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# Edit detail for SandBoxFrobeniusAlgebra revision 18 of 26

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Editor: Bill Page Time: 2011/02/15 23:29:55 GMT-8 Note: clean up notation

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

changed:
-Given two vectors $U=\{ u_i \}$ and $V=\{ v_j \}$
-\begin{axiom}
-U:T := unravel([script(u,[[i]]) for i in 1..n])
-V:T := unravel([script(v,[[i]]) for i in 1..n])
-\end{axiom}
-the tensor 'Y' operates on their tensor product to
-yield a vector $W=\{ w_k = {y_k}^{ji} u_i v_j \}$
-\begin{axiom}
-W:=contract(contract(Y,3,product(U,V),1),2,3)
-\end{axiom}
-Diagram::
-
-  U   V
-  2i  3j
Given two vectors $P=\{ p_i \}$ and $Q=\{ q_j \}$
\begin{axiom}
P:T := unravel([script(p,[[i]]) for i in 1..n])
Q:T := unravel([script(q,[[i]]) for i in 1..n])
\end{axiom}
the tensor $Y$ operates on their tensor product to
yield a vector $R=\{ r_k = {y_k}^{ji} p_i q_j \}$
\begin{axiom}
R:=contract(contract(Y,3,product(P,Q),1),2,3)
\end{axiom}
Pictorially::

Q   P
2j  3i

changed:
-    W
-
-or in a more convenient notation:
-\begin{axiom}
-W:=(Y*U)*V
-\end{axiom}
-The algebra 'Y' is commutative if the following tensor
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.
\begin{axiom}
R:=(Y*P)*Q
\end{axiom}
An algebra $Y$ is commutative if the tensor
$\Pi = \{ {\pi_k}^{ji} = {y_k}^{ji}-{y^k}^{ij} \}$

changed:
-KK:=Y-reindex(Y,[1,3,2])
XY:=Y-reindex(Y,[1,3,2])

changed:
-KB:=groebner(ravel(KK))
-\end{axiom}
-The algebra 'Y' is anti-commutative if the following tensor
groebner(ravel(XY))
\end{axiom}
The algebra 'Y' is anti-commutative if the tensor
$\Xi = \{ {\xi_k}^{ji} = {y_k}^{ji}+{y^k}^{ij} \}$

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

changed:
-KA:=groebner(ravel(AK))
groebner(ravel(XX))

changed:
-$A=\{ {a_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_r}^{kj} {y_s}^{ri} \}$
$\Psi = \{ {\psi_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} \}$

changed:
-AA := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(AA)
-\end{axiom}
-
-The Jacobi identity requires the following tensor to be zero::
YY := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(YY)
\end{axiom}

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::

changed:
-BA := AA - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(BA)
-\end{axiom}
-A scalar product is denoted by $U = \{ u^{ij} \}$
YYX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(XYY)
\end{axiom}
A scalar product is denoted by the (2,0)-tensor
$U = \{ u^{ij} \}$

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

changed:
-\begin{axiom}
-UA := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y
In other words, if the following tensor is zero
$\Omega = \{ \omega^{kji} = {Y_r}^{kj} U^{ri} - U^{kr} {Y_r}^{ji} \}$

\begin{axiom}
YU := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y

changed:
-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'.
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 $\Omega$. We are looking for
coefficients of the tensor $U$ such that 'K' transforms the
tensor $Y$ into $\Omega=0$ for any $Y$.

changed:
-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);
YUS:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel YU))
KS := jacobian(ravel(YUS),concat(map(variables,ravel(Y)))::List Symbol);

changed:
-UASS:T := unravel(map(x+->subst(x,SS),ravel UAS))
-\end{axiom}
-
-\begin{axiom}
-J := jacobian(ravel(UA),concat(map(variables,ravel(U)))::List Symbol);
-UU := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
-J::OutputForm * UU::OutputForm = 0
-\end{axiom}
-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'.
UASS:T := unravel(map(x+->subst(x,SS),ravel YUS))
\end{axiom}

\begin{axiom}
J := jacobian(ravel(YU),concat(map(variables,ravel(U)))::List Symbol);
uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
J::OutputForm * uu::OutputForm = 0
\end{axiom}
The matrix 'J' transforms the coefficients of the tensor $U$
into coefficients of the tensor $\Omega$. We are looking for
coefficients of the tensor $Y$ such that 'J' transforms the
tensor $U$ into $\Omega=0$ for any $U$.

