Quivers and quantum algebras

Fix a finite set I, which generalizes the set of simple roots of a Dynkin diagram. For any collection of Laurent polynomials zeta_{ij}(x) with i,j running over the set I, we may define a so-called shuffle algebra, following ideas of Feigin-Odesskii. The definition might seem a bit contrived at first, but we will observe that particular cases of this construction give rise to very interesting quantum algebras, such as
- quantum loop groups associated to finite type Cartan matrices
- cohomological (or more appropriately, K-theoretic) Hall algebras of quivers
- BPS algebras arising from quantum mechanics on a toric Calabi-Yau threefold
and the list goes on. The principle that we will repeatedly encounter is that the shuflle algebra gives a great computational tool to study the above algebraic objects. We will delve in the detailed study of shuffle algebras, with the concrete goal of obtaining a definition of quantum loop groups associated to any symmetric Cartan matrix. As challenging as this task will turn out to be, it pales in comparison to the (still unsolved) task of completely describing shuffle algebras for all possible collections of Laurent polynomials zeta_{ij}(x). One of the main goals of our course will be to provide interested mathematicians with tools they might need to tackle this difficult question.