Posts here often discuss operads, for which compositions are represented by a directed rooted tree and the root is the single output. Dually one can consider cooperads with multiple outputs. These give rise naturally to coalgebras rather than algebras. And yes, people have considered vertex operator coalgebras and so on. Recall that coalgebras are essential to the concept of Hopf algebra. An example of a Hopf algebra is a universal enveloping algebra of a semisimple Lie algebra. Deformations of these, also Hopf algebras, are the notorious quantum groups.

Now Ross Street has published a book, Quantum Groups: a Path to Current Algebra, which I would love to get my hands on. The book blurb says, “A key to understanding these new developments is categorical duality.” That simply means the duality given by turning a tree upside down, as drawn, so we need operad structures that combine upward branches and downward branches. Leaping ahead to triality, one expects branchings in three directions and no chosen root. Such diagrams are associated to cyclic operads, and we actually consider these with polygonal tilings, since the dual to a polygon is a branching tree with no real root.

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## Matti Pitkanen said,

July 28, 2007 @ 4:22 am

Thanks for refreshing my vague mental images about co-algebras.

I would like to understand how inclusions of hyperfinite factors of type II_1 relate to quantum groups and co-algebra structure and all that which is so difficult for me!

And how the replacement of complex rays of state space with N-rays (N is the included HFF and defines the resolution of quantum measurement since things can be measured only modulo action of N) brings in quantum group structure and exactly how this measurement resolution relates to co-product?

## kneemo said,

July 28, 2007 @ 6:33 pm

The Ross Street book looks excellent. I’m going to order it.