Hopf objects in monoidal catgories and their categories of representations

Micah Blake McCurdy – 27 August 2008

The definitions of a Hopf algebra can be given in a purely diagrammatic form which can be stated in any monoidal category. Many elementary notions of Hopf algebra theory can be formulated at this level. In particular, there are well known conditions that can be placed on a Hopf algebra H which ensure that Mod(H) has a variety of pleasant properties. I will focus on a few of these, chiefly conditions under which Mod(H) can be shown to be a non-degenerate, non-symmetric cyclic *-autonomous category. This ties together a remark of Joyal and Street concerning pivotal categories and remarks of Blute concerning Hopf algebras with involutive antipodes, and provides a host of examples of cyclic *-autonomous categories.