changed:
-in?(JP,ideal ravel AA)
-in?(JP,ideal ravel KK)
-in?(JP,ideal ravel AK)
-in?(JP,ideal ravel BA)
-\end{axiom}
in?(JP,ideal ravel YY)  -- associative
in?(JP,ideal ravel XY)  -- commutative
in?(JP,ideal ravel XX)  -- anti-commutative
in?(JP,ideal ravel YYX) -- Jacobi identity
\end{axiom}


An n-dimensional algebra is represented by a (2,1)-tensor viewed as an operator with two inputs i,j and one output k. For example in 2 dimensions

axiom
n:=2
 (1)
Type: PositiveInteger?
axiom
T:=CartesianTensor(1,n,FRAC POLY INT)
 (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]
)
 (3)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

Given two vectors and

axiom
P:T := unravel([script(p,[[i]]) for i in 1..n])
 (4)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))
axiom
Q:T := unravel([script(q,[[i]]) for i in 1..n])
 (5)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

the tensor operates on their tensor product to yield a vector

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

axiom
R:=(Y*P)*Q
 (7)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

An algebra is commutative if the tensor (the commutator) is zero:

   Y  -  X
Y


axiom
XY:=Y-reindex(Y,[1,3,2])
 (8)
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(XY))
 (9)
Type: List(Polynomial(Integer))

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

   Y  +  X
Y


axiom
XX:=Y+reindex(Y,[1,3,2])
 (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
groebner(ravel(XX))
 (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 is zero.

axiom
YY := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(YY)
 (12)
Type: List(Fraction(Polynomial(Integer)))

The Jacobi identity requires the following (3,1)-tensor to be zero:

  Y    -    Y  -   X
Y       Y       Y
Y

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


axiom
YYX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(XYY)
There are 1 exposed and 0 unexposed library operations named ravel
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op ravel
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel
with argument type(s)
Variable(XYY)
Perhaps you should use "@" to indicate the required return type,
or "\$" to specify which version of the function you need.

A scalar product is denoted by the (2,0)-tensor

axiom
U:T := unravel(concat
[[script(u,[[],[j,i]])
for i in 1..n]
for j in 1..n]
)
 (13)
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

axiom
YU := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y
 (14)
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(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
 (15)
Type: Equation(OutputForm?)

The matrix K transforms the coefficients of the tensor into coefficients of the tensor . We are looking for coefficients of the tensor such that K transforms the tensor into 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.

axiom
Kd:DMP(concat map(variables,ravel(U)),FRAC INT) := factor determinant(K)
 (16)
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]
 (17)
Type: DistributedMultivariatePolynomial?([*002u11,*002u12,*002u21,*002u22],Fraction(Integer))

therefore U must be symmetric.

axiom
nthFactor(Kd,1)
 (18)
Type: DistributedMultivariatePolynomial?([*002u11,*002u12,*002u21,*002u22],Fraction(Integer))
axiom
US:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel U))
 (19)
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
YUS:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel YU))
 (20)
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)
 (21)
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]))
 (22)
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
YS:T := unravel(map(x+->subst(x,SS),ravel Y))
 (23)
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 YUS))
 (24)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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
 (25)
Type: Equation(OutputForm?)

The matrix J transforms the coefficients of the tensor into coefficients of the tensor . We are looking for coefficients of the tensor such that J transforms the tensor into 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: PolynomialIdeals?(Fraction(Integer),IndexedExponents?(Symbol),Symbol,Polynomial(Fraction(Integer)))
axiom
#generators(%)
 (26)
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

axiom
in?(JP,ideal ravel YY)  -- associative
 (27)
Type: Boolean
axiom
in?(JP,ideal ravel XY)  -- commutative
 (28)
Type: Boolean
axiom
in?(JP,ideal ravel XX)  -- anti-commutative
 (29)
Type: Boolean
axiom
in?(JP,ideal ravel YYX) -- Jacobi identity
 (30)
Type: Boolean