Character varieties and pseudorepresentations

This is a pre-talk in preparation for Jinyue Luo's talk happening later in the week. It will also follow up on Kiumars Kaveh's recent talk.

One of the main motivating questions that I will discuss is the following. I will introduce the notion of a Cayley-Hamilton algebra: an algebra for which each of its elements is equipped with a sensible notion of characteristic polynomial called a “pseudorepresentation", and is a root of its characteristic polynomial. (For matrix algebras, this is the Cayley-Hamilton theorem.) Is every Cayley-Hamilton algebra a sub-algebra of some matrix algebra in a way that is compatible with the two notions of characteristic polynomial? This turns out to be related to an example in geometric invariant theory that Kiumars Kaveh introduced in his talk last semester.