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Frobenius Algebra, Vector Spaces and Polynomial Ideals

Edit detail for Frobenius Algebra, Vector Spaces and Polynomial Ideals revision 2 of 4

	1 2 3 4
Editor: Bill Page Time: 2011/05/13 11:28:33 GMT-7
Note: Use new LinearOperator domain

added:

From BillPage Fri May 13 11:28:33 -0700 2011
From: Bill Page
Date: Fri, 13 May 2011 11:28:33 -0700
Subject: Use new LinearOperator domain
Message-ID: <20110513112833-0700@axiom-wiki.newsynthesis.org>

SandBoxFrobeniusAlgebra updated

References

Frobenius algebras and 2D topological quantum field theories by Joachim Kock
Especially "Tensor calculus (linear algebra in coordinates)" in section 2.3.31, page 123.

See also:

http://ncatlab.org/nlab/show/Frobenius+algebra by John Baez, et al.
Ideals, Varieties, and Algorithms by David A. Cox, et al.

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

axiom

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

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

(3)

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

Given two vectors $P=\{ p^i \}$ and $Q=\{ q^j \}$

axiom

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)))

axiom

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 $R=\{ r_k = {y^k}_{ij} p^i q^j \}$

axiom

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

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

(6)

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

Pictorially:

  P Q
   Y
   R

  or more explicitly

  Pi Qj
   \/
    \
     Rk

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.

axiom

R:=(Y*P)*Q

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

(7)

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

An algebra is said to be associative if:

  Y    =    Y
   Y       Y

Note: the right hand side of the equation above is implicitly the mirror image of the left hand side:

  i   j   k   i  j     k   i     j  k
   \  |  /     \/     /     \     \/
    \ | /       \    /       \    /
     \|/    =    e  k    -    i  e
      |           \/           \/
      |            \           /
      l             l         l

This requires that the following (3,1)-tensor

$\label{eq8} \Psi = \{ {\psi_l}^{ijk} = {y^e}_{ij} {y^l}_{ek} - {y^l}_{ie} {y^e}_{jk} \}$

(8)

(associator) is zero.

axiom

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

$\label{eq9}\begin{array}{@{}l} \displaystyle \left[{-{{y_{1, \: 1}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{1}}}}, \:{{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}-{{y_{2, \: 1}^{1}}\ {y_{2, \: 1}^{2}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1, \: 2}^{1}}\ {y_{2, \: 1}^{2}}}+{{\left(-{y_{1, \: 2}^{2}}+{y_{1, \: 1}^{1}}\right)}\ {y_{2, \: 1}^{1}}}-{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{2, \: 1}^{1}}\ {y_{2, \: 2}^{2}}}+{{\left({y_{2, \: 1}^{2}}-{y_{1, \: 1}^{1}}\right)}\ {y_{2, \: 2}^{1}}}+{{y_{2, \: 1}^{1}}^2}}, \: \right. \ \ \displaystyle \left.{-{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{1, \: 2}^{2}}}}, \:{{\left(-{y_{2, \: 1}^{2}}+{y_{1, \: 2}^{2}}\right)}\ {y_{2, \: 2}^{1}}}, \: \right. \ \ \displaystyle \left.{{{y_{1, \: 2}^{1}}\ {y_{2, \: 2}^{2}}}+{{\left(-{y_{1, \: 2}^{2}}+{y_{1, \: 1}^{1}}\right)}\ {y_{2, \: 2}^{1}}}-{{y_{1, \: 2}^{1}}^2}}, \: \right. \ \ \displaystyle \left.{{\left({y_{2, \: 1}^{1}}-{y_{1, \: 2}^{1}}\right)}\ {y_{2, \: 2}^{1}}}, \:{-{{y_{1, \: 1}^{2}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1, \: 1}^{2}}\ {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, \: 1}^{2}}\ {y_{2, \: 1}^{1}}}-{{y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{1}}}}, \:{-{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{2, \: 1}^{1}}\ {y_{2, \: 1}^{2}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 2}^{2}}^2}-{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{\left(-{y_{2, \: 1}^{2}}+{y_{1, \: 2}^{2}}\right)}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 2}^{1}}\ {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_{1, \: 2}^{2}}}}, \:{{\left({y_{2, \: 1}^{2}}-{y_{1, \: 2}^{2}}\right)}\ {y_{2, \: 2}^{1}}}\right]$

(9)

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

The algebra is commutative if:

  Y = Y

  i   j     i  j     j  i
   \ /   =   \/   -   \/
    |         \       /
    k          k     k

This requires that the following (2,1)-tensor

$\label{eq10} \mathcal{C} = \{ {c^k}_{ij} = {y^k}_{ij} - {y^k}_{ji} \}$

(10)

