Edit detail for SandBox Grassmann Algebra Is Frobenius In Many Ways revision 6 of 8

-dimensional vector space representing Grassmann algebra with generators

An algebra is represented by a (2,1)-tensor viewed as a linear operator with two inputs and one output . For example:

n:=2

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dim:=2^n

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T := CartesianTensor(1,dim,EXPR INT)

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--T:=CartesianTensor(1,dim,FRAC POLY INT)
X:List T := [unravel [(i=j => 1;0) for j in 1..dim] for i in 1..dim]

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X(1),X(2)

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

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Generate structure constants for Grassmann Algebra

-- Construct a basis for the Grassmann algebra
GA:=AntiSymm(INT,[subscript(g,[i]) for i in 1..n])

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B:=[exp(reverse concat([0 for i in 1..n-length(x)],wholeRagits(x::RADIX(2))))$GA for x in 0..dim-1]

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-- Compute the multiplication table
M:=matrix [[B.i * B.j for j in 1..dim] for i in 1..dim]

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-- The structure constants of the algebra are given by the coefficients
-- of the polynomials in the multiplication table with respect to the basis
S(y)==map(x+->coefficient(x,y),M)

Yg:T:=unravel concat concat(map(S,B)::List List List FRAC POLY INT)

Compiling function S with type AntiSymm(Integer,[*01g1,*01g2]) -> 
      Matrix(Integer)

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

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

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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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ω := reindex(reindex(U,[2,1])*reindex(Yg,[1,3,2]),[3,2,1])-U*Yg

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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

This problem can be solved using linear algebra.

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial 
J := jacobian(ravel ω,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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nrows(J)

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ncols(J)

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

If the null space of the J matrix is not empty we can use the basis to find all non-trivial solutions for U:

NJ:=nullSpace(J)

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

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Ug:T := unravel(map(x+->subst(x,SS),ravel U))

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This defines a family of pre-Frobenius algebras:

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

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The scalar product must be non-degenerate:

Ud := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim]

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Definition 3

Co-pairing

Ωg:T:=unravel concat(transpose(1/Ud*adjoint([[Ug[i,j] for j in 1..dim] for i in 1..dim]).adjMat)::List List FRAC POLY INT)

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dimension
Ω
U

contract(contract(Ωg,1,Ug,1),1,2)

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Definition 4

Co-multiplication

λg:=reindex(contract(contract(Ug*Yg,1,Ωg,1),1,Ωg,1),[2,3,1]);

-- just for display
reindex(λg,[3,1,2])

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i  
λ=Ω

test(λg*X(1)=Ωg)

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Definition 5

  i 
  U
  

ιg:=X(1)*Ug

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Y=U
ι  

test(ιg * Yg = Ug)

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For example:

Ug0:T:=unravel eval(ravel Ug,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])

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Ωg0:T:=unravel eval(ravel Ωg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])

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λg0:T:=unravel eval(ravel λg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1]);

reindex(λg0,[3,1,2])

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