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, following the approach of Schiffler’s book (see bibliography below). This will include the concepts of quivers representations, path algebras, and partially the Auslander-Reiten theory. 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 transition to the theory of Hall algebras, as presented in Schiffmann’s lecture notes (see bibliography below). We will define Hall algebras associated with abelian categories, with a particular focus on the Hall algebras of quivers. This will allow us to connect the two main topics of the course, revealing how the combinatorial structure of quiver representations gives rise to rich algebraic structures.

Primary Texts:

• R. Schiffler. Quiver Representations. Springer (2014).

• O. 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.