Enveloping algebras via motivic Hall algebras

TL;DR

This paper provides a geometric construction of the universal enveloping algebra of Borcherds-Bozec and certain generalized Kac-Moody algebras using motivic Hall algebra techniques, linking algebraic structures to quiver representations.

Contribution

It introduces a new geometric realization of these algebras via motivic Hall algebras associated with quivers, extending previous work to more general algebraic structures.

Findings

Realization of Borcherds-Bozec algebra via quiver with loops

Geometric realization of generalized Kac-Moody algebra using acyclic quivers

Application of motivic semi-derived and Bridgeland's Hall algebras

Abstract

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].