Lectures on canonical and crystal bases of Hall algebras

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.

101 pages, Latex; minor revisions

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• A diagrammatic approach to categorification of quantum groups I
• 2-Kac-Moody algebras
• Canonical bases and Khovanov-Lauda algebras
• Categorification of integrable representations of quantum groups
• Canonical bases of loop algebras via quot schemes II

Cited by in corpus (9)

• Quivers with loops and perverse sheaves
• On the number of points of nilpotent quiver varieties over finite fields
• BPS Hall Algebra of Scattering Hall States
• Exact structures and degeneration of Hall algebras
• Microlocal characterization of Lusztig sheaves for affine quivers and $g$-loops quivers
• Hall algebras for odd periodic triangulated categories
• Hall categories and KLR categorification
• On the cohomology of quiver Grassmannians for acyclic quivers
• On the number of points of the Lusztig nilpotent variety over a finite field