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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
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SandBoxFrobeniusAlgebra

last edited 15 years ago by Bill Page

Edit detail for SandBoxFrobeniusAlgebra revision 19 of 26

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Editor: Bill Page Time: 2011/02/15 23:31:45 GMT-8
Note: clean up notation

changed:
-YYX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(XYY)
YYX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(YYX)

An n-dimensional algebra is represented by a (2,1)-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

fricas

(1) -> n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

fricas

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

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

(2)

Type: Type

fricas

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 $P=\{ p_i \}$ and $Q=\{ q_j \}$

fricas

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

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

(4)

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

fricas

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

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

(5)

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

the tensor operates on their tensor product to yield a vector $R=\{ r_k = {y_k}^{ji} p_i q_j \}$

fricas

R:=contract(contract(Y,3,product(P,Q),1),2,3)

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{{{p_{2}}\ {q_{2}}\ {y_{1}^{2, \: 2}}}+{{p_{1}}\ {q_{2}}\ {y_{1}^{2, \: 1}}}+{{p_{2}}\ {q_{1}}\ {y_{1}^{1, \: 2}}}+{{p_{1}}\ {q_{1}}\ {y_{1}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{p_{2}}\ {q_{2}}\ {y_{2}^{2, \: 2}}}+{{p_{1}}\ {q_{2}}\ {y_{2}^{2, \: 1}}}+{{p_{2}}\ {q_{1}}\ {y_{2}^{1, \: 2}}}+{{p_{1}}\ {q_{1}}\ {y_{2}^{1, \: 1}}}}\right]$

(6)

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

Pictorially:

  Q   P
  2j  3i
   \ /
    |
    1k
    R

In Axiom we may use the more convenient tensor inner product denoted by * that combines tensor product with a contraction on the last index of the first tensor and the first index of the second tensor.

fricas

R:=(Y*P)*Q

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{{{p_{2}}\ {q_{2}}\ {y_{1}^{2, \: 2}}}+{{p_{1}}\ {q_{2}}\ {y_{1}^{2, \: 1}}}+{{p_{2}}\ {q_{1}}\ {y_{1}^{1, \: 2}}}+{{p_{1}}\ {q_{1}}\ {y_{1}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{p_{2}}\ {q_{2}}\ {y_{2}^{2, \: 2}}}+{{p_{1}}\ {q_{2}}\ {y_{2}^{2, \: 1}}}+{{p_{2}}\ {q_{1}}\ {y_{2}^{1, \: 2}}}+{{p_{1}}\ {q_{1}}\ {y_{2}^{1, \: 1}}}}\right]$

(7)

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

An algebra is commutative if the tensor $\Pi = \{ {\pi_k}^{ji} = {y_k}^{ji}-{y^k}^{ij} \}$ (the commutator) is zero:

   Y  -  X
         Y

fricas

XY:=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:

fricas

groebner(ravel(XY))

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

(9)

Type: List(Polynomial(Integer))

The algebra Y is anti-commutative if the tensor $\Xi = \{ {\xi_k}^{ji} = {y_k}^{ji}+{y^k}^{ij} \}$ (the anti-commutator) is zero:

   Y  +  X
         Y

fricas

XX:=Y+reindex(Y,[1,3,2])

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {2 \ {y_{1}^{1, \: 1}}}&{{y_{1}^{2, \: 1}}+{y_{1}^{1, \: 2}}} \ {{y_{1}^{2, \: 1}}+{y_{1}^{1, \: 2}}}&{2 \ {y_{1}^{2, \: 2}}}$

(10)

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

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

fricas

groebner(ravel(XX))

$\label{eq11}\left[{y_{2}^{2, \: 2}}, \:{{y_{2}^{2, \: 1}}+{y_{2}^{1, \: 2}}}, \:{y_{2}^{1, \: 1}}, \:{y_{1}^{2, \: 2}}, \:{{y_{1}^{2, \: 1}}+{y_{1}^{1, \: 2}}}, \:{y_{1}^{1, \: 1}}\right]$

(11)

Type: List(Polynomial(Integer))

An algebra is associative if:

  Y    =    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 $\Psi = \{ {\psi_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} \}$ is zero.

