Edit detail for SandBoxFrobeniusAlgebra revision 13 of 26

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

(1) -> )library DEXPR

   DistributedExpression is now explicitly exposed in frame initial 
   DistributedExpression will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/DEXPR.NRLIB/DEXPR
n:=2

(1)

T:=CartesianTensor(1,n,DEXPR INT)

(2)

--T:=CartesianTensor(1,n,HDMP(concat[concat concat
--  [[[script(y,[[k],[j,i]])
--    for i in 1..n]
--      for j in 1..n]
--        for k in 1..n],
--          [script(u,[[i]]) for i in 1..n],
--            [script(v,[[i]]) for i in 1..n] ],FRAC INT))
Y:=unravel(concat concat
  [[[script(y,[[k],[j,i]])
    for i in 1..n]
      for j in 1..n]
        for k in 1..n]
          )$T

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Given two vectors and

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

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V:=unravel([script(v,[[i]]) for i in 1..n])$T

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the tensor Y operates on their tensor product to yield a vector

W:=contract(contract(Y,3,product(U,V),1),2,3)

(6)

Diagram:

  U   V
  2i  3j
   \ /
    |
    1k
    W

or in a more convenient notation:

W:=(Y*U)*V

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The algebra Y is commutative if the following tensor (the commutator) is zero

K:=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:

C:=groebner(ravel(K))

   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(DistributedExpression(Integer))

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

An algebra is associative if:

  Y I  =  I 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.

test(Y*Y = contract(product(Y,Y),3,4))

(9)

test(Y*Y = contract(Y,3,Y,1))

(10)

test(reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2]) = reindex(contract(product(Y,Y),1,5),[3,1,2,4]))

(11)

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

(12)

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

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AB:=groebner(ravel(AA))

   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(DistributedExpression(Integer))

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

The Jacobi identity requires the following tensor to be zero:

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

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

(14)

BB:=groebner(ravel(BA));

   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(DistributedExpression(Integer))

      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

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

(15)

We say that the scalar product is "associative" if the following tensor equation holds:

  Y I = I Y
   U     U

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

(16)