SUMMER SCHOOL
Geometry of quiver-representations
and preprojective algebras

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Topics for the participants' lectures

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.

  1. Affine quotient for a reductive group acting on an affine variety [LP, §§6.1-6.3] or [Ka2, §2.3].

  2. Representation spaces of quivers with relations, and invariants.

  3. Degenerations and Zwara's Theorem that degenerations of modules correspond to certain extensions [Z3].

  4. Stable and semistable points [Ki, §2].

  5. Moduli spaces of representations [Ki, §§3-4.2].

  6. Universal bundles [Ki, §5].

  7. General representations of quivers. Characterization of canonical decomposition and dimension vectors of subrepresentations in terms of the general dimensions of Ext spaces.

  8. Maps between general representations of quivers [CB1 §§1,2] and [S1, Theorem 5.2].

  9. Schubert varieties [GH §1.5, p193 - p197 (half way down)] and the connection with general representations of quivers which are stars.

  10. Schubert calculus [GH, §1.5, p197 (half way down) to p206]. See also [F1, §9.4].

  11. 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].

  12. Introduction to Kac-Moody algebras [Ka3].

  13. Kac's Theorem on indecomposable representations of quivers [Ka1, Ka2].

  14. Symplectic forms and the preprojective algebra. The moment map for the cotangent bundle of the representation space of a quiver.

  15. The nilpotent variety. Proof that the nilpotent variety has pure dimension, and maybe that it is Lagrangian [Lu1, §12].

  16. Crystal bases in quantized enveloping algebras [KS, §§2-3].

  17. Geometric construction of crystal bases [KS, §5].

  18. Construction of the positive part for Dynkin Lie algebras [Rie2].

  19. Constructible functions and enveloping algebras [Lu1, §§10.18-10.20 and §§12.10-12.13].

  20. Nakajima's quiver varieties [N1, N2].

  21. 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)