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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
SandBox Sedenion Algebra is Frobenius In Just One Way ←	↑	→ The Algebra of Complex Numbers Is Frobenius In Many Ways

SandBoxFrobeniusAlgebra

last edited 13 years ago by Bill Page

Edit detail for SandBoxFrobeniusAlgebra revision 14 of 26

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Editor: Bill Page Time: 2011/02/14 21:28:09 GMT-8
Note: scalar product of 2-d Frobenius algebra is symmetric

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

changed:
-Y:=unravel(concat concat
Y:T := unravel(concat concat

changed:
-          )$T
          )

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

changed:
-U:=unravel(concat
U:T := unravel(concat

changed:
-        )$T
        )

changed:
-We say that the scalar product is "associative" if the following tensor equation holds::
-
-  Y I = I Y
-   U     U
Definition 1

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

    Y I = I Y
     U     U

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

added:
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(UA),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 'UA'. We are looking for
coefficients of the tensor 'U' such that 'K' transforms 'Y'
into 'UA=0' for any 'Y'.

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.
\begin{axiom}
Kd:DMP(concat map(variables,ravel(U)),FRAC INT) := factor determinant(K)
\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..n] for i in 1..n]
\end{axiom}
therefore U must be symmetric.
\begin{axiom}
nthFactor(Kd,1)
US:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel U))
\end{axiom}

Pre-Frobenius Algebras
\begin{axiom}
UAS:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel UA))
--solve(ravel(UAS),removeDuplicates concat map(variables,ravel(US)))
KS := jacobian(ravel(UAS),concat(map(variables,ravel(Y)))::List Symbol);
NS:=nullSpace(KS)
--map(=,concat(map(variables,ravel(Y))),
entries reduce(+,[p[i]*NS.i for i in 1..#NS])
\end{axiom}

An n-dimensional algebra is represented by a (1,2)-tensor $Y=\{ {y_k}^{ji} \ i,j,k =1,2, ... n \}$ viewed as an operator with two inputs i,j and one output k. For example in 2 dimensions

axiom

)library DEXPR

   DistributedExpression is now explicitly exposed in frame initial
   DistributedExpression will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/DEXPR.NRLIB/DEXPR
n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

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

$\label{eq2}\hbox{\axiomType{CartesianTensor}\ } (1, 2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(2)

Type: Domain

axiom

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

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {y_{1}^{1, \: 1}}&{y_{1}^{1, \: 2}} \ {y_{1}^{2, \: 1}}&{y_{1}^{2, \: 2}}$

(3)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

Given two vectors $U=\{ u_i \}$ and $V=\{ v_j \}$

axiom

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

$\label{eq4}\left[{u_{1}}, \:{u_{2}}\right]$

(4)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

axiom

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

$\label{eq5}\left[{v_{1}}, \:{v_{2}}\right]$

(5)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

the tensor Y operates on their tensor product to yield a vector $W=\{ w_k = {y_k}^{ji} u_i v_j \}$

axiom

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

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{{{\left({{y_{1}^{2, \: 2}}\ {u_{2}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{1}^{1, \: 2}}\ {u_{2}}}+{{y_{1}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}, \: \right. \ \ \displaystyle \left.{{{\left({{y_{2}^{2, \: 2}}\ {u_{2}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{2}^{1, \: 2}}\ {u_{2}}}+{{y_{2}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}\right]$

(6)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

Diagram:

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

or in a more convenient notation:

axiom

W:=(Y*U)*V

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{{{\left({{y_{1}^{2, \: 2}}\ {u_{2}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{1}^{1, \: 2}}\ {u_{2}}}+{{y_{1}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}, \: \right. \ \ \displaystyle \left.{{{\left({{y_{2}^{2, \: 2}}\ {u_{2}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{2}^{1, \: 2}}\ {u_{2}}}+{{y_{2}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}\right]$

(7)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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

axiom

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

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 &{-{y_{1}^{2, \: 1}}+{y_{1}^{1, \: 2}}} \ {{y_{1}^{2, \: 1}}-{y_{1}^{1, \: 2}}}& 0$

(8)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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

axiom

C:=groebner(ravel(K))