fricas

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

$\label{eq12}\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]$

(12)

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

The Jacobi identity requires the following (3,1)-tensor $\Phi = \{ {\phi_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} - {y_s}^{ri} {y_r}^{jk} \}$ to be zero:

  Y    -    Y  -   X
   Y       Y       Y
                  Y

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

fricas

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

$\label{eq13}\begin{array}{@{}l} \displaystyle \left[{-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}^{2}}}, \: \right. \ \ \displaystyle \left.{{{\left(-{y_{1}^{2, \: 1}}-{y_{1}^{1, \: 2}}\right)}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}-{2 \ {y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}^{2}}- \ \ \displaystyle {{y_{1}^{1, \: 2}}\ {y_{1}^{2, \: 1}}}$

(13)

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

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

fricas

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

$\label{eq14}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{2, \: 1}}&{u^{2, \: 2}}$

(14)

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 following tensor is zero $\Omega = \{ \omega^{kji} = {Y_r}^{kj} U^{ri} - U^{kr} {Y_r}^{ji} \}$

fricas

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

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

(15)

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.

fricas

)expose MCALCFN

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

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

fricas

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

Type: Matrix(Polynomial(Integer))

fricas

K::OutputForm * yy::OutputForm = 0

$\label{eq16}\begin{array}{@{}l} \displaystyle {{\left[ \begin{array}{cccccccc} 0 & 0 & 0 & 0 &{{u^{2, \: 1}}-{u^{1, \: 2}}}& 0 & 0 & 0 \ {u^{1, \: 2}}& -{u^{1, \: 1}}& 0 & 0 &{u^{2, \: 2}}& -{u^{1, \: 2}}& 0 & 0 \ 0 &{u^{1, \: 1}}& -{u^{1, \: 1}}& 0 & 0 &{u^{2, \: 1}}& -{u^{1, \: 2}}& 0 \ 0 &{u^{1, \: 2}}& 0 & -{u^{1, \: 1}}& 0 &{u^{2, \: 2}}& 0 & -{u^{1, \: 2}} \ -{u^{2, \: 1}}& 0 &{u^{1, \: 1}}& 0 & -{u^{2, \: 2}}& 0 &{u^{2, \: 1}}& 0 \ 0 & -{u^{2, \: 1}}&{u^{1, \: 2}}& 0 & 0 & -{u^{2, \: 2}}&{u^{2, \: 2}}& 0 \ 0 & 0 & -{u^{2, \: 1}}&{u^{1, \: 1}}& 0 & 0 & -{u^{2, \: 2}}&{u^{2, \: 1}} \ 0 & 0 & 0 &{-{u^{2, \: 1}}+{u^{1, \: 2}}}& 0 & 0 & 0 & 0$

(16)

Type: Equation(OutputForm?)

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

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.

