Activities in Leeds

Throughout the Programme Grant, we will organise several workshops, seminars and research activities in Leeds.

Programme Grant Workshops

• The first Workshop: Combinatorial Representation Theory in Leeds, organised by Esentepe, Faber, Fedele and Parker took place 10-14 July 2023  at the University of  Leeds. See the webpage of the workshop for further information and registration.

Algebra Seminar

NB: from January 2025, the information on the Algebra Seminar, organised by Francesca Fedele and Ilaria Colazzo, will only be posted in the University of Leeds seminars website, click here for all the info.

The Algebra Seminar normally takes place during term-time at 3pm on Tuesdays in the MALL, School of Mathematics, University of Leeds.

Click here for information on the 2023/24 series (organised by Emine Yıldırım and Francesca Fedele). 

In this page, you can find the information on the seminar for the Autumn term 2024-25, organised by Francesca Fedele and Ilaria Colazzo (and Cristina Palmer-Anghel for the beginning of term) . See below the table for abstracts.

You can also open the table below in a new tab by clicking here.

Abstracts

• Senne Trappeniers (Vrije Universiteit Brussel): Skew braces and related Algebraic structures

Skew braces are algebraic structures that naturally arise in the study of solutions of the set-theoretic Yang--Baxter equation, which is equivalent with the braid equation. In this talk however, we will focus on its relation with other algebraic structures, namely Hopf--Galois structures and post-Lie rings.  In Hopf--Galois theory, skew braces play the same role that groups play in classical Galois theory. We will describe precisely how one obtains a Hopf--Galois structure from a skew brace, building upon a fundamental result by Greither-Pareigis and later work by Childs and Byott. This is based on joint work with Lorenzo Stefanello.  Post-Lie algebras appear, among others, naturally in the study of (simply transitive) affine actions of Lie groups, already hinting at a more general connection with skew braces. Indeed, results by Rump and Smoktunowicz give concrete settings where a (partial) correspondence exists between braces and pre-Lie rings. We will illustrate how skew braces are the natural group theoretic counterpart of post-Lie rings and how this perspective can be used to obtain a Lazard correspondence between the two structures.

• Atabey Kaygun (Istanbul Technical University): A Homotopical Dold-Kan for Crossed-Simplicial Groups

There is an equivalence between the category of simplicial abelian groups and the category of differential graded abelian groups called the Dold-Kan equivalence. There is also a class of curious objects called Crossed-Simplicial Groups defined by a distributive law between a collection of groups indexed by the natural numbers and the simplicial category \Delta. There have been attempts to extend the Dold-Kan to crossed-simplicial setting by explicitly constructing extensions of the differential graded side. But these are few and far between. In this talk, I'll start by a new but a weakened analogue of the Dold-Kan by using certain induction and restriction functors, and by passing to the homotopy categories on both sides. Then show that this homotopical version of the equivalence extends to the crossed-simplicial setting.

• Leandro Vendramin (Vrije Universiteit Brussel): Nichols Algebras over groups

Nichols algebras appear in several areas of mathematics, from Hopf algebras and quantum groups to Schubert calculus and conformal field theories. In this talk, I will review the main problems related to Nichols algebras and discuss some recent classification theorems.

• Rudolf Tange (University of Leeds): Invariants in divided power algebras

This is based on the equally named paper which is on arxiv and to appear in Journal of Algebra. I will discuss the connection with the representations of reductive groups and Lie algebras: the hyper algebra, the restricted enveloping algebra and their centres. Then I will move to divided power algebras, truncated polynomial algebras and their invariants. I will discuss some results (GL_n only) from my paper, a conjecture concerning the so-called “restriction property” and the connection with “special symmetrization map” from Okounkov-Olshanskii. If time allows, I will mention extensions of the above results to several matrices and vectors and covectors.

• Ezgi Kantarcı Oğuz (Galatasaray University) : A poset model for q-deformed Markov Numbers

The positive integers that come up in the solutions of the Diophantine equation $x^2+y^2+z^2=3xyz$ are called Markov numbers. They play an important role in the theory of rational expansions and come with a 100+ year old open problem called the Uniqueness conjecture. Recently discovered connections to cluster algebras have revitalized the theory, leading to generalizations and deformations motivated by the cluster model. In this talk, we will use oriented posets to construct a combinatorial model for q-deformed Markov numbers. We will also discuss future directions and further avenues of research.

• Valentine Soto (Université Grenoble Alpes): Generalized Kauer moves and derived equivalences of skew Brauer graph algebras.

Brauer graph algeras are finite dimensional algebras constructed from the combinatorial data of a graph called a Brauer graph. Kauer proved that derived equivalences of Brauer graph algebras can be obtained from the move of one edge in the corresponding Brauer graph. Moreover, this derived equivalence is entirely described thanks to a tilting object which can be interpreted in terms of silting mutation. In this talk, I will be interested in skew Brauer graph algebras which generalize the class of Brauer graph algebras. These algebras are constructed from the combinatorial data of a Brauer graph where some edges might be "degenerate". I will explain how Kauer's results can be generalized for the move of multiple edges and to the case of skew Brauer graph algebras.