$\label{eq9}\left[{{y_{2}^{2, \: 1}}-{y_{2}^{1, \: 2}}}, \:{{y_{1}^{2, \: 1}}-{y_{1}^{1, \: 2}}}\right]$

(9)

Type: List(Polynomial(Integer))

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 $A=\{ {a_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_r}^{kj} {y_s}^{ri} \}$ is zero.

axiom

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

$\label{eq10} \mbox{\rm true}$

(10)

Type: Boolean

axiom

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

$\label{eq11} \mbox{\rm true}$

(11)

Type: Boolean

axiom

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]))

$\label{eq12} \mbox{\rm true}$

(12)

Type: Boolean

axiom

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]))

$\label{eq13} \mbox{\rm true}$

(13)

Type: Boolean

axiom

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

$\label{eq14}\begin{array}{@{}l} \displaystyle \left[{{\left({y_{1}^{2, \: 1}}-{y_{1}^{1, \: 2}}\right)}\ {y_{2}^{1, \: 1}}}, \:{-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}+{{y_{1}^{1, \: 2}}^2}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \:{{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}^2}}, \: \right. \ \ \displaystyle \left.{{\left(-{y_{1}^{2, \: 1}}+{y_{1}^{1, \: 2}}\right)}\ {y_{1}^{2, \: 2}}}, \:{{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{2}^{1, \: 2}}^2}+{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{\left(-{y_{1}^{2, \: 1}}+{y_{1}^{1, \: 2}}\right)}\ {y_{2}^{1, \: 1}}}, \:{{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}+{{y_{2}^{2, \: 1}}^2}-{{y_{1}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{\left({y_{2}^{2, \: 1}}-{y_{2}^{1, \: 2}}\right)}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}}\right]$

(14)

Type: List(Fraction(Polynomial(Integer)))

axiom

AB:=groebner(ravel(AA))

$\label{eq15}\begin{array}{@{}l} \displaystyle \left[{{{\left({y_{2}^{2, \: 1}}-{y_{2}^{1, \: 2}}\right)}\ {y_{2}^{2, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{2}^{1, \: 2}}^2}+{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}^2}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{1, \: 2}}^2}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{2, \: 1}}^2}-{{y_{1}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 2}}^2}+{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{\left({{y_{2}^{1, \: 2}}^2}-{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}\right)}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 2}}^3}+{{y_{1}^{1, \: 1}}\ {{y_{2}^{1, \: 2}}^2}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}}, \:{{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {{y_{1}^{2, \: 1}}\ {{y_{2}^{1, \: 2}}^2}}+{{\left(-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}\right)}\ {y_{2}^{1, \: 2}}}+ \ \ \displaystyle {{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}$

(15)

Type: List(Polynomial(Integer))

axiom

#AB

$\label{eq16}16$

(16)

Type: PositiveInteger?

The Jacobi identity requires the following tensor to be zero:

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

axiom

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

$\label{eq17}\begin{array}{@{}l} \displaystyle \left[{-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}^2}}, \: \right. \ \ \displaystyle \left.{{{\left(-{y_{1}^{2, \: 1}}-{y_{1}^{1, \: 2}}\right)}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}-{2 \ {y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}^2}- \ \ \displaystyle {{y_{1}^{1, \: 2}}\ {y_{1}^{2, \: 1}}}$

(17)

Type: List(Fraction(Polynomial(Integer)))

axiom

BB:=groebner(ravel(BA));

Type: List(Polynomial(Integer))

axiom

#BB

$\label{eq18}26$

(18)

Type: PositiveInteger?