(commutator) is zero.

axiom

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

$\label{eq11}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 &{{y_{2, \: 1}^{1}}-{y_{1, \: 2}^{1}}} \ {-{y_{2, \: 1}^{1}}+{y_{1, \: 2}^{1}}}& 0$

(11)

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

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

axiom

groebner(ravel(YC))

$\label{eq12}\left[{{y_{2, \: 1}^{2}}-{y_{1, \: 2}^{2}}}, \:{{y_{2, \: 1}^{1}}-{y_{1, \: 2}^{1}}}\right]$

(12)

Type: List(Polynomial(Integer))

The algebra is anti-commutative if:

  Y = -Y

  i   j     i  j     j  i
   \ /   =   \/   =   \/
    |         \       /
    k          k     k

This requires that the following (2,1)-tensor

$\label{eq13} \mathcal{A} = \{ {a^k}_{ij} = {y^k}_{ij} + {y^k}_{ji} \}$

(13)

(anti-commutator) is zero.

axiom

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

$\label{eq14}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {2 \ {y_{1, \: 1}^{1}}}&{{y_{2, \: 1}^{1}}+{y_{1, \: 2}^{1}}} \ {{y_{2, \: 1}^{1}}+{y_{1, \: 2}^{1}}}&{2 \ {y_{2, \: 2}^{1}}}$

(14)

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

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

axiom

groebner(ravel(YA))

$\label{eq15}\left[{y_{2, \: 2}^{2}}, \:{y_{2, \: 2}^{1}}, \:{{y_{2, \: 1}^{2}}+{y_{1, \: 2}^{2}}}, \:{{y_{2, \: 1}^{1}}+{y_{1, \: 2}^{1}}}, \:{y_{1, \: 1}^{2}}, \:{y_{1, \: 1}^{1}}\right]$

(15)

Type: List(Polynomial(Integer))

The Jacobi identity is:

            X
  Y =  Y + Y
   Y  Y     Y

  i     j     k  i      j     k  i     j      k   i  j   k
   \    |    /    \    /     /    \     \    /     \  \ /
    \   |   /      \  /     /      \     \  /       \  0
     \  |  /        \/     /        \     \/         \/ \
      \ | /          \    /          \    /           \  \
       \|/     =      e  k      -     i  e       -     e  j
        |              \/              \/               \/
        |               \              /                /
        l                l            l                 l

An algebra satisfies the Jacobi identity if and only if the following (3,1)-tensor

$\label{eq16} \Theta = \{ {\theta^l}_{ijk} = {y^l}_{ek} {y^e}_{ij} - {y^l}_{ie} {y^e}_{jk} - {y^l}_{ej} {y^e}_{ik} \}$

(16)

is zero.

axiom

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

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

(17)

Type: List(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..n]
      for j in 1..n]
        )

$\label{eq18}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{2, \: 1}}&{u^{2, \: 2}}$

(18)

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

$\label{eq19} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(19)

(three-point function) is zero.

axiom

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

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

(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 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);

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 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.

Theorem 1

All 2-dimensional pre-Frobenius algebras are symmetric.

Proof: Consider the determinant of the matrix K above.

axiom

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

$\label{eq22}{{\left({u^{1, \: 2}}-{u^{2, \: 1}}\right)}^4}\ {{\left({{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}\right)}^2}$

(22)

Type: Factored(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)))

Theorem 2

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

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

axiom

YUS:T :=  reindex(reindex(US,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-US*Y

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

(26)

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

axiom

KS := jacobian(ravel(YUS),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

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

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

(28)

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

axiom

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

$\label{eq29}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{{{u^{1, \: 1}}\ {u^{1, \: 2}}\ {p_{4}}}+{{\left(-{{u^{1, \: 1}}\ {u^{2, \: 2}}}+{{u^{1, \: 2}}^2}\right)}\ {p_{3}}}-{{u^{1, \: 2}}\ {u^{2, \: 2}}\ {p_{2}}}+{{{u^{1, \: 1}}^2}\ {p_{1}}}}\over{{u^{1, \: 2}}^2}}&{{{{u^{1, \: 2}}\ {p_{4}}}-{{u^{2, \: 2}}\ {p_{3}}}+{{u^{1, \: 1}}\ {p_{1}}}}\over{u^{1, \: 2}}} \ {{{{u^{1, \: 2}}\ {p_{4}}}-{{u^{2, \: 2}}\ {p_{3}}}+{{u^{1, \: 1}}\ {p_{1}}}}\over{u^{1, \: 2}}}&{p_{1}}$

(29)

