Higher Kac-Moody algebras

It was conjectured by Kac, and proved by Hausel, that the constant term of the Kac polynomials, counting absolutely indecomposable representations of a set quiver with varying dimension vectors, record the root multiplicities of the associated Kac-Moody Lie algebra. This begs the question: is there some Lie-theoretic interpretation of the other coefficients of the Kac polynomials? The answer to this question comes via Donaldson-Thomas theory. It was shown by Mozgovoy that the Kac polynomials themselves can be considered as refined DT invariants of special quivers (endowed with potential). The Jacobi algebras associated to such data can be thought of as “nc-Landau-Ginzburg models”. Recent work with Sven Meinhardt on the categorification of DT theory shows how to upgrade this statement to a Lie-theoretic interpretation for the entire Kac polynomials (not just the constant coefficients). If there is time I will explain how recent work of McGerty and Nevins, along with purity results on the Borel-Moore homology of preprojective stacks, suggests a conjectural approach to “Borcherdsifying” the resulting extended Kac-Moody Lie algebra.