A scalar product is denoted by $U = \{ u^{ij} \}$

axiom

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

$\label{eq19}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{2, \: 1}}&{u^{2, \: 2}}$

(19)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

Definition 1

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

    Y I = I Y
     U     U

axiom

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

$\label{eq20}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {y_{2}^{1, \: 1}}}&{-{{u^{1, \: 2}}\ {y_{2}^{1, \: 2}}}+{{u^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{u^{1, \: 1}}\ {y_{1}^{1, \: 2}}}+{{u^{1, \: 2}}\ {y_{1}^{1, \: 1}}}} \ {-{{u^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{u^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{u^{1, \: 1}}\ {y_{1}^{2, \: 1}}}+{{u^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}&{-{{u^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{u^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{u^{1, \: 1}}\ {y_{1}^{2, \: 2}}}+{{u^{1, \: 2}}\ {y_{1}^{1, \: 2}}}}$

(20)

Type: CartesianTensor?(1,2,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) 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.

axiom

)expose MCALCFN

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

Type: Matrix(Fraction(Polynomial(Integer)))

axiom

YY := transpose matrix [concat(map(variables,ravel(Y)))::List
Symbol];

Type: Matrix(Polynomial(Integer))

axiom

K::OutputForm * YY::OutputForm = 0

$\label{eq21}\begin{array}{@{}l} \displaystyle {{\left[ \begin{array}{cccccccc} 0 & 0 & 0 & 0 &{{u^{2, \: 1}}-{u^{1, \: 2}}}& 0 & 0 & 0 \ {u^{1, \: 2}}& -{u^{1, \: 1}}& 0 & 0 &{u^{2, \: 2}}& -{u^{1, \: 2}}& 0 & 0 \ 0 &{u^{1, \: 1}}& -{u^{1, \: 1}}& 0 & 0 &{u^{2, \: 1}}& -{u^{1, \: 2}}& 0 \ 0 &{u^{1, \: 2}}& 0 & -{u^{1, \: 1}}& 0 &{u^{2, \: 2}}& 0 & -{u^{1, \: 2}} \ -{u^{2, \: 1}}& 0 &{u^{1, \: 1}}& 0 & -{u^{2, \: 2}}& 0 &{u^{2, \: 1}}& 0 \ 0 & -{u^{2, \: 1}}&{u^{1, \: 2}}& 0 & 0 & -{u^{2, \: 2}}&{u^{2, \: 2}}& 0 \ 0 & 0 & -{u^{2, \: 1}}&{u^{1, \: 1}}& 0 & 0 & -{u^{2, \: 2}}&{u^{2, \: 1}} \ 0 & 0 & 0 &{-{u^{2, \: 1}}+{u^{1, \: 2}}}& 0 & 0 & 0 & 0$

(21)

Type: Equation(OutputForm?)

The matrix K transforms the coefficients of the tensor Y into coefficients of the tensor UA. We are looking for coefficients of the tensor U such that K transforms Y into UA=0 for any Y.

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.

axiom

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

$\label{eq22}\begin{array}{@{}l} \displaystyle {{{u^{1, \: 1}}^2}\ {{u^{1, \: 2}}^4}\ {{u^{2, \: 2}}^2}}-{4 \ {{u^{1, \: 1}}^2}\ {{u^{1, \: 2}}^3}\ {u^{2, \: 1}}\ {{u^{2, \: 2}}^2}}+ \ \ \displaystyle {6 \ {{u^{1, \: 1}}^2}\ {{u^{1, \: 2}}^2}\ {{u^{2, \: 1}}^2}\ {{u^{2, \: 2}}^2}}-{4 \ {{u^{1, \: 1}}^2}\ {u^{1, \: 2}}\ {{u^{2, \: 1}}^3}\ {{u^{2, \: 2}}^2}}+ \ \ \displaystyle {{{u^{1, \: 1}}^2}\ {{u^{2, \: 1}}^4}\ {{u^{2, \: 2}}^2}}-{2 \ {u^{1, \: 1}}\ {{u^{1, \: 2}}^5}\ {u^{2, \: 1}}\ {u^{2, \: 2}}}+ \ \ \displaystyle {8 \ {u^{1, \: 1}}\ {{u^{1, \: 2}}^4}\ {{u^{2, \: 1}}^2}\ {u^{2, \: 2}}}-{{12}\ {u^{1, \: 1}}\ {{u^{1, \: 2}}^3}\ {{u^{2, \: 1}}^3}\ {u^{2, \: 2}}}+ \ \ \displaystyle {8 \ {u^{1, \: 1}}\ {{u^{1, \: 2}}^2}\ {{u^{2, \: 1}}^4}\ {u^{2, \: 2}}}-{2 \ {u^{1, \: 1}}\ {u^{1, \: 2}}\ {{u^{2, \: 1}}^5}\ {u^{2, \: 2}}}+ \ \ \displaystyle {{{u^{1, \: 2}}^6}\ {{u^{2, \: 1}}^2}}-{4 \ {{u^{1, \: 2}}^5}\ {{u^{2, \: 1}}^3}}+{6 \ {{u^{1, \: 2}}^4}\ {{u^{2, \: 1}}^4}}- \ \ \displaystyle {4 \ {{u^{1, \: 2}}^3}\ {{u^{2, \: 1}}^5}}+{{{u^{1, \: 2}}^2}\ {{u^{2, \: 1}}^6}}$

