Hopf algebras, Hopf monads and derived categories of sheaves

The aim of this thesis is to explain how the theory of Hopf monads on monoidal categories can be used to investigate the Hopf algebra object in a category of which objects are complexes of sheaves on a smooth complex projective variety.

In particular, we associate to a smooth complex projective variety X the category of orbits of the bounded derived category of coherent sheaves of the variety under the double shift functor and discuss its structure. We explain why the derived functors on the level of bounded derived categories of coherent sheaves induce functors on the level of categories of orbits. We prove that if the variety is of even dimension and has trivial canonical bundle, then the Serre functor on the orbit category is trivialised. As a direct application of this, we obtain functors on the level of orbit categories which have the same left and right adjoint functor.

Next, we work in a general categorical level considering monoidal categories with duals and a pair of adjoint functors defined between them. Moreover, we assume that the right adjoint is a strong symmetric monoidal functor and has a right quasi-inverse which is also a strong monoidal functor. We prove that every such pair of adjoint functors defines an augmented Hopf monad. From the theory of augmented Hopf monads of Brugières, Lack and Virelizier we obtain that these Hopf monads are equivalent to central Hopf algebras.

We explain why the orbit category of an even dimensional smooth complex projective variety X with trivial canonical bundle is a symmetric monoidal category with duals. In addition we explain why the diagonal embedding map of the variety gives rise to a pair of adjoint functors on the level orbit categories such that the right adjoint is strong symmetric monoidal and has a right quasi-inverse. As a result, we obtain a Hopf monad on the orbit category and pin down the Hopf algebra with explicit morphisms.