Lectures on Hall algebras

TL;DR

This paper provides an overview of Hall algebras, covering their definitions, properties, and applications to quivers, coherent sheaves, and derived categories, serving as comprehensive lecture notes.

Contribution

It offers a detailed exposition of Hall algebras, including classical, quiver, and coherent sheaves cases, and explores their role in derived categories, filling a gap in educational resources.

Findings

Detailed definitions and properties of Hall algebras

Analysis of Hall algebras of quivers and coherent sheaves

Discussion of Hall algebras in derived categories

Abstract

These are notes for a minicourse on Hall algebras given at the ICTP in Trieste in January 2006. After giving the definition and first properties of Hall algebras, we study in some details the classical Hall algebra, the Hall algebra of quivers, and the Hall algebra of coherent sheaves on smooth projective curves. The last section deals with the Hall algebras in the context of derived categories. (The figures only seem to come out nicely in the .ps file)