### Paul-André Melliès (CNRS, Université Paris 7), Braided notions of dialogue categories

#### Schedule

- March 1, 2012, 16:30 - 17:30

#### Abstract

A dialogue category is a monoidal category equipped with an exponentiating object bot called its tensorial pole. In a dialogue category, every object x is thus equipped with a left negation x --o bot and a right negation bot o-- x. An important point of the definition is that the object x is not required to coincide with its double negation. In this talk, I will define three non-commutative notions of dialogue categories -- called helical, cyclic and balanced dialogue categories. I will then explain how to see the category of left H-modules of arbitrary dimension on a ribbon Hopf algebra H as a balanced dialogue category Mod(H) whose tensorial pole bot is provided by the underlying field k. I will also explain how to recover from this basic observation the well-known fact that the full subcategory of finite dimensional left H-modules defines a ribbon category.