Quiver varieties and Hall algebras

In this paper, we give a geometric construction of the quantum group Ut(G) using Nakajima categories, which were developed in Scherotzke ('Quiver varieties and self-injective algebras', Preprint, 2014, arxiv.org/abs/1405.4729). Our methods allow us to establish a direct connection between the algebraic realization of the quantum group as Hall algebra by Bridgeland ('Quantum groups via Hall algebras of complexes', Annals of Maths. 177 (2013) 739-759) and its geometric counterpart by Qin ('Quantum groups via cyclic quiver varieties I', Compos. Math., Preprint, 2013, arxiv.org/abs/1312.1101).