Arithmetic and representation theoretic properties of certain Hopf algebras

This thesis deals with arithmetic and representation theoretic properties of certain semisimple Hopf algebras. It consists of two parts. The first one is devoted to the study of Hopf orders. The initial goal of this part is to prove that for any finite non-abelian simple group G there is a twist for CG, arising from a 2-cocycle on an abelian subgroup of G, such that the associated twisted group algebra does not admit a Hopf order over any number ring. For showing this we prove the non-existence result for a key family of simple groups G and combine it with two theorems of Thompson and Barry and Ward on minimal simple groups. In addition, we prove the non-existence of Hopf orders for twists of group algebras of direct products of Frobenius groups, subject to some technical conditions. The second part of the thesis takes place in the finite W-algebras framework. A finite W-algebra H_l is an algebra constructed from a reductive Lie algebra g and a nilpotent element e in g. We interpret Goodwin’s translation functors as an action of a subcategory of U(g)-representations on the category of finitely generated H_l-modules. This action is obtained by transporting the tensor product of U(g)-modules through Skryabin's equivalence. Recently, Genra and Juillard studied sufficient conditions to apply the Hamiltonian reduction by stages to finite W-algebras, getting a Skryabin equivalence by stages. We show that the latter is an equivalence of U(g)-module categories.