Hall Algebras and Edge Contractions with Loops

TL;DR

This paper generalizes the theory of Hall algebras and edge contractions to include vertices with multiple edges and loops, establishing new algebraic embeddings that preserve core structures.

Contribution

It introduces a broader framework for Hall algebra embeddings via edge contractions, extending previous work to more complex graph configurations.

Findings

Established new embeddings among Hall algebras for graphs with loops

Proved these embeddings preserve Hopf algebra structures

Extended the applicability of Hall algebra techniques to more general graph contractions

Abstract

We extend the study of Hall algebras and edge contractions by generalizing Yiqiang Li's work to contraction along vertices with multiple edges. Using the edge contractions, we establish new embeddings among Hall algebras in this broader setting. Our results demonstrate that these embeddings preserve key algebraic structures, including Hopf algebra operations.