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

This defines a 4-parameter family of 2-d pre-Frobenius algebras

axiom

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

$\label{eq30} \mbox{\rm true}$

(30)

Type: Boolean

Alternatively we may consider

axiom

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

$\label{eq31}\begin{array}{@{}l} \displaystyle {{\left[ \begin{array}{cccc} 0 & -{y_{1, \: 1}^{2}}&{y_{1, \: 1}^{2}}& 0 \ -{y_{2, \: 1}^{1}}&{-{y_{2, \: 1}^{2}}+{y_{1, \: 1}^{1}}}& 0 &{y_{1, \: 1}^{2}} \ {{y_{2, \: 1}^{1}}-{y_{1, \: 2}^{1}}}& -{y_{1, \: 2}^{2}}&{y_{2, \: 1}^{2}}& 0 \ -{y_{2, \: 2}^{1}}&{-{y_{2, \: 2}^{2}}+{y_{2, \: 1}^{1}}}& 0 &{y_{2, \: 1}^{2}} \ {y_{1, \: 2}^{1}}& 0 &{{y_{1, \: 2}^{2}}-{y_{1, \: 1}^{1}}}& -{y_{1, \: 1}^{2}} \ 0 &{y_{1, \: 2}^{1}}& -{y_{2, \: 1}^{1}}&{-{y_{2, \: 1}^{2}}+{y_{1, \: 2}^{2}}} \ {y_{2, \: 2}^{1}}& 0 &{{y_{2, \: 2}^{2}}-{y_{1, \: 2}^{1}}}& -{y_{1, \: 2}^{2}} \ 0 &{y_{2, \: 2}^{1}}& -{y_{2, \: 2}^{1}}& 0$

(31)

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

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)))

axiom

#generators(%)

$\label{eq32}51$

(32)

Type: PositiveInteger?

Theorem 3

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

Proof

Consider the ideals of the associator, commutator, anti-commutator and Jacobi identity

axiom

YYI:=ideal ravel YY;

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

axiom

in?(JP,YYI)  -- associative

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean

axiom

YCI:=ideal ravel YC;

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

axiom

in?(JP,YCI)  -- commutative

$\label{eq34} \mbox{\rm true}$

(34)

Type: Boolean

axiom

YAI:=ideal ravel YA;

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

axiom

in?(JP,YAI)  -- anti-commutative

$\label{eq35} \mbox{\rm true}$

(35)

Type: Boolean

axiom

YXI:=ideal ravel YX;

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

axiom

in?(JP,YXI) -- Jacobi identity

$\label{eq36} \mbox{\rm true}$

(36)

Type: Boolean

Y-forms

Three traces of two graftings of an algebra gives six (2,0)-forms.

Left snail and right snail:

  LS                    RS

  Y /\                    /\ Y
   Y  )                  (  Y
    \/                    \/

  i  j                        j  i
   \/                          \/
    \    /\              /\    /
     e  f  \            /  f  e
      \/    \          /    \/
       \    /          \    /
        f  /            \  f
         \/              \/

$\label{eq37} LS = \{ {y^e}_{ij} {y^f}_{ef} \} \ RS = \{ {y^f}_{fe} {y^e}_{ji} \}$

(37)

axiom

LS:=contract(Y*Y,1,2)

$\label{eq38}\left[ \begin{array}{cc} {{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{2}}}+{{y_{1, \: 1}^{1}}^2}}&{{{y_{2, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{2, \: 1}^{1}}\ {y_{2, \: 1}^{2}}}+{{\left({y_{1, \: 2}^{2}}+{y_{1, \: 1}^{1}}\right)}\ {y_{2, \: 1}^{1}}}} \ {{{y_{1, \: 2}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 2}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{1, \: 2}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{1}}}}&{{{y_{2, \: 2}^{2}}^2}+{{y_{2, \: 1}^{1}}\ {y_{2, \: 2}^{2}}}+{{\left({y_{1, \: 2}^{2}}+{y_{1, \: 1}^{1}}\right)}\ {y_{2, \: 2}^{1}}}}$

(38)

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

axiom

RS:=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,2),[2,1])

$\label{eq39}\left[ \begin{array}{cc} {{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{1}}}+{{y_{1, \: 1}^{1}}^2}}&{{{y_{2, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{\left({y_{2, \: 1}^{1}}+{y_{1, \: 2}^{1}}\right)}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{2, \: 1}^{1}}}} \ {{{y_{1, \: 2}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 2}^{1}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 2}^{1}}\ {y_{1, \: 2}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{1}}}}&{{{y_{2, \: 2}^{2}}^2}+{{y_{1, \: 2}^{1}}\ {y_{2, \: 2}^{2}}}+{{\left({y_{2, \: 1}^{2}}+{y_{1, \: 1}^{1}}\right)}\ {y_{2, \: 2}^{1}}}}$

