Partial representations of Hopf algebras

In this work, the notion of a partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra H  , one can associate to it a Hopf algebroid HparHpar which has the universal property that each partial representation of H   can be factorized by an algebra morphism from HparHpar. We define also the category of partial modules over a Hopf algebra H  , which is the category of modules over its associated Hopf algebroid HparHpar. The Hopf algebroid structure of HparHpar enables us to enhance the category of partial H-modules with a monoidal structure and such that the algebra objects in this category are the symmetric partial actions. Some examples of categories of partial H  -modules are explored. In particular we can describe fully the category of partially Z2Z2-graded modules.