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Edit detail for SandBox Grassmann Algebra Is Frobenius In Many Ways revision 4 of 8

1 2 3 4 5 6 7 8
Editor: Bill Page
Time: 2011/04/05 15:36:06 GMT-7
Note: Frobenius pairing

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

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

added:

Alternatively we may consider

removed:
-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}

changed:
---uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
---J::OutputForm * uu::OutputForm = 0
uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
J::OutputForm * uu::OutputForm = 0
nrows(J)
ncols(J)

changed:
-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}
-
If the null space of the 'J' matrix is not empty we can use
the basis to find all non-trivial solutions for U:
\begin{axiom}
NJ:=nullSpace(J)
SS:=map((x,y)+->x=y,concat map(variables,ravel U),
  entries reduce(+,[p[i]*NJ.i for i in 1..#NJ]))
Ug:T := unravel(map(x+->subst(x,SS),ravel U))
\end{axiom}

This defines a family of pre-Frobenius algebras:
\begin{axiom}
test(unravel(map(x+->subst(x,SS),ravel YU))$T=0*YU)
\end{axiom}

The scalar product must be non-degenerate:
\begin{axiom}
Ud := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim]
\end{axiom}


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:

axiom
n:=2

\label{eq1}2(1)
Type: PositiveInteger?
axiom
dim:=2^n

\label{eq2}4(2)
Type: PositiveInteger?
axiom
T:=CartesianTensor(1,dim,FRAC POLY INT)

\label{eq3}\hbox{\axiomType{CartesianTensor}\ } (1, 4, \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..dim]
      for j in 1..dim]
        for k in 1..dim]
          )

\label{eq4}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 
\begin{array}{cccc}
{y_{1, \: 1}^{1}}&{y_{2, \: 1}^{1}}&{y_{3, \: 1}^{1}}&{y_{4, \: 1}^{1}}
\
{y_{1, \: 2}^{1}}&{y_{2, \: 2}^{1}}&{y_{3, \: 2}^{1}}&{y_{4, \: 2}^{1}}
\
{y_{1, \: 3}^{1}}&{y_{2, \: 3}^{1}}&{y_{3, \: 3}^{1}}&{y_{4, \: 3}^{1}}
\
{y_{1, \: 4}^{1}}&{y_{2, \: 4}^{1}}&{y_{3, \: 4}^{1}}&{y_{4, \: 4}^{1}}
(4)
Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

Generate structure constants for Grassmann Algebra

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

\label{eq5}\hbox{\axiomType{AntiSymm}\ } (\hbox{\axiomType{Integer}\ } , [ <em> 01 g 1, </em> 01 g 2 ])(5)
Type: Type
axiom
B:=[exp(reverse concat([0 for i in 1..n-length(x)],wholeRagits(x::RADIX(2))))$GA for x in 0..dim-1]

\label{eq6}\left[ 1, \:{g_{1}}, \:{g_{2}}, \:{{g_{1}}\ {g_{2}}}\right](6)
Type: List(AntiSymm?(Integer,[*01g1,*01g2]))
axiom
-- Compute the multiplication table
M:=matrix [[B.i * B.j for j in 1..dim] for i in 1..dim]

\label{eq7}\left[ 
\begin{array}{cccc}
1 &{g_{1}}&{g_{2}}&{{g_{1}}\ {g_{2}}}
\
{g_{1}}& 0 &{{g_{1}}\ {g_{2}}}& 0 
\
{g_{2}}& -{{g_{1}}\ {g_{2}}}& 0 & 0 
\
{{g_{1}}\ {g_{2}}}& 0 & 0 & 0 
(7)
Type: Matrix(AntiSymm?(Integer,[*01g1,*01g2]))
axiom
-- 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)
Type: Void
axiom
Yg:T:=unravel concat concat(map(S,B)::List List List FRAC POLY INT)
axiom
Compiling function S with type AntiSymm(Integer,[*01g1,*01g2]) -> 
      Matrix(Integer)

\label{eq8}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(8)
Type: CartesianTensor?(1,4,Fraction(Polynomial(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{eq9}\left[ 
\begin{array}{cccc}
{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}}
(9)
Type: CartesianTensor?(1,4,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{eq10}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (10)
(three-point function) is zero.

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

\label{eq11}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 
\begin{array}{cccc}
0 & 0 & 0 & 0 
\
{{u^{2, \: 1}}-{u^{1, \: 2}}}&{u^{2, \: 2}}&{{u^{2, \: 3}}-{u^{1, \: 4}}}&{u^{2, \: 4}}
\
{{u^{3, \: 1}}-{u^{1, \: 3}}}&{{u^{3, \: 2}}+{u^{1, \: 4}}}&{u^{3, \: 3}}&{u^{3, \: 4}}
\
{{u^{4, \: 1}}-{u^{1, \: 4}}}&{u^{4, \: 2}}&{u^{4, \: 3}}&{u^{4, \: 4}}
(11)
Type: CartesianTensor?(1,4,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 U = U(Y)

This problem can be solved using linear algebra.

Alternatively we may consider

axiom
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel(YU),concat(map(variables,ravel(U)))::List Symbol);
Type: Matrix(Fraction(Polynomial(Integer)))
axiom
uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
Type: Matrix(Polynomial(Integer))
axiom
J::OutputForm * uu::OutputForm = 0

\label{eq12}\begin{array}{@{}l}
\displaystyle
{{\left[ 
\begin{array}{cccccccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 (12)
Type: Equation(OutputForm?)
axiom
nrows(J)

\label{eq13}64(13)
Type: PositiveInteger?
axiom
ncols(J)

\label{eq14}16(14)
Type: PositiveInteger?

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.

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

axiom
NJ:=nullSpace(J)

\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}\right] (15)
Type: List(Vector(Fraction(Polynomial(Integer))))
axiom
SS:=map((x,y)+->x=y,concat map(variables,ravel U),
  entries reduce(+,[p[i]*NJ.i for i in 1..#NJ]))

\label{eq16}\begin{array}{@{}l}
\displaystyle
\left[{{u^{1, \: 1}}={p_{1}}}, \:{{u^{1, \: 2}}={p_{2}}}, \:{{u^{1, \: 3}}={p_{3}}}, \:{{u^{1, \: 4}}={p_{4}}}, \:{{u^{2, \: 1}}={p_{2}}}, \: \right.
\
\
\displaystyle
\left.{{u^{2, \: 2}}= 0}, \:{{u^{2, \: 3}}={p_{4}}}, \:{{u^{2, \: 4}}= 0}, \:{{u^{3, \: 1}}={p_{3}}}, \:{{u^{3, \: 2}}= -{p_{4}}}, \: \right.
\
\
\displaystyle
\left.{{u^{3, \: 3}}= 0}, \:{{u^{3, \: 4}}= 0}, \:{{u^{4, \: 1}}={p_{4}}}, \:{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{4, \: 4}}= 0}\right] 
(16)
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
Ug:T := unravel(map(x+->subst(x,SS),ravel U))

\label{eq17}\left[ 
\begin{array}{cccc}
{p_{1}}&{p_{2}}&{p_{3}}&{p_{4}}
\
{p_{2}}& 0 &{p_{4}}& 0 
\
{p_{3}}& -{p_{4}}& 0 & 0 
\
{p_{4}}& 0 & 0 & 0 
(17)
Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

This defines a family of pre-Frobenius algebras:

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

\label{eq18} \mbox{\rm true} (18)
Type: Boolean

The scalar product must be non-degenerate:

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

\label{eq19}-{{p_{4}}^4}(19)
Type: Fraction(Polynomial(Integer))