One can associate to any finite graph $Q$ the skew-symmetic Kac-Moody Lie algebra. While this algebra is always infinite, unless $Q$ is a Dynkin diagram of type ADE, $mathfrak{g}_Q$ shares a lot of the nice features of a semisimple Lie algebra. In particular, the cohomology of Nakajima quiver varieties associated to $Q$ gives a geometric representations of $mathfrak{g}_Q$. Encouraged by this story, one could hope to define the Yangian of $mathfrak{g}_Q$, for general $Q$, as a subalgebra of the algebra of endomorphisms of cohomology of quiver varieties. In fact there are two approaches to doing this: firstly via the stable envelope construction of Maulik and Okounkov, secondly via the preprojective Hall algebra of Schiffmann and Vasserot. Via another Hall algebra construction, due to Kontsevich and Soibelman, and work of myself and Meinhardt on BPS Lie algebras, these approaches turn out to be more or less the same. In showing this we show that the correct Lie algebra of endomorphisms associated to $Q$ is cohomologically graded, with zeroeth piece the old Kac-Moody Lie algebra - the best way to build a (Borcherds) Lie algebra out of $Q$ is directly from geometric representation theory.

Ben Davison is a mathematician from Watford. He is currently a reader at Edinburgh University, having previously worked at Glasgow and held postdocs in Paris, Bonn, Lausanne and Vienna. His interests include Donaldson-Thomas theory, Yangians, Higgs bundles and cluster algebras.