(39)

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

axiom

test(LS=RS)

$\label{eq40} \mbox{\rm false}$

(40)

Type: Boolean

Left and right deer:

   RD                 LD

   \ /\/              \/\ /
    Y /\              /\ Y
     Y  )            (  Y
      \/              \/

   i            j    i            j
    \    /\    /      \    /\    /
     \  f  \  /        \  /  f  /
      \/    \/          \/    \/
       \    /\          /\    /
        e  /  \        /  \  e
         \/    \      /    \/
          \    /      \    /
           f  /        \  f
            \/          \/

$\label{eq41} RD = \{ {y^e}_{if} {y^f}_{ej} \} \ LD = \{ {y^f}_{ie} {y^e}_{fj} \}$

(41)

Left and right deer forms are identical but different from snails.

axiom

RD:=contract(Y*Y,1,3)

$\label{eq42}\left[ \begin{array}{cc} {{{y_{1, \: 2}^{2}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{1}}}+{{y_{1, \: 1}^{1}}^2}}&{{{y_{2, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{2, \: 1}^{1}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{2, \: 1}^{1}}}} \ {{{y_{1, \: 2}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{1, \: 2}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{1}}}}&{{{y_{2, \: 2}^{2}}^2}+{{\left({y_{2, \: 1}^{2}}+{y_{1, \: 2}^{2}}\right)}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{2, \: 1}^{1}}}}$

(42)

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

axiom

LD:=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,3),[2,1])

$\label{eq43}\left[ \begin{array}{cc} {{{y_{1, \: 2}^{2}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{1}}}+{{y_{1, \: 1}^{1}}^2}}&{{{y_{2, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{2, \: 1}^{1}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{2, \: 1}^{1}}}} \ {{{y_{1, \: 2}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{1, \: 2}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{1}}}}&{{{y_{2, \: 2}^{2}}^2}+{{\left({y_{2, \: 1}^{2}}+{y_{1, \: 2}^{2}}\right)}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{2, \: 1}^{1}}}}$

(43)

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

axiom

test(LD=RD)

$\label{eq44} \mbox{\rm true}$

(44)

Type: Boolean

axiom

test(RD=RS)

$\label{eq45} \mbox{\rm false}$

(45)

Type: Boolean

axiom

test(RD=LS)

$\label{eq46} \mbox{\rm false}$

(46)

Type: Boolean

Left and right turtles:

  RT                   LT

   /\ / /               \ \ /\
  (  Y /                 \ Y  )
   \  Y                   Y  /
    \/                     \/

           i     j      i     j
    /\    /     /        \     \    /\
   /  f  /     /          \     \  f  \
  /    \/     /            \     \/    \
  \     \    /              \    /     /
   \     e  /                \  e     /
    \     \/                  \/     /
     \    /                    \    /
      \  f                      f  /
       \/                        \/

$\label{eq47} RT = \{ {y^e}_{fi} {y^f}_{ej} \} \ LT = \{ {y^f}_{ie} {y^e}_{jf} \}$

(47)

axiom

RT:=contract(Y*Y,1,4)

$\label{eq48}\left[ \begin{array}{cc} {{{y_{2, \: 1}^{2}}^2}+{2 \ {y_{1, \: 1}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 1}^{1}}^2}}&{{{y_{2, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{1}}}} \ {{{y_{2, \: 1}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{2}}\ {y_{2, \: 1}^{1}}}+{{y_{1, \: 1}^{1}}\ {y_{1, \: 2}^{1}}}}&{{{y_{2, \: 2}^{2}}^2}+{2 \ {y_{1, \: 2}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}^2}}$

(48)

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

axiom

LT:=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,4),[2,1])

$\label{eq49}\left[ \begin{array}{cc} {{{y_{1, \: 2}^{2}}^2}+{2 \ {y_{1, \: 1}^{2}}\ {y_{1, \: 2}^{1}}}+{{y_{1, \: 1}^{1}}^2}}&{{{y_{1, \: 2}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{2, \: 1}^{1}}}} \ {{{y_{1, \: 2}^{2}}\ {y_{2, \: 2}^{2}}}+{{y_{1, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{1, \: 2}^{1}}\ {y_{2, \: 1}^{2}}}+{{y_{1, \: 1}^{1}}\ {y_{2, \: 1}^{1}}}}&{{{y_{2, \: 2}^{2}}^2}+{2 \ {y_{2, \: 1}^{2}}\ {y_{2, \: 2}^{1}}}+{{y_{2, \: 1}^{1}}^2}}$