricas

in?(JP,ideal ravel YC)  -- commutative

(27)

Type: Boolean

fricas

in?(JP,ideal ravel YA)  -- anti-commutative

(28)

Type: Boolean

fricas

in?(JP,ideal ravel YX) -- Jacobi identity

(29)

Type: Boolean