help links subscribe changes refresh edit

	Frobenius Algebra, Vector Spaces and Polynomial Ideals
Octonion Algebra is Frobenius in Just One Way ←	↑	→ SandBox Quaternion Algebra is Frobenius in Many Ways

SandBox Grassmann Algebra Is Frobenius In Many Ways

Edit detail for SandBox Grassmann Algebra Is Frobenius In Many Ways revision 1 of 8

	1 2 3 4 5 6 7 8
Editor: Bill Page Time: 2011/04/05 09:37:32 GMT-7
Note: draft

changed:
-
$2^n$-dimensional vector space representing Grassmann algebra with $n$ generators

An algebra is represented by a (2,1)-tensor
$Y=\{ {y^k}_{ij} \ i,j,k =1,2, ... dim \}$
viewed as a linear operator with two inputs $i,j$ and one
output $k$. For example:
\begin{axiom}
n:=2
dim:=2^n
T:=CartesianTensor(1,n,FRAC POLY INT)
Y:T := unravel(concat concat
  [[[script(y,[[i,j],[k]])
    for i in 1..diim]
      for j in 1..dim]
        for k in 1..dim]
          )
\end{axiom}

A scalar product is denoted by the (2,0)-tensor
$U = \{ u_{ij} \}$
\begin{axiom}
U:T := unravel(concat
  [[script(u,[[],[j,i]])
    for i in 1..dim]
      for j in 1..dim]
        )
\end{axiom}
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

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

\begin{axiom}
YU := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y
\end{axiom}
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.
\begin{axiom}
)expose MCALCFN
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
\end{axiom}
The matrix 'K' transforms the coefficients of the tensor $Y$
into coefficients of the tensor $\Phi$. We are looking for
coefficients of the tensor $U$ such that 'K' transforms the
tensor $Y$ into $\Phi=0$ for any $Y$.

A necessary condition for the equation to have a non-trivial
solution is that the matrix 'K' be degenerate.

Consider the determinant of the matrix 'K' above.
\begin{axiom}
--Kd := factor(determinant(K)::DMP(concat map(variables,ravel(U)),FRAC INT))
\end{axiom}
The scalar product must also be non-degenerate
\begin{axiom}
--Ud:DMP(concat map(variables,ravel(U)),FRAC INT) := determinant [[U[i,j] for j in 1..dim] for i in 1..dim]
\end{axiom}

The basis of the null space of the 'K' matrix
\begin{axiom}
--YUS:T :=  reindex(reindex(US,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-US*Y
--KS := jacobian(ravel(YUS),concat(map(variables,ravel(Y)))::List Symbol);
--NS:=nullSpace(KS)
--SS:=map((x,y)+->x=y,concat map(variables,ravel Y),
--  entries reduce(+,[p[i]*NS.i for i in 1..#NS]))
--YS:T := unravel(map(x+->subst(x,SS),ravel Y))
\end{axiom}
This defines a family of pre-Frobenius algebras
\begin{axiom}
--test(unravel(map(x+->subst(x,SS),ravel YUS))$T=0*YU)
\end{axiom}

Alternatively we may consider
\begin{axiom}
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
\end{axiom}
The matrix 'J' transforms the coefficients of the tensor $U$
into coefficients of the tensor $\Phi$. We are looking for
coefficients of the tensor $Y$ such that 'J' transforms the
tensor $U$ into $\Phi=0$ for any $U$.

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.
\begin{axiom}
--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(%)
\end{axiom}

-dimensional vector space representing Grassmann algebra with generators

An algebra is represented by a (2,1)-tensor $Y=\{ {y^k}_{ij} \ i,j,k =1,2, ... dim \}$ viewed as a linear operator with two inputs and one output . For example:

axiom

n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

dim:=2^n

$\label{eq2}4$

(2)

Type: PositiveInteger?

axiom

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

$\label{eq3}\hbox{\axiomType{CartesianTensor}\ } (1, 2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(3)

Type: Type

axiom

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

   The upper bound in a loop must be an integer.

A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$

axiom

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

$\label{eq4}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{1, \: 3}}&{u^{1, \: 4}}$

(4)

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

$\label{eq5} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(5)

(three-point function) is zero.

axiom

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

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

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.

axiom

)expose MCALCFN

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

   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(YU)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
yy := transpose matrix [concat(map(variables,ravel(Y)))::List Symbol];

Type: Matrix(Polynomial(Integer))

axiom

K::OutputForm * yy::OutputForm = 0

$\label{eq6}{K \ {\left[ \begin{array}{c} Y$

(6)

Type: Equation(OutputForm?)

The matrix K transforms the coefficients of the tensor into coefficients of the tensor $\Phi$ . We are looking for coefficients of the tensor such that K transforms the tensor into $\Phi=0$ for any .

A necessary condition for the equation to have a non-trivial solution is that the matrix K be degenerate.

Consider the determinant of the matrix K above. Axiom output parse error!

The scalar product must also be non-degenerate Axiom output parse error!

The basis of the null space of the K matrix Axiom output parse error!

This defines a family of pre-Frobenius algebras Axiom output parse error!

Alternatively we may consider

axiom

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

   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(YU)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];

Type: Matrix(Polynomial(Integer))

axiom

J::OutputForm * uu::OutputForm = 0

$\label{eq7}\begin{array}{@{}l} \displaystyle {J \ {\left[ \begin{array}{c} {u^{1, \: 1}} \ {u^{1, \: 2}} \ {u^{1, \: 3}} \ {u^{1, \: 4}} \ {u^{2, \: 1}} \ {u^{2, \: 2}} \ {u^{2, \: 3}} \ {u^{2, \: 4}} \ {u^{3, \: 1}} \ {u^{3, \: 2}} \ {u^{3, \: 3}} \ {u^{3, \: 4}} \ {u^{4, \: 1}} \ {u^{4, \: 2}} \ {u^{4, \: 3}} \ {u^{4, \: 4}}$

(7)

Type: Equation(OutputForm?)

The matrix J transforms the coefficients of the tensor into coefficients of the tensor $\Phi$ . We are looking for coefficients of the tensor such that J transforms the tensor into $\Phi=0$ 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. Axiom output parse error!