Some examples of weak Hopf algebras

Micah Blake McCurdy – 9 September 2009

A very gentle talk to give some examples of weak Hopf algebras. First, I discuss a known exaqmple from Pastro and Street, namely, the category-algebra over a field which gives rise to a weak bialgebra and the groupoid-algebra which gives rise to a weak Hopf algebra. This example has a number of pleasant features, especially noting that the failure of these weak bialgebras to be bialgebras neatly matches the "failure" of the categories or groupoids to be monoids or groups, respectively. The basic idea of this example can be extended to deal with some weak Hopf algebras from combinatorics; in particular, there is an obvious generalization of the Grossman-Larson Hopf algebra of rooted trees to a Grossman-Larson _weak_ Hopf algebra of _coloured_ rooted trees. In fact, it appears that Grossman and Larson, who discuss coloured rooted trees in their work, carefully avoided discovering weak Hopf algebras by artificially restricting the roots of their coloured trees to not be coloured. God and time willing, I might even discuss a "shuffle weak Hopf algebra" which has objects composable sequences of morphisms in a category in place of the usual sequences of letters in an alphabet that are the objects of the shuffle Hopf algebra.