Hopf algebras and monoidal categories

Abstract

In this thesis we study the correspondence between categorical notions and bialgebra notions, and make a kind of dictionary and grammar book for translation between these notions. We will show how to obtain an antipode, and how to define braidings and quantizations. The construction is done in two ways. First we use the properties of a bialgebra to define a monoidal structure on (co)modules over this bialgebra. Then we go from a (strict) monoidal category and use a certain monoidal functor from this category to reconstruct bialgebra and (co)module structures. We will show that these constructions in a sense are inverse to each other. In some cases the correspondence is 1-1, and in the final Part we conjecture when this is the case for the category of comodules that are finitely generated and projective over the base ring k. We also briefly discuss how to transfer the results to non-strict categories.