Edit detail for SandBoxFrobeniusAlgebra revision 22 of 26

References

• Frobenius algebras and 2D topological quantum field theories by Joachim Kock

  Especially "Tensor calculus (linear algebra in coordinates)" in section 2.3.31, page 123.

See also:

• http://ncatlab.org/nlab/show/Frobenius+algebra by John Baez, et al.
• Ideals, Varieties, and Algorithms by David A. Cox, et al.

An n-dimensional algebra is represented by a (2,1)-tensor Y=\{ {y^k}_{ij} \ 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,[[i,j],[k]])
 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}_{ij} p^i q^j \}

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

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.

R:=(Y*P)*Q

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

(associator) is zero.

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

The algebra Y is commutative if:

 Y = Y
 i j i j j i
 \ / = \/ - \/
 | \ /
 k k k

This requires that the following (2,1)-tensor

(commutator) is zero.

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

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

groebner(ravel(YC))

The algebra Y is anti-commutative if:

 Y = -Y
 i j i j j i
 \ / = \/ = \/
 | \ /
 k k k

This requires that the following (2,1)-tensor

(anti-commutator) is zero.

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

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

groebner(ravel(YA))

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

is zero.

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

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 (3,0)-tensor:

 i j k i j k i j k
 \ | / \/ / \ \/
 \|/ = \ / - \ /
 0 0 0

(three-point function) is zero.

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 \Phi. We are looking for coefficients of the tensor U such that K transforms the tensor Y into \Phi=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

All 2-dimensional pre-Frobenius algebras are symmetric.

Proof: Consider the determinant of the matrix K above.

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

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 commutative.

Proof: The basis of the null space of the symmetric K matrix are all symmetric

YUS:T := reindex(reindex(US,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-US*Y

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 defines a 4-parameter family of 2-d pre-Frobenius algebras

test(unravel(map(x+->subst(x,SS),ravel YUS))$T=0*YU)

Alternatively we may consider

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 \Phi. We are looking for coefficients of the tensor Y such that J transforms the tensor U into \Phi=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

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

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

Y-forms

Left snail and right snail:

 LS RS
 Y A A Y
 Y ) ( Y
 U U
 i j j i
 \/ \/
 \ /\ /\ /
 e f \ / f e
 \/ \ / \/
 \ / \ /
 f / \ f
 \/ \/

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)

Left and right deer:

 LD RD
 \ /\/ \/\ /
 Y /\ /\ Y
 Y ) ( Y
 \/ \/
 i j i j
 \ /\ / \ /\ /
 \ f \ / \ / f /
 \/ \/ \/ \/
 \ /\ /\ /
 e / \ / \ e
 \/ \ / \/
 \ / \ /
 f / \ f
 \/ \/

LD:=contract(Y*Y,1,3)

RD:=reindex(contract(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),1,3),[2,1])

test(LD=RD)