• Roberto Civino (Università degli studi dell'Aquila): Unrefinable partitions into distinct parts

Unrefinable partitions, arising quite unexpectedly in a combinatorial problem in group theory, represent a special subset of integer partitions into distinct parts, constrained by an additional additive relationship between the parts. Despite being a natural combinatorial object, they remain relatively unexplored in the literature, with only a few known properties and results.

In this talk, we explore the foundational aspects of unrefinable partitions, showing some of their initial properties. We will present an algorithm designed to efficiently test for unrefinability in a given partition. By establishing a bound on the largest part in such partitions, we introduce the concept of maximal unrefinable partitions, a subclass with its own distinctive structure. We will show how to count such maximal unrefinable partitions using explicit bijections, providing a clearer understanding of their combinatorial structure.

• Alexandre Mikhailov (University of Leeds): Commutative Poisson algebras from deformations of noncommutative algebras and non-Abelian Hamiltonian systems.

See here for the abstract.

• Arne Van Antwerpen (University of Ghent): On groups and algebras related to the Yang-Baxter equation.

See here for the abstract.

• Alice Dell'Arciprete (University of York): Quiver presentations for Hecke categories and KLR algebras.

We discuss the algebraic structure of KLR algebras by way of the diagrammatic Hecke categories of maximal parabolics of finite symmetric groups. Combinatorics (in the shape of Dyck tableaux) plays a huge role in understanding the structure of these algebras. Instead of looking only at the sets of Dyck tableaux (which enumerate the q-decomposition numbers) we look at the relationships for passing between these Dyck tableaux. In fact, this “meta-Kazhdan-Lusztig combinatorics” is sufficiently rich as to completely determine the Ext-quiver and relations presentation of these algebras.

• Marina Godinho (University of Glasgow): A twist on ring morphisms.

Abstract: In this talk, I will show that a ring morphism p: A ⟶B satisfying certain mild assumptions induces a derived endomorphism of A and a derived endomorphism of B, which are closely related. In fact, the derived endomorphism of A is the twist around the restriction of scalars functor, and the derived endomorphism of B is the corresponding cotwist. These endomorphisms are autoequivalences in certain settings, one of which is that of Frobenius exact categories. More precisely, assume that A is the endomorphism algebra of an object in a Frobenius exact category satisfying mild assumptions and B is the corresponding stable endomorphism algebra. Then, if B is "n-relatively spherical", I will show that both the twist and cotwist are equivalences. In fact, when B is finite dimensional,  "3-relatively spherical" is equivalent to self-injective, and the cotwist turns out to be a shift of the Nakayama autoequivalence of B.  This technology can be used to construct new derived autoequivalences for very singular varieties.

• Geoffrey Janssens (UCLouvain and VUBrussel): Bridging representation theories through cluster algebras.

Abstract: see below or here for a pdf.

Given a (Dynkin) quiver Q one can associate both a simple Lie algebra 𝔤 and the category of representations Rep(Q) of Q. Early on it was realised that both associated objects are related, as for example beautifully illustrated by Gabriel's theorem. In this talk we will consider two associated categories of representations: (i) (some quotient of) the derived category D^b(Rep(Q)) and (ii) the finite dimensional representations of the quantum loop algebra U_q(L𝔤). Although both look quit differently, we will delve into wished to be understood ties. The presented story will be one of categorifications of a common algebra with a rich combinatorial structure: a cluster algebra. The category (i) yields a so-called additive categorification and (ii) a monoidal one.  In the first half of the talk we will give a gentle and minimalistic introduction to the various objects and concepts mentioned. In the second half we will give an intuitive overview of recent conjectural connections and then finish by (very briefly) mentioning some ongoing contributions.

Internal Algebra Seminar & Research Discussions

On Wednesdays, typically biweekly, we have our group research discussions to be updated on what everyone is up to. This was organised in the academic year 2022-23 by Francesca Fedele, in the first half of the academic year 2023-24 by Özgür Esentepe and after that (and currently) by Amit Hazi.

 Sometimes, instead of the discussions we have an extra seminar or a series of seminars for few weeks. The following is a list of some of these talks.

• Amit Hazi (8/05/2024, 22/05/2024, 29/05/24)

• Michelle Daher (1/05/24, 15/05/24): Smoothing theory

• Michael Tsironis, Vrije Universiteit Amsterdam, Netherlands (22/11/23): Skein relations for punctured surfaces

• Juan Pablo Maldonado, Mar del Plata, Argentina (08/02/23): Determinants arising from frieze patterns (Slides)

• Emine Yıldırım (19/10/22, 26/10/22, 02/11/22): In the landscape of algebra and combinatorics: Cluster Categories, Associahedra, The Coxeter Transformation. (Seminar notes)

• Francesca Fedele (09/11/22, 16/11/22, 23/11/22): An introduction to higher homological algebra. (Seminar notes)

• Özgür Esentepe (30/11/22, 11/01/22, 25/01/23): Cluster tilting for Cohen-Macaulay modules (Seminar notes)