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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
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SandBoxFrobeniusAlgebra

last edited 13 years ago by Bill Page

Edit detail for SandBoxFrobeniusAlgebra revision 17 of 26

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Editor: Bill Page Time: 2011/02/15 21:20:12 GMT-8
Note: clean up notation

removed:
-)library DEXPR

removed:
---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))

changed:
-(the commutator) is zero
(the commutator) is zero::

   Y  -  X
         Y


changed:
-(the commutator) is zero
(the anti-commutator) is zero::

   Y  +  X
         Y


changed:
-  Y I  =  I Y
  Y    =    Y

added:

  Y    -    Y  -   X
   Y       Y       Y
                  Y

changed:
-    Y I = I Y
    Y   =   Y

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

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

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:

   Y  -  X
         Y

axiom

KK:=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

KB:=groebner(ravel(KK))

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

(9)

Type: List(Polynomial(Integer))

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

   Y  +  X
         Y

axiom

AK:=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:

axiom

KA:=groebner(ravel(AK))

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

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

axiom

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

$\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 tensor to be zero:

  Y    -    Y  -   X
   Y       Y       Y
                  Y

  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)

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

(13)

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

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

axiom

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 following tensor equation holds:

    Y   =   Y
     U     U

axiom

UA := 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.

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

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

(16)

Type: Equation(OutputForm?)

The matrix K transforms the coefficients of the tensor 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)

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

(17)

Type: DistributedMultivariatePolynomial?([*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]

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

(18)

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

therefore U must be symmetric.

axiom

nthFactor(Kd,1)

$\label{eq19}{u^{1, \: 2}}-{u^{2, \: 1}}$

(19)

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

axiom

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

$\label{eq20}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{1, \: 2}}&{u^{2, \: 2}}$

(20)

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

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

axiom

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

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

(21)

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)

$\label{eq22}\begin{array}{@{}l} \displaystyle \left[{\left[{{{u^{1, \: 1}}^2}\over{{u^{1, \: 2}}^2}}, \:{{u^{1, \: 1}}\over{u^{1, \: 2}}}, \:{{u^{1, \: 1}}\over{u^{1, \: 2}}}, \: 1, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ -{{u^{2, \: 2}}\over{u^{1, \: 2}}}, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[{{-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^2}}\over{{u^{1, \: 2}}^2}}, \: -{{u^{2, \: 2}}\over{u^{1, \: 2}}}, \: -{{u^{2, \: 2}}\over{u^{1, \: 2}}}, \: 0, \: 0, \: 1, \: 1, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[{{u^{1, \: 1}}\over{u^{1, \: 2}}}, \: 1, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1 \right]}\right]$

(22)

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

axiom

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

$\label{eq23}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle {y_{1}^{1, \: 1}}={{\left( \begin{array}{@{}l} \displaystyle {{u^{1, \: 1}}\ {u^{1, \: 2}}\ {p_{4}}}+{{\left(-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^2}\right)}\ {p_{3}}}- \ \ \displaystyle {{u^{1, \: 2}}\ {u^{2, \: 2}}\ {p_{2}}}+{{{u^{1, \: 1}}^2}\ {p_{1}}}$

(23)

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

axiom

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

$\label{eq24}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{{{u^{1, \: 1}}\ {u^{1, \: 2}}\ {p_{4}}}+{{\left(-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^2}\right)}\ {p_{3}}}-{{u^{1, \: 2}}\ {u^{2, \: 2}}\ {p_{2}}}+{{{u^{1, \: 1}}^2}\ {p_{1}}}}\over{{u^{1, \: 2}}^2}}&{{{{u^{1, \: 2}}\ {p_{4}}}-{{u^{2, \: 2}}\ {p_{3}}}+{{u^{1, \: 1}}\ {p_{1}}}}\over{u^{1, \: 2}}} \ {{{{u^{1, \: 2}}\ {p_{4}}}-{{u^{2, \: 2}}\ {p_{3}}}+{{u^{1, \: 1}}\ {p_{1}}}}\over{u^{1, \: 2}}}&{p_{1}}$

(24)

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.

axiom

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

$\label{eq25}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 & 0 \ 0 & 0$

(25)

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

axiom

J := jacobian(ravel(UA),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

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

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Type: Equation(OutputForm?)

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

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) ];

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

axiom