(22)

Type: DistributedMultivariatePolynomial?([*002u11,*002u12,*002u21,*002u22],Fraction(Integer))

The scalar product must also be non-degenerate

axiom

Ud:DMP(concat map(variables,ravel(U)),FRAC INT) := determinant
[[U[i,j] for j in 1..n] for i in 1..n]

$\label{eq23}{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}$

(23)

Type: DistributedMultivariatePolynomial?([*002u11,*002u12,*002u21,*002u22],Fraction(Integer))

therefore U must be symmetric.

axiom

nthFactor(Kd,1)

$\label{eq24}{u^{1, \: 2}}-{u^{2, \: 1}}$

(24)

Type: DistributedMultivariatePolynomial?([*002u11,*002u12,*002u21,*002u22],Fraction(Integer))

axiom

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

$\label{eq25}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{1, \: 2}}&{u^{2, \: 2}}$

(25)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

Pre-Frobenius Algebras

axiom

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

$\label{eq26}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 &{-{{u^{1, \: 2}}\ {y_{2}^{1, \: 2}}}+{{u^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{u^{1, \: 1}}\ {y_{1}^{1, \: 2}}}+{{u^{1, \: 2}}\ {y_{1}^{1, \: 1}}}} \ {-{{u^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{u^{1, \: 2}}\ {y_{2}^{1, \: 2}}}-{{u^{1, \: 1}}\ {y_{1}^{2, \: 1}}}+{{u^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}&{-{{u^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{u^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{u^{1, \: 1}}\ {y_{1}^{2, \: 2}}}+{{u^{1, \: 2}}\ {y_{1}^{1, \: 2}}}}$

(26)

Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

axiom

--solve(ravel(UAS),removeDuplicates concat map(variables,ravel(US)))
KS := jacobian(ravel(UAS),concat(map(variables,ravel(Y)))::List
Symbol);

Type: Matrix(Fraction(Polynomial(Integer)))

axiom

NS:=nullSpace(KS)

$\label{eq27}\begin{array}{@{}l} \displaystyle \left[{\left[{{{u^{1, \: 1}}^2}\over{{u^{1, \: 2}}^2}}, \:{{u^{1, \: 1}}\over{u^{1, \: 2}}}, \:{{u^{1, \: 1}}\over{u^{1, \: 2}}}, \: 1, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ -{{u^{2, \: 2}}\over{u^{1, \: 2}}}, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[{{-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^2}}\over{{u^{1, \: 2}}^2}}, \: -{{u^{2, \: 2}}\over{u^{1, \: 2}}}, \: -{{u^{2, \: 2}}\over{u^{1, \: 2}}}, \: 0, \: 0, \: 1, \: 1, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[{{u^{1, \: 1}}\over{u^{1, \: 2}}}, \: 1, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1 \right]}\right]$

(27)

Type: List(Vector(Fraction(Polynomial(Integer))))

axiom

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

$\label{eq28}\begin{array}{@{}l} \displaystyle \left[{{\left( \begin{array}{@{}l} \displaystyle {{u^{1, \: 1}}\ {u^{1, \: 2}}\ {p_{4}}}+{{\left(-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^2}\right)}\ {p_{3}}}- \ \ \displaystyle {{u^{1, \: 2}}\ {u^{2, \: 2}}\ {p_{2}}}+{{{u^{1, \: 1}}^2}\ {p_{1}}}$

(28)

Type: List(Fraction(Polynomial(Integer)))