M Theory Lesson 114

The observant M theorist will have noticed the large variety of hexagons that keep popping up. In particular, there is the broken pentagon hexagon, labelled by elements of $S_3$, and also the Yang-Baxter hexagon, with its alternating associator and braiding maps. But if the permutation maps (eg. $ABC \rightarrow BAC$) are considered as braid generators then these two hexagons look very similar. When the four-leaved trees undergo a flip from outside edge to outside edge, we may take that to be an associator on three elements.

Generalising permutations to braidings is very natural in category theory. Ordinary algebras are vector spaces, associated with the symmetric monoidal category Vect, but the symmetry arises as an additional constraint on the underlying braiding of a braided monoidal category. The Loday-Ronco maps between $A_n$ and $S_n$, and their Hopf algebras, should therefore have a generalisation to braided monoidal objects, such as the weak Hopf algebras of Robert Coquereaux, considered recently by Street and Pastro.