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

## TL;DR

This paper constructs an embedding of quantum generalized Kac--Moody algebras into Hall algebras of complexes, addressing challenges from infinite-dimensional projectives by focusing on finitely-presented representations and complexes with finite homology.

## Contribution

It introduces a novel embedding of quantum Kac--Moody algebras into Hall algebras of complexes, overcoming issues related to infinite-dimensional projectives.

## Key findings

- Established an embedding of quantum Kac--Moody algebras into Hall algebras.
- Developed methods to handle infinite-dimensional projectives.
- Connected representation theory with Hall algebra structures.

## 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.