Geometric and homological methods in representation theory
57 January 2022, online
Geometric and homological methods have long been two key approaches in representation theory. Geometric models have been developed in a wide variety of settings and underpin links between representation theory, cluster theory, combinatorics and symplectic geometry. Homological algebra provides powerful techniques to obtain invariants for mathematical objects. The utility of these methods has been recognised well beyond their birthplace in algebraic topology, occurring in representation theory, algebraic and symplectic geometry, Lie theory and mathematical physics, e.g. homological mirror symmetry, among many other subjects.
The purpose of this workshop is to bring together researchers focusing on geometric and homological methods applied in representation theory and related areas, with particular focus on recent advances in geometric models and clustertilting theory in both positive and negative CalabiYau settings.
Programme
All times are in GMT.
Wednesday 5th January
 10:0011:00: Osamu Iyama (Tokyo), "Singularity categories and cluster tilting" (Iyama  slides)
 11:0011:30: Anya Nordskova (Hasselt), "Derived Picard groups of representationfinite symmetric algebras" (Nordskova  slides)
 11:3012:00: Break
 12:0013:00: Joe Grant (University of East Anglia), "On periodic trivial extension algebras"
 14:4515:45: Peter Jørgensen (Aarhus), "The index with respect to a rigid subcategory of a triangulated category"
 15:4516:00: Break
 16:0016:30: Jeremy Brightbill (UC Santa Barbara), "The Nstable Category" (Brightbill  slides)
Thursday 6th January
 10:0011:00: Aaron Chan (Nagoya), "Categorification of curves and quasitriangulations on unpunctured nonorientable marked surfaces" (Chan  slides)
 11:0011:30: Haibo Jin (Cologne), "Cutting surfaces and recollements of gentle algebras" (Jin  slides)
 11:3012:00: Break
 12:0013:00: Alexandra Zvonareva (Stuttgart), "tstructures and thick subcategories of discrete cluster categories" (Zvonareva  slides)
 14:4515:45: Ana Garcia Elsener (Glasgow), "Extensions in Jacobian algebras via punctured skein relations" (GarciaElsener  slides)
 15:4516:00: Break
 16:0017:00: Alex Dugas (University of the Pacific), "Semisimplehomology resolutions of modules" (Dugas  slides)
Friday 7th January
 10:0011:00: Karin Baur (Leeds/Graz), "Cluster structures for Grassmannians" (Baur  slides)
 11:0011:30: Francesca Fedele (Verona), "Generalised cluster categories from nCalabiYau triples" (Fedele  slides)
 11:3012:00: Break
 12:0013:00: Yadira ValdiviesoDíaz (UDLAP, Mexico), "CalderoChapoton functions for orbifolds and snake graphs"
 14:3015:00: Dixy Msapato (Leeds), "The Karoubi envelope of an extriangulated category" (Msapato  slides)
 15:0015:15: Break
 15:1516:15: Steffen König (Stuttgart), "Almost selfinjective algebras"
Abstracts
List of abstracts Accordion

Karin Baur, "Cluster structures for Grassmannians"
The category of maximal CohenMacaulay modules over a certain quotient of a preprojective algebra is known to provide a categorification of Scott's cluster algebra structure of the Grassmannian Gr(k,n), by work of Jensen, King and Su. This category is of infinite type in general, with finite types corresponding to the ADE Dynkin diagrams. We study this category in the infinite types. It is known to be tauperiodic and we show that it is a tubular category. This makes it a very interesting family of categories of infinite types and allows us to characterise some of the small rank modules. Joint work with Dusko Bogdanic, Jianrong Li and with Ana Garcia Elsener.

Jeremy Brightbill, "The Nstable category"
A wellknown theorem of Buchweitz provides equivalences between three categories: the stable category of projective modules over a Gorenstein algebra, the homotopy category of acyclic projective complexes, and the singularity category. The latter two categories generalize easily to the setting of Ncomplexes; we identify a candidate for the "Nstable" category and adapt Buchweitz's theorem to the setting of Ncomplexes over a Frobenius exact abelian category. We will also discuss how the Nstable category can be used to construct fractional CalabiYau categories. Prior familiarity with Ncomplexes will not be assumed. Based on a joint work with Vanessa Miemietz.

Aaron Chan, "Categorification of curves and quasitriangulations on unpunctured nonorientable marked surfaces"
To an unpunctured nonorientable marked surface, Dupont and Palesi introduced a commutative algebra analogous to the cluster algebras associated to orientable surfaces of FominShapiroThruston. The role of clusters in such an algebra is given by quasitriangulations, that is, maximal collections of arcs and onesided simple closed curves. In the case of a triangulation, we can associate it with a quiver with potential (even, with a gentle quiver). This quiver comes with an antiinvolution that allows us to categorify curves on the surface by indecomposable symmetric representations  in the sense of DerksenWeyman and BoosIrelli Cerulli. As a consequence, we can categorify quasitriangulation in a similar way as the orientable case where triangulations are categorified by clustertilting objects. If time permits, I will also talk about our progress on finding the cluster character for the algebras of DupontPalesi. This talk is based on a joint work in progress with Veronique BazierMatte and Kayla Wright.

