Mathematisches Institut der Universität Bonn

Graduate Seminar on Hall Algebras (WS 2019/20)

Dozentin: Prof. Dr. Catharina Stroppel

Termin

Thursdays 16-18.00 , SR 1.008

(note that the time changed due to overlapping events)

The seminar will be about several aspects of Hall algebras. These are algebras which arise ubiquitous in classical and modern representation theory.

A few sentences about what Hall algebras are (good for)

Given a finite field k=Fq and a k-linear category C with appropriate finiteness conditions one can assign a Hall algebra. It has a basis given by the isomorphism classes of objects in C. The multiplication of two basis vectors is given by counting extensions between the objects representing the basis vectors. The resulting structure constants of the algebra depend on the cardinality q of the field. It turns however out that they depend in a polynomial way on q, so that one can just treat q as in indeterminate. The result is the (generic) Hall algebra defined over rational functions C(q).

One basic example which is already important and interesting is when one takes a Dynkin diagram of a semisimple complex Lie algebra g, views it as a quiver Q by putting some orientation and then considers the category of finite dimensional representations of Q over k. The resulting generic Hall algebra is (by a theorem of Ringel) isomorphic to Uq(n). Here Uq(n) is a certain C(q)-deformation of the universal enveloping algebra of the positive part of g in a triangular decomposition. In particular one might expect that Hall algebras are not only algebras, but in fact Hopf algebras which is indeed true thanks to a nontrivial theorem of Green.

Hall algebras are directly connected with the moduli problem of classifying objects in a category up to isomorphism, but also directly to enumerative geometry. One of the most famous classification problems is the Jordan problem which describes endomorphisms of a fixed finite dimensional vector space up to conjugation. The corresponding Hall algebra is particularly easy and concrete. It is just a ring of symmetric polynomials. The beautiful feature of Hall algebras is that it allows to interpret most of the known bases of this ring via certain classes of modules. These are ideas which are important in modern representation theory and the basic principal of a general concept called nowadays categorification.

Goal of the seminar
The goal of the seminar is to get familiar with the definition of Hall algebras with several applications in mind. The arguments will be mostly algebraic using often also some (basic) categorical language. The seminar will show several ideas and concepts which are crucial for modern representation theory.

Main references

The main references for the seminar will be Schiffmann: Lectures on Hall algebras
Dyckerhoff: Higher categorical aspects of Hall algebras

Prerequisites: A solid knowledge in algebra is absolutely necessary (Algebra I and Algebra II). Knowledge about or some experience with the representation theory of quivers and basic definitions on semisimple Lie algebras will be helpful to understand the main examples and some of the motivations. The basic definitions and first consequences should however be sufficient. Familiarity with standard homological algebra concepts (Exts and projectives) and basic categorical notions will be assumed.

The seminar should supplement well some of the aspects treated in the representation theory II class and in Oberdieck's selected topics on Enumerative geometry. Neither of them is however required. It is rather the converse: the seminar might provide some deeper understanding of some aspects of the lecturer courses.