An ndimensional 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
n:=2
axiom
T:=CartesianTensor(1,n,FRAC POLY INT)
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]
)
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
Given two vectors and
axiom
U:T := unravel([script(u,[[i]]) for i in 1..n])
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
axiom
V:T := unravel([script(v,[[i]]) for i in 1..n])
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)
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
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
The algebra Y
is commutative if the following tensor
(the commutator) is zero:
Y  X
Y
axiom
KK:=Yreindex(Y,[1,3,2])
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
A basis for the ideal defined by the coefficients of the
commutator is given by:
axiom
KB:=groebner(ravel(KK))
Type: List(Polynomial(Integer))
The algebra Y
is anticommutative if the following tensor
(the anticommutator) is zero:
Y + X
Y
axiom
AK:=Y+reindex(Y,[1,3,2])
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
A basis for the ideal defined by the coefficients of the
anticommutator is given by:
axiom
KA:=groebner(ravel(AK))
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
AA := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])Y*Y; ravel(AA)
Type: List(Fraction(Polynomial(Integer)))
The Jacobi identity requires the following tensor to be zero:
Y  Y  X
Y Y Y
Y
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)
Type: List(Fraction(Polynomial(Integer)))
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]
)
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
Definition 1
We say that the scalar product is "associative" if the following
tensor equation holds:
Y = Y
U U
axiom
UA := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])U*Y
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
Definition 2
An algebra with a nondegenerate associative scalar product is
called ''preFrobenius''.
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
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 nontrivial
solution is that the matrix K
be degenerate.
Theorem 1
The scalar product of all 2dimensional preFrobenius
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)
Type: DistributedMultivariatePolynomial
?([*002u11,
*002u12,*002u21,*002u22],Fraction(Integer))
The scalar product must also be nondegenerate
axiom
Ud:DMP(concat map(variables,ravel(U)),FRAC INT) := determinant [[U[i,j] for j in 1..n] for i in 1..n]
Type: DistributedMultivariatePolynomial
?([*002u11,
*002u12,*002u21,*002u22],Fraction(Integer))
therefore U must be symmetric.
axiom
nthFactor(Kd,1)
Type: DistributedMultivariatePolynomial
?([*002u11,
*002u12,*002u21,*002u22],Fraction(Integer))
axiom
US:T := unravel(map(x+>subst(x,U[2,1]=U[1,2]),ravel U))
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
Theorem 2
All 2dimensional 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))
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)
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]))
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
YS:T := unravel(map(x+>subst(x,SS),ravel Y))
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
This is a 4parameter family of 2d preFrobenius algebras with
a given admissible (i.e. symmetric) scalar product.
axiom
UASS:T := unravel(map(x+>subst(x,SS),ravel UAS))
Type: CartesianTensor
?(1,
2,Fraction(Polynomial(Integer)))
axiom
J := jacobian(ravel(UA),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
Type: Equation(OutputForm
?)
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
.
A necessary condition for the equation to have a nontrivial
solution is that all 70 of the 4x4 submatrices of J
are
degenerate. To this end we can form the polynomial ideal of
the determinants of these submatrices.
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(%)
Theorem 3
A 2d algebra is preFrobenius if it is associative,
commutative, anticommutative or if it satisfies the
Jacobi identity.
Proof
axiom
in?(JP,ideal ravel AA)
Type: Boolean
axiom
in?(JP,ideal ravel KK)
Type: Boolean
axiom
in?(JP,ideal ravel AK)
Type: Boolean
axiom
in?(JP,ideal ravel BA)
Type: Boolean