Cohomological Hall algebras and affine quantum groups

Abstract

We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in Yang and Zhao (The cohomological Hall algebra of a preprojective algebra. arXiv: 1407.7994v5 , 2015) for any quiver Q and any one-parameter formal group $${\mathbb {G}}$$ . In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra $$\mathfrak {g}_Q$$ associated to Q, whose quantization comes from the formal group $${\mathbb G}$$ . We prove, when the group $${\mathbb G}$$ is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian.