Lectures on canonical and crystal bases of Hall algebras

TL;DR

This paper provides an exposition on the theory of canonical and crystal bases for Hall algebras, including constructions by Lusztig and Kashiwara-Saito, with some new results and conjectures, aimed at a mathematical audience.

Contribution

It offers a comprehensive exposition of Lusztig's and Kashiwara-Saito's geometric constructions of bases and crystal graphs for Hall algebras, with new results on Hall algebras of curves.

Findings

Contains new results and conjectures on Hall algebras of curves.

Provides detailed exposition of Lusztig's construction of canonical bases.

Explains Kashiwara and Saito's geometric approach to crystal graphs.

Abstract

These are the notes for a series of lectures given on the theory of canonical and crystal bases for Hall algebras (for a summer school in Grenoble in 2008). It may be viewed as a follow-up to arXiv:math/0611617. It covers the construction, due to Lusztig, of the canonical bases for the Hall algebra of a quiver Q in terms of a certain category of perverse sheaves over the moduli space of representations of Q. It also contains an exposition of Kashiwara and Saito's geometric construction of the crystal graph in terms of irreducible components of Lusztig's lagrangian in the cotangent bundle to the above moduli spaces. The last section deals with the Hall algebras of curves. It contains a few new results and conjectures. Apart from these, the text is purely expositional.