Edit detail for SandBoxFrobeniusAlgebra revision 18 of 26

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

n:=2

T:=CartesianTensor(1,n,FRAC POLY 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]
 )

Given two vectors P=\{ p_i \} and Q=\{ q_j \}

P:T := unravel([script(p,[[i]]) for i in 1..n])

Q:T := unravel([script(q,[[i]]) for i in 1..n])

the tensor Y operates on their tensor product to yield a vector R=\{ r_k = {y_k}^{ji} p_i q_j \}

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

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.

R:=(Y*P)*Q

An algebra Y is commutative if the tensor \Pi = \{ {\pi_k}^{ji} = {y_k}^{ji}-{y^k}^{ij} \} (the commutator) is zero:

 Y - X
 Y

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

A basis for the ideal defined by the coefficients of the commutator is given by:

groebner(ravel(XY))

The algebra Y is anti-commutative if the tensor \Xi = \{ {\xi_k}^{ji} = {y_k}^{ji}+{y^k}^{ij} \} (the anti-commutator) is zero:

 Y + X
 Y

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

A basis for the ideal defined by the coefficients of the anti-commutator is given by:

groebner(ravel(XX))

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 \Psi = \{ {\psi_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} \} is zero.

YY := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(YY)

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:

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

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
 to learn more about the available operations. Perhaps 
 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 U = \{ u^{ij} \}

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

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 \Omega = \{ \omega^{kji} = {Y_r}^{kj} U^{ri} - U^{kr} {Y_r}^{ji} \}

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

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.

)expose MCALCFN

 MultiVariableCalculusFunctions is now explicitly exposed in frame 
 initial 
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

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.

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.

Kd:DMP(concat map(variables,ravel(U)),FRAC INT) := factor determinant(K)

The scalar product must also be non-degenerate

Ud:DMP(concat map(variables,ravel(U)),FRAC INT) := determinant [[U[i,j] for j in 1..n] for i in 1..n]

therefore U must be symmetric.

nthFactor(Kd,1)

US:T := unravel(map(x+->subst(x,U[2,1]=U[1,2]),ravel U))

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

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);

NS:=nullSpace(KS)

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.

UASS:T := unravel(map(x+->subst(x,SS),ravel YUS))

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

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.

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.

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) ];

#generators(%)

Theorem 3

A 2-d algebra is pre-Frobenius if it is associative, commutative, anti-commutative or if it satisfies the Jacobi identity.

Proof

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