Geoffrey Janssens (UCLouvain and VUBrussel) – Bridging representation theories through cluster algebras

Given a (Dynkin) quiver $Q$ one can associate both a simple Lie algebra 𝔤 and the category of representations $Rep(Q)$ of $Q$. Early on it was realised that both associated objects are related, as for example beautifully illustrated by Gabriel's theorem. In this talk we will consider two associated categories of representations: (i) (some quotient of) the derived category $D^b(Rep(Q))$ and (ii) the finite dimensional representations of the quantum loop algebra $U_q(L𝔤)$. Although both look quit differently, we will delve into wished to be understood ties. The presented story will be one of categorifications of a common algebra with a rich combinatorial structure: a cluster algebra. The category (i) yields a so-called additive categorification and (ii) a monoidal one.  In the first half of the talk we will give a gentle and minimalistic introduction to the various objects and concepts mentioned. In the second half we will give an intuitive overview of recent conjectural connections and then finish by (very briefly) mentioning some ongoing contributions.