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# Edit detail for SandBoxFrobeniusAlgebra revision 24 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: 2011/02/18 09:08:03 GMT-8 Note: associativity implies turtle is snail, jacobi implies snails vanish

changed:
-viewed as an operator with two inputs $i,j$ and one
viewed as a linear operator with two inputs $i,j$ and one

changed:
-\begin{axiom}
-in?(JP,ideal ravel YY)  -- associative
-in?(JP,ideal ravel YC)  -- commutative
-in?(JP,ideal ravel YA)  -- anti-commutative
-in?(JP,ideal ravel YX) -- Jacobi identity
-\end{axiom}

Consider the ideals of the associator, commutator, anti-commutator
and Jacobi identity
\begin{axiom}
YYI:=ideal ravel YY;
in?(JP,YYI)  -- associative
YCI:=ideal ravel YC;
in?(JP,YCI)  -- commutative
YAI:=ideal ravel YA;
in?(JP,YAI)  -- anti-commutative
YXI:=ideal ravel YX;
in?(JP,YXI) -- Jacobi identity
\end{axiom}

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

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

changed:
-\end{axiom}
test(RD=RS)
test(RD=LS)
\end{axiom}

changed:
-
The turles are symmetric
\begin{axiom}
test(RT=reindex(RT,[2,1])
test(LT=reindex(LT,[2,1])
\end{axiom}
Five of the six forms are independent.
\begin{axiom}
test(RT=RS)
test(RT=LS)
test(RT=RD)
test(LT=RS)
test(LT=LS)
test(LT=RD)
\end{axiom}
Associativity implies right turtle equals right snail
and left turtle equals left snail.
\begin{axiom}
in?(ideal ravel(RT-RS),YYI)
in?(ideal ravel(LT-LS),YYI)
\end{axiom}
If the Jacobi identity holds then both snails are zero
\begin{axiom}
in?(ideal ravel(RS),YXI)
in?(ideal ravel(LS),YXI)
\end{axiom}
and right turtle and deer have opposite signs
\begin{axiom}
in?(ideal ravel(RT+RD),YXI)
\end{axiom}


References

An n-dimensional algebra is represented by a (2,1)-tensor viewed as a linear 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

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

axiom
YYI:=ideal ravel YY;
Type: PolynomialIdeals?(Fraction(Integer),IndexedExponents?(Symbol),Symbol,Polynomial(Fraction(Integer)))
axiom
in?(JP,YYI)  -- associative (33)
Type: Boolean
axiom
YCI:=ideal ravel YC;
Type: PolynomialIdeals?(Fraction(Integer),IndexedExponents?(Symbol),Symbol,Polynomial(Fraction(Integer)))
axiom
in?(JP,YCI)  -- commutative (34)
Type: Boolean
axiom
YAI:=ideal ravel YA;
Type: PolynomialIdeals?(Fraction(Integer),IndexedExponents?(Symbol),Symbol,Polynomial(Fraction(Integer)))
axiom
in?(JP,YAI)  -- anti-commutative (35)
Type: Boolean
axiom
YXI:=ideal ravel YX;
Type: PolynomialIdeals?(Fraction(Integer),IndexedExponents?(Symbol),Symbol,Polynomial(Fraction(Integer)))
axiom
in?(JP,YXI) -- Jacobi identity (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
\/              \/ (37)

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

Left and right deer:

   RD                 LD

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

i            j    i            j
\    /\    /      \    /\    /
\  f  \  /        \  /  f  /
\/    \/          \/    \/
\    /\          /\    /
e  /  \        /  \  e
\/    \      /    \/
\    /      \    /
f  /        \  f
\/          \/ (41)
Left and right deer forms are identical but different from snails.
axiom
RD:=contract(Y*Y,1,3) (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]) (43)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))
axiom
test(LD=RD) (44)
Type: Boolean
axiom
test(RD=RS) (45)
Type: Boolean
axiom
test(RD=LS) (46)
Type: Boolean

Left and right turtles:

  RT                   LT

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

i     j      i     j
/\    /     /        \     \    /\
/  f  /     /          \     \  f  \
/    \/     /            \     \/    \
\     \    /              \    /     /
\     e  /                \  e     /
\     \/                  \/     /
\    /                    \    /
\  f                      f  /
\/                        \/ (47)

axiom
RT:=contract(Y*Y,1,4) (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]) (49)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))
axiom
test(LT=RT) (50)
Type: Boolean

The turles are symmetric

axiom
test(RT=reindex(RT,[2,1])
Line   1: test(RT=reindex(RT,[2,1])
....A...................B
Error  A: Missing mate.
Error  B: syntax error at top level
Error  B: Possibly missing a )
3 error(s) parsing
test(LT=reindex(LT,[2,1])
Line   2: test(LT=reindex(LT,[2,1])
....A...................B
Error  A: Missing mate.
Error  B: syntax error at top level
Error  B: Possibly missing a )
3 error(s) parsing

Five of the six forms are independent.

axiom
test(RT=RS) (51)
Type: Boolean
axiom
test(RT=LS) (52)
Type: Boolean
axiom
test(RT=RD) (53)
Type: Boolean
axiom
test(LT=RS) (54)
Type: Boolean
axiom
test(LT=LS) (55)
Type: Boolean
axiom
test(LT=RD) (56)
Type: Boolean

Associativity implies right turtle equals right snail and left turtle equals left snail.

axiom
in?(ideal ravel(RT-RS),YYI) (57)
Type: Boolean
axiom
in?(ideal ravel(LT-LS),YYI) (58)
Type: Boolean

If the Jacobi identity holds then both snails are zero

axiom
in?(ideal ravel(RS),YXI) (59)
Type: Boolean
axiom
in?(ideal ravel(LS),YXI) (60)
Type: Boolean

and right turtle and deer have opposite signs

axiom
in?(ideal ravel(RT+RD),YXI) (61)
Type: Boolean