Representation Theory A

Course Description: Following Kac and Moody, a diagram can be used to define a Lie algebra. For instance, the \(A_N\) Dynkin diagram corresponds to the special linear Lie algebra \(\mathfrak{sl}(N+1)\). By orienting the diagram, we obtain a quiver, which in turn defines an abelian category of quiver representations. This raises a natural question: how do the “algebraic” properties of the Lie algebra relate to the “categorical” properties of its corresponding representations? A first answer is given by Gabriel’s theorem. It establishes a fundamental link by creating a bijection between the positive roots of the Lie algebra and the isomorphism classes of indecomposable representations of the quiver. These classes form a basis for the Grothendieck group of the category. However, the Grothendieck group is only a “shadow” of the category, discarding rich information about morphisms and extensions. To establish a deeper connection that involves the category’s full structure, one must turn to Hall algebras. The Hall algebra is constructed directly from the extension structure within the abelian category. This powerful framework reveals a profound relationship: the Hall algebra of a quiver is directly related to a (quantum) deformation of the universal enveloping algebra of the corresponding Lie algebra. This provides a “categorification” of the Lie algebra, where the algebra’s structure is realized through the “categorical” structure of its representations.

This course offers an introduction to the representation theory of quivers and the theory of Hall algebras. The course is designed for master’s level students with a background in linear algebra and basic homological algebra theory. We will begin with a study of quiver representations. This will include the concepts of quivers representations and path algebras. We will explore Gabriel’s theorem, which classifies quivers of finite representation type, and delve into the geometric aspects of quiver representations.

The second part of the course will introduce the theory of derived categories and derived functors, culminating in a presentation of Beilinson’s theorem. This result establishes an explicit relation between the bounded derived category of projective space and the bounded derived category of a quiver with relations.

The third part of the course will then offer a concise introduction to Hall algebras. We will define Hall algebras associated with abelian categories, with particular emphasis on those arising from quivers. This perspective will allow us to connect the two central themes of the course, showing how the combinatorial structure of quiver representations gives rise to rich algebraic structures. In addition, we will discuss Cramer’s theorem, which relates Hall algebras to derived equivalences.

Primary Texts:

Quiver representations:

  • A systematic introduction to the theory of quiver representations from the perspective of associative algebras and their module categories: Assem, Simson, Skowroński, *Elements of Representation Theory of Associative algebras. Volume 1. Techniques of Representation Theory. Cambridge University Press. 2006

  • More accessible introductions to quiver representations: Crawley-Boevey, Lectures on representations of quivers, 1992 and Schiffler, Quiver Representations, Springer, 2014.

Derived categories:

  • Comprehensive overviews: Milicic, Lectures on derived categories, 2014 and Gelfand and Manin, Methods of homological algebra, Springer, 2003.

  • A gentler introduction: Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Science Publications, 2006.

Hall algebras:

  • Schiffmann. Lectures on Hall Algebras. arXiv:math/0611617.

Prerequisites: Linear algebra and basic homological algebra (Ext groups). Familiarity with basic category theory is helpful but not required.

List of possible topics for the final exam:

  • Braid groups and derived equivalences:
    • Y. Sekiya, K. Yamaura. Tilting theoretical approach to moduli spaces over preprojective algebras. published version
    • P. Seidel, R. Thomas. Braid group actions on derived categories of coherent sheaves. published version
    • A. Ishii, H. Uehara. Autoequivalences of derived categories on the minimal resolutions of An-singularities on surfaces. published version
  • Bridgeland’s stability conditions:
    • T. Bridgeland. Stability Conditions on Triangulated Categories. published version
    • T. Bridgeland. Stability Conditions and Kleinian Singularities. arXiv version
  • derived McKay equivalence:
  • derived categories in the topological setting:
    • M. Kashiwara, P. Shapira. Sheaves on manifold.
  • Joyce’s Hall algebra:
    • D. Joyce. Configurations in abelian categories—I: Basic properties and moduli stacks. published version
    • D. Joyce. Configurations in abelian categories. II. Ringel–Hall algebras. published version
    • T. Bridgeland. An introduction to motivic Hall algebras. published version
  • Hall algebras of quivers:
    • Ringel and Green’s theorems on Hall algebras of quivers without edge-loops following Olivier Schiffmann’s lecture notes
    • S.-J. Kang. Ringel–Hall algebra construction of quantum Borcherds–Bozec algebras. published version
  • Hall algebras of curves:
    • I. Burban, O. Schiffmann. On the Hall algebra of an elliptic curve, I published version
    • O. Schiffmann, E. Vasserot. Hall algebras of curves, commuting varieties and Langlands duality. published version
  • Hall algebras and derived categories: