Köln Algebra and Representation Theory Seminar

Abstract: An associahedron is a polytope arising from combinatorics of Catalan-type objects (for example, from a collection of all triangulations of a given polygon). Fomin and Zelevinsky found a way to construct the same combinatorial structure from considering the Coxeter group of type A_n. This allowed them to define a generalized associahedron for every finite reflection group. For generalized associahedra arising from crystallographic reflection groups, it was also shown that they can be realized as polytopes. We use the folding technique to construct polytopal realisations of generalized associahedra for all non-simply-laced root systems, including non-crystallographic ones. This is a joint work with Pavel Tumarkin and Emine Yildrim.

Abstract: With (almost) every finite mutation class of quivers we can associate two groups: an extended affine Weyl group of type A or D, and a certain quotient of a Coxeter group which behaves nicely with respect to mutations. I will discuss a connection between these groups and a (conjectural) characterization of mutation-finite quivers in terms of positive semidefinite symmetric matrices. The talk is based on joint works (some still in progress) with Anna Felikson, John Lawson and Michael Shapiro.

Abstract: I will discuss a replacement for homotopy cardinality in situations where it is a priori ill-defined, including Z/2-graded dg-categories. A key ingredient are Calabi-Yau structures and their relative generalizations. As an application we obtain a Hall algebra for many pre-triangulated dg-categories for which it was previously undefined. Another application is the proof of a conjecture of Ng-Rutherford-Shende-Sivek expressing the ruling polynomial of a Z/2m-graded Legendrian knot (which is part of the HOMFLY polynomial if m=1) in terms of the homotopy cardinality of its augmentation category. All this is joint work with Mikhail Gorsky, arxiv:2409.10154. 

Abstract: In a Hom-finite 2-Calabi-Yau category with a cluster tilting object, there are various cluster multiplication formulas, beside the one corresponding to the exchange relation in cluster algebras. For example, the cluster multiplication formulas in higher Ext-dimensions given by Chen-Ding-Zhang in the cluster category of an acyclic quiver; the refined cluster multiplication formula given by Keller-Plamondon-Qin. In this talk, I will introduce the motivation for defining motivic weighted cluster characters, and present motivic cluster multiplication formulas, also the refined version. As an application, we can use the motivic cluster multiplication formulas to recover all cluster multiplication formulas above, after taking suitable weighted functions. 

Abstract: I will discuss the connection between projective presentations of maximal rank and generically $\tau$-regular components of module varieties. Then I will present some classification results for generically $\tau$-regular components. This is joint work with Grzegorz Bobinski.

Abstract: Differential graded algebras were defined in 1954 by Cartan and proved to be highly useful in connection with homological algebra methods in various subjects. However, the ring theory of these algebras remained largely unexplored until very recently. I propose a notion of a differential graded division algebra. After having given some examples where they occur naturally, I show the link to the more classical notion of graded division algebras given by Nastacescu and van Oystaen. Finally, I give a complete classification of differential graded division algebras.

Abstract: Chuang and Lazarev introduced the concept of rank functions on triangulated categories as a generalisation of classical work by Cohn and Schofield on Sylvester rank functions. In this talk, we propose a generalisation of this notion to the broader framework of higher homological algebra. 

Abstract: Highest weight categories are an abstraction of the representation theory of semisimple Lie algebras introduced by Cline, Parshall and Scott in the late 1980s.  There are by now many characterisations of when an abelian category is highest weight, but most are hard to verify in practice. I’ll discuss two new criteria - one numerical in terms of the Grothendieck group, and one in terms of Bridegland stability conditions - which are easier to verify. I’ll also relate these to a criterion by Green and Schroll for when modules over a monomial algebra are highest weight. This is joint work with Alessio Cipriani (arXiv:241013728). 

Abstract: Noncommutative plane curves are plane curves on noncommutative projective planes, defined by a homogeneous central element in a 3-dimensional Artin--Schelter regular algebra. They have been studied and classified in degree 2, and some examples have been considered in degree 3, but a general theory does not exist yet. Introducing the notion of central curves for orders on surfaces and using the fact that the noncommutative projective planes in question are finite over their centers, we can translate the algebraic questions into geometric questions and solve them using geometric methods. This also gives further insight into the dictionary between noncommutative algebra and algebraic stacks. This is joint work with Thilo Baumann and Okke van Garderen.

Abstract: n-representation infinite (= n-RI) algebras are an analog of hereditary representation infinite algebras in Iyama's higher Auslander-Reiten theory. An n-RI algebra is of type

à if its higher preprojective algebra is a skew-group algebra R*G, where G < SL_{n+1} is a finite abelian group acting on the polynomial ring R in n+1 variables. However, there is no perfect correspondence between n-RI algebras of type à and abelian subgroups G, so describing all n-RI of type à is a non-trivial problem.

The theme of this talk is a solution to this problem. First, I will explain how to rephrase the problem in terms of combinatorics on the McKay quiver. The n-RI algebras naturally fall into higher tilting classes, and we will describe those classes as certain lattice points in a cone. Finally, I want to show how every tilting class can be given the structure of a finite distributive lattice.

Time permitting, I will explain why a lattice cone should be expected from the perspective of toric geometry, and what all of this has to do with the associated quotient singularity.

This is based on joint work with Oleksandra Gasanova. 

Abstract: 'Simple-minded objects' are generalisations of simple modules. They satisfy Schur's lemma and a version of the Jordan-Hölder theorem, depending on context. In this talk we will consider simple-minded systems (SMSs), which were introduced  by Koenig-Liu as an abstraction of non-projective simple modules in stable module categories, and simple-minded collections (SMCs), introduced by Koenig-Yang as an abstraction of the simple objects in hearts of t-structures.

We will discuss when mutation of an SMS/SMC is possible and gives another SMS/SMC. This provides a conceptual understanding and unifies results by Dugas, Jorgensen and Koenig-Yang. The main tools we use are simple-minded reduction, which allows us to see that SMC mutation is compatible with SMS mutation via a singularity category construction, and simple HRS tilts of length hearts, which is of particular importance in the study of stability conditions. This talk is based on joint work with Nathan Broomhead, David Pauksztello and Jon Woolf.

Abstract: Motivated by the theory of non-commutative resolutions and the results on Auslander correspondence, we study the category of reflexive modules over (commutative or non-commutative) Noetherian rings. One well-established sufficient condition for this category to behave well is that the ring should be (commutative) normal. We will explain that these nice behaviors are governed by the Auslander-type conditions which are some requirements on the minimal injective resolution of the ring. We will also discuss a Morita-type result characterizing the category of reflexive modules.

Abstract: We show that if an abelian category is hereditary, then its m-extended category is (3m+1)-abelian in the sense of Jasso.  Under weak additional assumptions, we also prove the converse.  There is also a higher version, which starts with a n-cluster tilting subcategory over an n-hereditary algebra.

Abstract: The space of representations from a surface group into PSL(2,R) has the structure of a classic cluster algebra. This structure has implications to positivity and gives nice coordinate charts of the space. More recently Berenstein and Retakh developed a noncommutative cluster algebra associated to a surface. This structure parameterizes “maximal representations” into the symplectic group over a noncommutative ring with involution. We generalize this story further to “polygonal cluster algebras” which have points in algebras with 2 involutions like Clifford algebras. In this way we are able to parametrize the next class of  theta positive representations into Spin(p,q). This work is joint with D. Kaufman, M. Niemeyer, and A. Wienhard.