1. Lie algebras and graphs (Nick)
- describe Dynkin diagram associated to a finite-dimensional simply laced Lie algebra [Sc09a, Section A.1]
- give the ADE classification [Sc09a, Section A.1], [McG05. Section 1]
- define the (simply laced) Kac-Moody algebra associated to a graph [Sc09a, Section A.2]
- real and imaginary roots
2. Representations of quivers (Rob)
- define quivers and their representations [Sa06, Section 4]
- Gabriel's theorem: ADE classification of quivers of finite representation type [McG05. Section 2], [Sc09a, Theorem 3.7], [Sa06, Theorem 4.5]
- Kac's theorem: generalizing Gabriel's theorem to arbitrary quivers [Sc09a, Theorem 3.13], [Sa06, Theorem 4.6]
3. Hopf algebras and quantum groups (Josiah and John)
- definition of a Hopf algebra [CP, Section 4.1]
- modules and tensor products
- examples: U(g) [CP, Example 4.1.8] and C[G] [CP, Example 4.1.6]
- topological duality between U(g) and C[G] [CP, Example 4.1.17]
- special case: graded duality between U(n) and C[N]
- definition of a quantum Kac-Moody algebra [Sc09a, Section A.4]
- maybe a little bit about its representation theory?
- maybe something about the quantum Yang-Baxter equation?
4. Hall algebras (Josiah and Michael)
- define the Hall algebra of a quiver (or of a finitary category) [Sc09a, Lecture 1.3], [TL09]
- show that it is naturally a Hopf algebra [Sc09a, Lecture 1], [Gr95, Theorem 1]
5. Ringel's construction (David and Joe)
- check Serre relations and prove injectivity, along with surjectivity in finite type [Sc09a, Lecture 3.3], [Ri93], [Sa06, Section 5]
- explain how to do it directly over C with constructible functions [Lu91, Prop 10.20], [Lu91-ICM, Section 19], [TL09]
6. The canonical basis and crystals (AJ and Dylan)
- PBW basis = {characteristic functions} = {simple modules} = {constant sheaves on orbits} depends on orientations [Sc09a, Lecture 3.4]
- some vague words about perverse sheaves; define canonical basis = {simple perverse sheaves}
- the canonical basis is independent of orientations [Lu90], [Lu91-ICM, Section 19]
- Kashiwara's construction of the canonical basis [Ka91, Section 6]
- introduction to the theory of crystals as motivation for the canonical basis [Ka91], [Sc09b, Lecture 4.1]
7. The preprojective algebra and the semicanonical basis (Nick)
- give Lusztig's construction of U(n) [Lu91, Theorem 12.13] and [Sa06, Section 6.1]
- show that it is not sensitive to changes in the orientation of Q [Lu91, Theorem 12.15]
- prove that the components of the nilpotent variety index a basis [Lu91, Conjecture 12.14], [Lu00, Theorem 2.7]
- still descends to a basis of every finite-dimensional simple module [Lu00, Section 3]
8. Nakajima's construction (Matt and Dan)
- define quiver varieties and their cores [Na94]
- explain why core components index the semicanonical basis of an irrep [Na94, Section 10], [Sa06, Section 6]
9. Multiplication formulas for the dual semicanonical basis (Dylan and Michael)
- explain the GLS formulas for products of dual semicanonical basis elements [GLS05, Theorem 1], [GLS07a, Theorem 1.1]
- prove that the canonical and semicanonical bases differ [GLS07a, Sections 7 and 13]
- a few words about 2CY categories [GLS07a, Sections 1.5, 7.1, 8]
10. Prepare problem set
[CP] Chari and Pressley, A guide to quantum groups
[GLS05] Geiss, Leclerc, and Schröer, Semicanonical bases and preprojective algebras
[GLS07a] Geiss, Leclerc, and Schröer, Semicanonical bases and preprojective algebras II: a multiplication formula
[Gr95] Green, Hall algebras, hereditary algebras and quantum groups
[Ka91] Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras
[Lu91] Lusztig,
Quivers, perverse sheaves, and quantized enveloping algebras
[Lu91-ICM] Lusztig,
Intersection cohomology methods in representation theory
[Lu00] Lusztig,
Semicanonical bases arising from enveloping algebras
[McG05] McGerty, Quivers and lattices (video and notes)
[Na94] Nakajima, Instantons on ALE spaces, quiver varieties,
and Kac-Moody algebras
[Ri93] Ringel, The Hall algebra approach to quantum groups
[Sa06] Savage, Finite-dimensional algebras and quivers
[Sc09a] Schiffmann, Lectures on Hall algebras
[Sc09b] Schiffmann, Lectures on canonical and crystal bases of Hall algebras
[TL09] Toledano Laredo, Hall algebras (video and notes)
Note: The references listed above are (almost) a subset of those listed here.