Enveloping algebras via motivic Hall algebras

We give a geometric realization of the whole universal enveloping algebra of the Borcherds-Bozec algebra using quiver with loops via the motivic semi-derived Hall algebra approach. In particular, using acyclic quivers, we give a geometric realization of the whole universal enveloping algebra of a certain generalized Kac-Moody algebra using the motivic Bridgeland’s Hall algebra given in [12].