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# Edit detail for SandBoxFrobeniusAlgebra revision 15 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: 2011/02/14 21:51:28 GMT-8 Note: 4-parameter family of 2-d pre-Frobenius algebras

changed:
-Pre-Frobenius Algebras
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

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

This is a 4-parameter family of 2-d pre-Frobenius algebras with
a given admissible (i.e. symmetric) scalar product.
\begin{axiom}
UASS:T := unravel(map(x+->subst(x,SS),ravel UAS))
\end{axiom}


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

axiom
)library DEXPR
DistributedExpression is now explicitly exposed in frame initial
DistributedExpression will be automatically loaded when needed from
/var/zope2/var/LatexWiki/DEXPR.NRLIB/DEXPR
n:=2
 (1)
Type: PositiveInteger?
axiom
T:=CartesianTensor(1,n,FRAC POLY INT)
 (2)
Type: Domain
axiom
--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))
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
U:T := unravel([script(u,[[i]]) for i in 1..n])
 (4)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))
axiom
V:T := unravel([script(v,[[i]]) for i in 1..n])
 (5)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

the tensor Y operates on their tensor product to yield a vector

axiom
W:=contract(contract(Y,3,product(U,V),1),2,3)
 (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
 (7)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

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

axiom
K:=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
C:=groebner(ravel(K))
 (9)
Type: List(Polynomial(Integer))

An algebra is associative if:

  Y I  =  I 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
test(Y*Y = contract(product(Y,Y),3,4))
 (10)
Type: Boolean
axiom
test(Y*Y = contract(Y,3,Y,1))
 (11)
Type: Boolean
axiom
test(reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2]) = reindex(contract(product(Y,Y),1,5),[3,1,2,4]))
 (12)
Type: Boolean
axiom
test(reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2]) = reindex(contract(Y,1,Y,2),[3,1,2,4]))
 (13)
Type: Boolean
axiom
AA := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(AA)
 (14)
Type: List(Fraction(Polynomial(Integer)))
axiom
AB:=groebner(ravel(AA))
 (15)
Type: List(Polynomial(Integer))
axiom
#AB
 (16)
Type: PositiveInteger?

The Jacobi identity requires the following tensor to be zero:

  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)
 (17)
Type: List(Fraction(Polynomial(Integer)))
axiom
BB:=groebner(ravel(BA));
Type: List(Polynomial(Integer))
axiom
#BB
 (18)
Type: PositiveInteger?

A scalar product is denoted by

axiom
U:T := unravel(concat
[[script(u,[[],[j,i]])
for i in 1..n]
for j in 1..n]
)
 (19)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))

## Definition 1

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

    Y I = I Y
U     U


axiom
UA := 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 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
 (21)
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)
 (22)
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]
 (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 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))
 (26)
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)
 (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 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))
 (30)
Type: CartesianTensor?(1,2,Fraction(Polynomial(Integer)))