VUB–Leeds Algebra School (2026)

Combinatorial surface models for cluster algebras and cluster categories

Instructor: Karin Baur

Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras and cluster categories, gentle algebras, etc. In this course, I will show how triangulations give rise to cluster algebras and to cluster categories. We will also study the combinatorial approach to Grassmannian cluster algebras and cluster categories.

Time permitted, we will also discuss how tilings of surfaces give gentle algebras and module categories.

Symmetric tensor categories

Instructor: Pavel Etingof

Lecture 1: Algebra and representation theory without vector spaces.

A modern view of representation theory is that it is a study not just of individual representations (say, finite dimensional representations of an affine group or, more generally, supergroup scheme \(G\) over an algebraically closed field \(k\)) but also of the category \(Rep(G)\) formed by them. The properties of \(Rep(G)\) can be summarized by saying that it is a symmetric tensor category (shortly, STC) which uniquely determines \(G\). A STC is a natural home for studying any kind of linear algebraic structures (commutative algebras, Lie algebras, Hopf algebras, modules over them, etc.); for instance, doing so in \(Rep(G)\) amounts to studying such structures with a \(G\)-symmetry. It is therefore natural to ask: does the study of STC reduce to group representation theory, or is it more general? In other words, do there exist STC other than \(Rep(G)\)? If so, this would be interesting, since algebra in such STC would be a new kind of algebra, one "without vector spaces". Luckily, the answer turns out to be ``yes". I will discuss examples in characteristic zero and \(p>0\), and also Deligne's theorem, which puts restrictions on the kind of examples one can have.

Lecture 2: Representation theory in non-integral rank.

Examples of symmetric tensor categories over complex numbers which are not representation categories of supergroups were given by Deligne-Milne in 1981. These very interesting categories are interpolations of representation categories of classical groups \(GL(n)\), \(O(n)\), \(Sp(n)\) to arbitrary complex values of n. Deligne later generalized them to symmetric groups and also to characteristic $p$, where, somewhat unexpectedly, one needs to interpolate n to $p$-adic integer values rather than elements of the ground field. I will review some of the recent results on these categories and discuss algebra and representation theory in them.

Lecture 3. Symmetric tensor categories of moderate growth and modular representation theory.

Deligne categories discussed in Lecture 2 violate an obvious necessary condition for a symmetric tensor category (STC) to have any realization by finite dimensional vector spaces (and in particular to be of the form $Rep(G)$): for each object $X$ the length of the n-th tensor power of $X$ grows at most exponentially with $n$. We call this property "moderate growth". So it is natural to ask if there exist STC of moderate growth other than $Rep(G)$. In characteristic zero, the negative answer is given by the remarkable theorem of Deligne (2002), discussed in Lecture 1. Namely Deligne's theorem says that a STC of moderate growth can always be realized in supervector spaces. However, in characteristic $p$ the situation is much more interesting. Namely, Deligne's theorem is known to fail in any characteristic $p>0$. The simplest exotic symmetric tensor category of moderate growth (i.e., not of the form $Rep(G)$) for $p>3$ is the semisimplification of the category of representations of $Z/p$, called the Verlinde category. For example, for $p=5$, this category has an object $X$ such that $X^2=X+1$, so $X$ cannot be realized by a vector space (as its dimension would have to equal the golden ratio). I will discuss some aspects of algebra in these categories, in particular failure of the PBW theorem for Lie algebras (and how to fix it) and a generalization of Deligne's theorem in characteristic p obtained by Coulembier, Ostrik and myself, with applications to modular representation theory. I will also discuss a family of non-semisimple exotic categories in characteristic p constructed in my joint work with Dave Benson and Victor Ostrik, and their relation to the representation theory of groups $(Z/p)^n$ over a field of characteristic $p$.

Lecture 4. Oligomorphic groups and tensor categories.

I will try to give a brief introduction to the work of Harman and Snowden about symmetric tensor categories attached to oligomorphic groups, in particular discuss the Delannoy category.

Algebraic and categorical structures for modular functors and quantum codes

Instructor: Christoph Schweigert

We explain how the Kitaev model assigns, to a finite-dimensional semisimple Hopf algebra $H$ and a surface equipped with a triangulation, a quantum code. The protected subspaces of this code carry representations of the mapping class group of the surface, which can be used to construct quantum gates. This structure naturally organizes in terms of a modular functor. Depending on time, we will extend this discussion to modular functors with defects. We will sketch some of their applications in representation theory and mathematical physics.

Homology of racks and quandles

Instructor: Markus Szymik

This series of lectures will delve into the homological aspects of racks and quandles, starting with the foundational concepts. We will define the canonical chain complexes and explore their low-dimensional homology and cohomology. By working through some examples by hand, we will uncover the strengths and limitations of the direct approach. We will then transition into a more conceptual and flexible framework inspired by Quillen's general theory. We explain a proof that this leads to the same result and show that it allows us to compute the entire integral homology of all permutation racks, for example. Throughout the exposition, we will compare the homology theories of racks and quandles with those of groups and group actions, illuminating their similarities and differences. This field is lush with unanswered questions and untrodden challenges. As we progress, we will highlight as many of them as possible to engage the audience with the theory and enable them to contribute to ongoing research.