SUMMER SCHOOL
Geometry of quiver-representations
and preprojective algebras
Home page
This is a list of the topics for lectures in the first
half of the Summer School.
Further details about the lectures are available here.
Change of plan: all 21 lectures are to be
50 minutes maximum.
- Affine quotient for a reductive group acting on an affine variety
[LP, §§6.1-6.3] or
[Ka2, §2.3].
- Representation spaces of quivers with relations, and invariants.
- Degenerations and Zwara's Theorem that degenerations of modules correspond to
certain extensions [Z3].
- Stable and semistable points [Ki, §2].
- Moduli spaces of representations [Ki, §§3-4.2].
- Universal bundles [Ki, §5].
- General representations of quivers. Characterization of
canonical decomposition and dimension vectors of
subrepresentations in terms of the general dimensions of Ext
spaces.
- Maps between general representations of quivers
[CB1 §§1,2]
and [S1, Theorem 5.2].
- Schubert varieties [GH §1.5,
p193 - p197 (half way down)] and the connection with general
representations of quivers which are stars.
- Schubert calculus [GH,
§1.5, p197 (half way down) to p206]. See also
[F1, §9.4].
- Chern class calculations. Basic properties of Chern classes.
Chern classes for the universal quotient bundle for a
Grassmannian. Chern class calculations for general representations
of quivers, as in [CB2].
- Introduction to Kac-Moody algebras [Ka3].
- Kac's Theorem on indecomposable representations of quivers
[Ka1, Ka2].
- Symplectic forms and the preprojective algebra. The moment
map for the cotangent bundle of the representation space of a
quiver.
- The nilpotent variety. Proof that the nilpotent variety
has pure dimension, and maybe that it is Lagrangian
[Lu1, §12].
- Crystal bases in quantized enveloping algebras
[KS, §§2-3].
- Geometric construction of crystal bases
[KS, §5].
- Construction of the positive part for Dynkin
Lie algebras [Rie2].
- Constructible functions and enveloping algebras
[Lu1, §§10.18-10.20 and §§12.10-12.13].
- Nakajima's quiver varieties [N1, N2].
- Construction of integrable representations
[N1, N2].
This page is maintained by
William Crawley-Boevey
Original version 6 December 1999
Revised 12 January 2000 (Schubert calculus lectures reorganized)
Revised 5 April 2000 (Schubert calculus lectures reorganized again)
Revised 14 April 2000 (All lectures 50 minutes maximum)