Description
Group actions and representations are some of the most fundamental concepts in algebra and geometry.
An action of a group encodes how the group elements describe (geometric) symmetries of the object upon which the group act.
(Finite dimensional) representations of a group allow to translate group(action)s in terms of linear algebra and (invertible) matrices.
In this talk, I will discuss a recent refinement of these notions, called /partial/ actions and /partial/ representations. In geometric terms, a partial action does not describe symmetries of an object as a whole, but between certain sub-objects, so one could think of these as “local” (or better, “partial") symmetries rather than the usual “global” ones.
Similarly, in a partial representation, group elements are mapped to not necessarily invertible matrices.
These notions are motivated by natural examples and several results that show how they provide an useful and powerful additional tool to study groups. Moreover there is a close relationship with (inverse) semigroups and groupoids.
In the second part of the talk I will discuss how these notions generalize to partial actions and representations of Hopf algebras. I will give a sketch of main results and some open questions in this area.