Two questions came up at the Fields Institute conference towards the end of September 2002. They both led to more examples of braidings. In his lecture, Bill Lawvere suggested that, while Eilenberg-Kelly [LaJolla] had determined the scarcity of symmetries on graded modules over a commutative ring K, the situation for doubly graded modules would be richer. In fact, there is a braiding for each choice of four invertible elements a, b, c, d of K; it is induced by convolution from the braiding
amnbmn'cm'ndm'n': (m,m') + (n,n') → (n,n') + (m,m')
on the free monoidal ModK-category K*(Z x Z) on the discrete monoidal (under addition) Z x Z. The braiding is a symmetry iff
a2 = bd = d2 = 1.
In particular, for doubly graded abelian groups (where a, b, c, d must be 1 or -1), there are 16 braidings, 8 of them symmetries. Of course, not all of these are braided monoidally inequivalent.
After my lecture at the Fields Institute, Peter Schauenburg asked whether my formal representation theory extended to an explanation of the double/centre construction. I still have no proper answer for this. However, the question led me to consider a generalization of the centre construction. I call it the "external centre of a pseudomonoid"; the pseudomonoid can live in any braided monoidal bicategory, while the construction yields an ordinary braided monoidal category. When the braided monoidal category is the cartesian monoidal bicategory Cat, so that a pseudomonoid is a monoidal category V, the construction yields the centre Z(V) of V as per Joyal-Street [Tortile Yang-Baxter operators in tensor categories, JPAA 71 (1991) 43-51].