Quantum generalized Kac--Moody algebras via Hall algebras of complexes

Abstract

We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac--Moody algebra into a localized Hall algebra of ${\mathbb{Z}}_2$-graded complexes of representations of a quiver with (possible) loops. To overcome difficulties resulting from the existence of infinite dimensional projective objects, we consider the category of finitely-presented representations and the category of ${\mathbb{Z}}_2$-graded complexes of projectives with finite homology.