Francesco Sala (University of Tokyo) : Kac's polynomials, quantum groups, and cohomological Hall algebras

Given a quiver Q and a dimension vector d, Kac proved the existence of a polynomial A(t) with integer coefficients, depending on Q and d, with the property that the evaluation of such a polynomial at a power q of a prime coincides with the number of absolutely indecomposable representations of Q with dimension d over a finite field with q elements.
As conjectured by Kac and proved by Hausel, the constant term of A(t) gives the dimension of the root space of the (derived) Kac--Moody Lie algebra of Q associated with d.
The question concerning a Lie theoretic interpretation of the full Kac's polynomial is deeply related to the recently developed theories of Maulik-Okounkov Yangians and of cohomological Hall algebras of preprojective algebras of quivers à la Schiffmann-Vasserot.

During the first part of the present talk, I will revise the state of the art of the relation between the Kac's polynomial and Schiffmann-Vasserot cohomological Hall algebras. The second part of the talk is devoted to the curve case: there exists a curve analog of the Kac's polynomial, so it makes sense to address the question of a Lie theoretic interpretation of it.
I will discuss a possible way to answer this question by using some new cohomological Hall algebras introduced in my works with Olivier Schiffmann and Mauro Porta, respectively.