fricas

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

$\label{eq17}\begin{array}{@{}l} \displaystyle {{{u^{1, \: 1}}^{2}}\ {{u^{1, \: 2}}^{4}}\ {{u^{2, \: 2}}^{2}}}-{4 \ {{u^{1, \: 1}}^{2}}\ {{u^{1, \: 2}}^{3}}\ {u^{2, \: 1}}\ {{u^{2, \: 2}}^{2}}}+ \ \ \displaystyle {6 \ {{u^{1, \: 1}}^{2}}\ {{u^{1, \: 2}}^{2}}\ {{u^{2, \: 1}}^{2}}\ {{u^{2, \: 2}}^{2}}}-{4 \ {{u^{1, \: 1}}^{2}}\ {u^{1, \: 2}}\ {{u^{2, \: 1}}^{3}}\ {{u^{2, \: 2}}^{2}}}+ \ \ \displaystyle {{{u^{1, \: 1}}^{2}}\ {{u^{2, \: 1}}^{4}}\ {{u^{2, \: 2}}^{2}}}-{2 \ {u^{1, \: 1}}\ {{u^{1, \: 2}}^{5}}\ {u^{2, \: 1}}\ {u^{2, \: 2}}}+ \ \ \displaystyle {8 \ {u^{1, \: 1}}\ {{u^{1, \: 2}}^{4}}\ {{u^{2, \: 1}}^{2}}\ {u^{2, \: 2}}}-{{12}\ {u^{1, \: 1}}\ {{u^{1, \: 2}}^{3}}\ {{u^{2, \: 1}}^{3}}\ {u^{2, \: 2}}}+ \ \ \displaystyle {8 \ {u^{1, \: 1}}\ {{u^{1, \: 2}}^{2}}\ {{u^{2, \: 1}}^{4}}\ {u^{2, \: 2}}}-{2 \ {u^{1, \: 1}}\ {u^{1, \: 2}}\ {{u^{2, \: 1}}^{5}}\ {u^{2, \: 2}}}+ \ \ \displaystyle {{{u^{1, \: 2}}^{6}}\ {{u^{2, \: 1}}^{2}}}-{4 \ {{u^{1, \: 2}}^{5}}\ {{u^{2, \: 1}}^{3}}}+{6 \ {{u^{1, \: 2}}^{4}}\ {{u^{2, \: 1}}^{4}}}- \ \ \displaystyle {4 \ {{u^{1, \: 2}}^{3}}\ {{u^{2, \: 1}}^{5}}}+{{{u^{1, \: 2}}^{2}}\ {{u^{2, \: 1}}^{6}}}$

(17)

Type: DistributedMultivariatePolynomial?([u[;1,1],u[;1,2],u[;2,1],u[;2,2]],Fraction(Integer))

The scalar product must also be non-degenerate

fricas

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

$\label{eq18}{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}$

(18)

Type: DistributedMultivariatePolynomial?([u[;1,1],u[;1,2],u[;2,1],u[;2,2]],Fraction(Integer))

therefore U must be symmetric.

fricas

nthFactor(Kd,1)

   There are no exposed library operations named nthFactor but there 
      are 3 unexposed operations with that name. Use HyperDoc Browse or
      issue
                            )display op nthFactor
      to learn more about the available operations.

   Cannot find a definition or applicable library operation named 
      nthFactor with argument type(s) 
DistributedMultivariatePolynomial([u[;1,1],u[;1,2],u[;2,1],u[;2,2]],Fraction(Integer))
                               PositiveInteger

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

Theorem 2

All 2-dimensional algebras with associative scalar product are symmetric.

Proof: The basis of the null space of the symmetric K matrix are all symmetric

fricas

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

$\label{eq19}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 &{-{{u^{1, \: 2}}\ {y_{2}^{1, \: 2}}}+{{u^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{u^{1, \: 1}}\ {y_{1}^{1, \: 2}}}+{{u^{1, \: 2}}\ {y_{1}^{1, \: 1}}}} \ {-{{u^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{u^{1, \: 2}}\ {y_{2}^{1, \: 2}}}-{{u^{1, \: 1}}\ {y_{1}^{2, \: 1}}}+{{u^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}&{-{{u^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{u^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{u^{1, \: 1}}\ {y_{1}^{2, \: 2}}}+{{u^{1, \: 2}}\ {y_{1}^{1, \: 2}}}}$

(19)

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

fricas

KS := jacobian(ravel(YUS),concat(map(variables,ravel(Y)))::List Symbol);

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

fricas

NS:=nullSpace(KS)

$\label{eq20}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\frac{{u^{1, \: 1}}^{2}}{{u^{1, \: 2}}^{2}}}, \:{\frac{u^{1, \: 1}}{u^{1, \: 2}}}, \:{\frac{u^{1, \: 1}}{u^{1, \: 2}}}, \: 1, \: 0, \: 0, \: 0, \: 0 \right]$

(20)

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

fricas

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

$\label{eq21}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle {y_{1}^{1, \: 1}}= \ \ \displaystyle {\frac{{{u^{1, \: 1}}\ {u^{1, \: 2}}\ {p_{4}}}+{{\left(-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^{2}}\right)}\ {p_{3}}}-{{u^{1, \: 2}}\ {u^{2, \: 2}}\ {p_{2}}}+{{{u^{1, \: 1}}^{2}}\ {p_{1}}}}{{u^{1, \: 2}}^{2}}}$

