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# Edit detail for SandBoxFrobeniusAlgebra revision 21 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: 2011/02/16 17:21:57 GMT-8 Note: Y-forms: snails

added:

Y-forms

Left snail and right snail::

LS                    RS

Y A                   A Y
Y )                 ( Y
U                   U

i  j                          j  i
\/     0                0     \/
\    / \              / \    /
e  f   \            /   f  e
\/     \          /     \/
\     /          \     /
\   /            \   /
\ /              \ /
0                0

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

\begin{axiom}
LS=contract(Y*Y,1,2)
RS=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,2),[2,1])
test(LS=RS)
\end{axiom}


References

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

  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
 (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

 (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)
 (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

 (10)
(commutator) is zero.
axiom
YC:=Y-reindex(Y,[1,3,2])
 (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))
 (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

 (13)
(anti-commutator) is zero.
axiom
YA:=Y+reindex(Y,[1,3,2])
 (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))
 (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

 (16)
is zero.

axiom
YX := YY - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(YX)
 (17)
Type: List(Fraction(Polynomial(Integer)))

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]
)
 (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


 (19)
(three-point function) is zero.

axiom
YU := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y
 (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
 (21)
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

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))
 (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]
 (23)
Type: DistributedMultivariatePolynomial?([*002u11,*002u12,*002u21,*002u22],Fraction(Integer))

therefore U must be symmetric.

axiom
nthFactor(Kd,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))
 (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
 (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)
 (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]))
 (28)
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
YS:T := unravel(map(x+->subst(x,SS),ravel Y))
 (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)
 (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
 (31)
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(%)
 (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

axiom
in?(JP,ideal ravel YY)  -- associative
 (33)
Type: Boolean
axiom
in?(JP,ideal ravel YC)  -- commutative
 (34)
Type: Boolean
axiom
in?(JP,ideal ravel YA)  -- anti-commutative
 (35)
Type: Boolean
axiom
in?(JP,ideal ravel YX) -- Jacobi identity
 (36)
Type: Boolean

Y-forms

Left snail and right snail:

  LS                    RS

Y A                   A Y
Y )                 ( Y
U                   U

i  j                          j  i
\/     0                0     \/
\    / \              / \    /
e  f   \            /   f  e
\/     \          /     \/
\     /          \     /
\   /            \   /
\ /              \ /
0                0


 (37)

axiom
LS=contract(Y*Y,1,2)
 (38)
Type: Equation(CartesianTensor?(1,2,Fraction(Polynomial(Integer))))
axiom
RS=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,2),[2,1])
 (39)
Type: Equation(CartesianTensor?(1,2,Fraction(Polynomial(Integer))))
axiom
test(LS=RS)
 (40)
Type: Boolean