Knots, groups, cospans and spans III

This post is the third of a series, the first being In praise of composition: knot groups and cospans and the second Knots, groups, cospans and spans.

I want to give some details,  and a slight extension to the remarks of the second post. In brief I will show how the Frobenius equations characterize the existence of inverses, and how the tensor product of Frobenius objects gives a semi-direct product of groups.



The starting point is a group in a category with finite limits, but let's just think of the category of sets and functions, so that we are just talking about a usual group G.

In the previous posts I considered spans, but these are just generalized relations. To simplify I will consider here just relations, the composition of relations, the product of relations and so on. We will write a relation R from X to Y as R : X -|-> Y, the composition with S : Y -|-> Z as RS : X -|-> Z, the product of R with T : U -|-> V as RxT : XxU -|-> YxV, the opposite of R as R^ : Y-|-> X, a function f : X->Y considered as a relation as f : X-|-> Y, the identity relation as 1 : X-|-> X.

Turning back to the group G, the operations of multiplication m and identity e give rise to four relations  (I a one point set)
m: GxG-|->G, e: I-|->G, m^:G-|->GxG, and e^:G-|->I which clearly satisfy the associative law (mx1)m=(1xm)m:GxGxG-|->G, the coassociative law G-|->GxGxG and identity and coidentity laws. But they also satisfy the laws introduced by Carboni and Walters in 1987 exactly studying relations, now called the Frobenius equations, namely
(1xm^)(mx1) = mm^ = (m^x1)(1xm): GxG -|-> GxG.

I did not mention so far the operation of inverse in the group. So all the definitions of the last paragraph could be made just for a monoid. However the Frobenius equations are then satisfied if and only if the monoid is a group.

Let's see why that is true. Call the three relations equated above R, S, and T respectively. Then
(x,y)R(z,w) if there exists a u in G such that x=zu (ie x=m(z,u)) and uy=w. If the monoid is a group this is equivalent to equations inverse(z) x=w inverse(y), that is, xy=zw. But this is exactly the second relation.  So a group satisfies the Frobenius equations.

For the converse, suppose only that G is a monoid and that the Frobenius equations are satisfied. Then R=S means that there exists a u such that x=zu and uy=w if and only if xy=zw. Put x=e=w  the identity of the monoid and y=z. Clearly xy=zw(= y=z). Hence there exists a u in G such that zu=1=uz.

Next we want to discuss commutativity. The group G was not assumed commutative but in a certain twisted sense in the category of relations the Frobenius operations of a group are commutative. To say an operation m is commutative we usually say that m composed with the twist operation (x,y) |-> (y,x) is the same as m (that is m(y,x)=m(x,y)). But there is an unusual twist operation GxG -|-> GxG, namely the function twist: ((x,y) |-> (xy inverse(x) , x). One sees immediately that
m(twist(x,y))=m(xy inverse(x)x)=m(x,y). So in this twisted sense G is commutative and also cocommutative. (It was exactly this twist that was used by Artin to represent braids.) This "braid-twist" satisfies the equations satisfied by the transposition permutation except that it is not its own inverse.

Now it is a general fact that the (tensor) product of two Frobenius algebras in the presence of such a braid-twist has also the structure of a Frobenius algebra.  So GxG has an induced Frobenius operations (written, I hope without confusion, also as m and m^) defined as follows:
m=(1x twist x 1)(m x m): (GxG)x(GxG)-|->GxG.
In more elementary terms this is:
m((x,y), (z,w))=(xyz inverse(y),yw), the multiplication of a semidirect product of G with G.

Kirchhoff's law I have mentioned in the second post an example involving invariants of knots. I hinted in the first post at connections with electric circuit theory, discussed in  P. Katis, N. Sabadini, R.F.C. Walters, On the algebra of systems with feedback & boundary, Rendiconti del Circolo Matematico di Palermo Serie II, Suppl. 63 (2000), 123-156. Notice that taking the group G to be the real numbers the component m^ : G -|-> GxG is the fanning out of a wire into two, satisfying the Kirchhoff law for currents, and m is the corresponding joining of two wires. Repeated use of m and m^ can describe any Kirkhoff connector. Since the reals are commutative the braid-twist is the usual twist.