(21)

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

fricas

YS:T := unravel(map(x+->subst(x,SS),ravel Y))

$\label{eq22}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {\frac{{{u^{1, \: 1}}\ {u^{1, \: 2}}\ {p_{4}}}+{{\left(-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^{2}}\right)}\ {p_{3}}}-{{u^{1, \: 2}}\ {u^{2, \: 2}}\ {p_{2}}}+{{{u^{1, \: 1}}^{2}}\ {p_{1}}}}{{u^{1, \: 2}}^{2}}}&{\frac{{{u^{1, \: 2}}\ {p_{4}}}-{{u^{2, \: 2}}\ {p_{3}}}+{{u^{1, \: 1}}\ {p_{1}}}}{u^{1, \: 2}}} \ {\frac{{{u^{1, \: 2}}\ {p_{4}}}-{{u^{2, \: 2}}\ {p_{3}}}+{{u^{1, \: 1}}\ {p_{1}}}}{u^{1, \: 2}}}&{p_{1}}$

(22)

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

This is a 4-parameter family of 2-d pre-Frobenius algebras with a given admissible (i.e. symmetric) scalar product.

fricas

UASS:T := unravel(map(x+->subst(x,SS),ravel YUS))

$\label{eq23}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 & 0 \ 0 & 0$

(23)

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

fricas

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

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

fricas

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

Type: Matrix(Polynomial(Integer))

fricas

J::OutputForm * uu::OutputForm = 0

$\label{eq24}\begin{array}{@{}l} \displaystyle {{\left[ \begin{array}{cccc} 0 & -{y_{2}^{1, \: 1}}&{y_{2}^{1, \: 1}}& 0 \ -{y_{1}^{1, \: 2}}&{-{y_{2}^{1, \: 2}}+{y_{1}^{1, \: 1}}}& 0 &{y_{2}^{1, \: 1}} \ {-{y_{1}^{2, \: 1}}+{y_{1}^{1, \: 2}}}& -{y_{2}^{2, \: 1}}&{y_{2}^{1, \: 2}}& 0 \ -{y_{1}^{2, \: 2}}&{-{y_{2}^{2, \: 2}}+{y_{1}^{1, \: 2}}}& 0 &{y_{2}^{1, \: 2}} \ {y_{1}^{2, \: 1}}& 0 &{{y_{2}^{2, \: 1}}-{y_{1}^{1, \: 1}}}& -{y_{2}^{1, \: 1}} \ 0 &{y_{1}^{2, \: 1}}& -{y_{1}^{1, \: 2}}&{{y_{2}^{2, \: 1}}-{y_{2}^{1, \: 2}}} \ {y_{1}^{2, \: 2}}& 0 &{{y_{2}^{2, \: 2}}-{y_{1}^{2, \: 1}}}& -{y_{2}^{2, \: 1}} \ 0 &{y_{1}^{2, \: 2}}& -{y_{1}^{2, \: 2}}& 0$

(24)

Type: Equation(OutputForm?)

The matrix J transforms the coefficients of the tensor into coefficients of the tensor $\Omega$ . We are looking for coefficients of the tensor such that J transforms the tensor into $\Omega=0$ 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.

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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) ];

Type: PolynomialIdeal?(Fraction(Integer),IndexedExponents?(Symbol),Symbol,Polynomial(Fraction(Integer)))

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#generators(%)

$\label{eq25}51$

(25)

Type: PositiveInteger?

Theorem 3

A 2-d algebra is pre-Frobenius if it is associative, commutative, anti-commutative or if it satisfies the Jacobi identity.

Proof

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in?(JP,ideal ravel YY)  -- associative

$\label{eq26} \mbox{\rm true}$

(26)

Type: Boolean

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in?(JP,ideal ravel XY)  -- commutative

$\label{eq27} \mbox{\rm true}$

(27)

Type: Boolean

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in?(JP,ideal ravel XX)  -- anti-commutative

$\label{eq28} \mbox{\rm true}$

(28)

Type: Boolean

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in?(JP,ideal ravel YYX) -- Jacobi identity

$\label{eq29} \mbox{\rm true}$

(29)

Type: Boolean