Edit detail for SandBoxFrobeniusAlgebra revision 18 of 26

An n-dimensional algebra is represented by a (2,1)-tensor viewed as an operator with two inputs i,j and one output k. For example in 2 dimensions

n:=2

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

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

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Given two vectors and

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

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

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the tensor operates on their tensor product to yield a vector

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

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Pictorially:

  Q   P
  2j  3i
   \ /
    |
    1k
    R

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

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An algebra is commutative if the tensor (the commutator) is zero:

   Y  -  X
         Y

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

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

groebner(ravel(XY))

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The algebra Y is anti-commutative if the tensor (the anti-commutator) is zero:

   Y  +  X
         Y

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

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

groebner(ravel(XX))

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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 is zero.

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

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The Jacobi identity requires the following (3,1)-tensor to be zero:

  Y    -    Y  -   X
   Y       Y       Y
                  Y

  2    3 6   2 5    6   2 6  3   
   \  / /     \ \  /     \ \/   
    \/ /       \ \/       \/\   
     \/    -    \/    -    \/   
      \         /           \   
       4       1             4  

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

   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(XYY)

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

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]
        )

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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 following tensor is zero

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

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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 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);

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

The scalar product of all 2-dimensional pre-Frobenius algebras is symmetric.

Proof: Consider the determinant of the matrix K above.

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

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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]

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therefore U must be symmetric.

nthFactor(Kd,1)

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US:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel U))

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Theorem 2

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

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

YUS:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel YU))

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KS := jacobian(ravel(YUS),concat(map(variables,ravel(Y)))::List Symbol);

NS:=nullSpace(KS)

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SS:=map((x,y)+->x=y,concat map(variables,ravel Y),
  entries reduce(+,[p[i]*NS.i for i in 1..#NS]))

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YS:T := unravel(map(x+->subst(x,SS),ravel Y))

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This is a 4-parameter family of 2-d pre-Frobenius algebras with a given admissible (i.e. symmetric) scalar product.

UASS:T := unravel(map(x+->subst(x,SS),ravel YUS))

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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

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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(%)

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Theorem 3

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

Proof

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

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

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

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

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