Alex Dugas, "Semisimplehomology resolutions of modules"
Let A be a noetherian ring. Of course, when A has infinite global dimension, not every Amodule will have a finite projective resolution. We propose thus to study finite length projective resolutions which are not exact, but have relatively small homology. To this end, we say that an Amodule M admits a semisimplehomology (ssh) resolution if there exists a complex of finitelygenerated projectives
0 → P_{n} → ... → P_{0} → 0
such that H_{0}(P_{∙}) ≅ M and H_{i}(P_{∙}) is semisimple for all i > 0. Such resolutions were previously studied over group algebras by Benson and Carlson [1], where it is shown that every finitelygenerated kGmodule can be so resolved (here, G is a finite group and k a field of characteristic p dividing G). We extend this result in several directions, showing that sshresolutions exist for all finitelength Amodules whenever A is commutative; a cohomologically noetherian finitedimensional algebra; or selfinjective of Loewy length 3. However, beyond these cases it remains unclear when sshresolutions exist, even for simple modules, and what form they take when they do exist, leading to many interesting open questions.
[1] D. J. Benson and J. F. Carlson. Complexity and multiple complexes. Math. Z. 195 (1987), no. 2, 221238.

Francesca Fedele, "Generalised cluster categories from nCalabiYau triples"
The original definition of cluster algebras has been categorified and generalised in several ways over the past 20 years. In this talk, we focus on Iyama and Yang’s generalised cluster categories T/N coming from nCalabiYau triples. In such a construction, T is a triangulated category with a certain triangulated subcategory N and silting subcategory M. Using a different approach from the original one, we give a deeper understanding of T/N and reprove it is a generalised cluster category. In order to do so, we use more classic homological tools such as limits, colimits, direct and inverse systems. A similar approach can also be adopted to study negative cluster categories, coming from (n)CalabiYau triples. In this case, M is a simple minded collection instead of a silting subcategory.

Ana Garcia Elsener, "Extensions in Jacobian algebras via punctured skein relations"
Given a (nonnecessarily gentle nor skew gentle) Jacobian algebra arising from a marked surface and a local configuration resembling the punctured disk, we show how to find nonsplit extensions over this configuration using the tagged arcs and skein relations. These tagged arcs and geometric moves were used in a commutative setting to find bases for cluster algebras.

Joe Grant, "On periodic trivial extension algebras"
Consider a finitedimensional algebra with finite global dimension. A recent result of Chan, Darpö, Iyama, and Marczinzik states that this algebra is fractionally CalabiYau if and only if its trivial extension algebra is periodic. I will present some work towards finding a different proof of this result.

Osamu Iyama, "Singularity categories and cluster tilting"
There are a number of important dCalabiYau triangulated categories which have cluster tilting objects if d is positive, and simpleminded systems if d is negative. In this talk, I will explain families of such CalabiYau triangulated categories given by the singularity categories of Gorenstein isolated singularities for positive d, and nonpositive Gorenstein dg algebras for negative d. I will also explain a general method how to prove that such singularity categories are equivalent to cluster categories in the case d is positive. A part of this talk is based on joint works with Norihiro Hanihara and Haibo Jin.

Haibo Jin, "Cutting surfaces and recollements of gentle algebras"
It is known that (graded) gentle algebras are in bijection with admissible dissections of (graded) marked oriented surfaces. For a graded marked surface with an admissible dissection corresponding to a graded gentle algebra A, we show that there is a recollement of derived categories D(A) by D(B) and D(C), where B, C are graded gentle algebras obtained by cutting the surface of A along the given dissection and the dual dissection respectively. Furthermore, if all three algebras are homologically smooth and proper, this recollement restricts to a recollement of partially wrapped Fukuya categories.
My talk is divided into two parts. I will start with a more general algebraic setting, that is, I will consider recollements for graded quadratic monomial algebras. Then I will apply the algebraic results to the geometric model. This is joint work with Wen Chang and Sibylle Schroll.

Peter Jørgensen, "The index with respect to a rigid subcategory of a triangulated category"
If A is a finitedimensional algebra over a field, M a finitely generated Amodule, and P_{1} → P_{0} → M → 0 is a minimal projective presentation, then the difference [P_{0}][P_{1}] in the Grothendieck group K_{0}(proj A) is an early incarnation of the socalled index. It was used by Auslander and Reiten to investigate when M is determined by its composition factors. Later, the index was imported into the theory of 2CalabiYau categories. It was used by Dehy, Keller, Palu, Plamondon, and others to categorify gvectors and to provide a key factor in the CalderoChapoton formula.
The index in 2CalabiYau categories was recently interpreted by Padrol, Palu, Pilaud, and Plamondon in terms of extriangulated categories. In joint work with Amit Shah, we generalise the extriangulated viewpoint and define the index of an object of a triangulated category with respect to a rigid subcategory. We prove that on triangles, this version of the index is additive with an error term. This generalises the crucial property of previous versions of the index.

Steffen Koenig, "Almost selfinjective algebras"
By definition, all isomorphism classes of indecomposable projective modules over a selfinjective algebra consist of injective modules, and there is a faithful projectiveinjective module. Derived equivalences between selfinjective algebras induce stable equivalences of Morita type and preserve global dimension and dominant dimension.
By definition, at most one isomorphism class of indecomposable projective modules over an almost selfinjective algebra does not consist of injective modules, and there is a faithful strongly projectiveinjective module. Do derived equivalences between almost selfinjective algebras induce stable equivalences and do they preserve some homological dimensions?
(Joint work with Ming Fang and Wei Hu.)

Dixy Msapato, "The Karoubi envelope of an extriangulated category"
An additive category is idempotent complete if every idempotent morphism has a kernel. Idempotent completeness is often a desirable property in homological algebra, for example; the KrullRemakSchmidt property on additive categories is equivalent to idempotent completeness with the property that the endomorphism ring of every object is semiperfect [Krause 2015].
A related notion is that of being weakly idempotent complete. This is when every retraction has a kernel. This also has many interesting consequences, especially concerning exact model structures. For example, for weakly idempotent exact categories, there is a correspondence between exact model structures and complete cotorsion pairs [Gillespie 2011].
For an additive category C, a closely related idempotent complete category is its Karoubi envelope C’ (or idempotent completion) . It’s been shown that if C is also a triangulated category or exact category, the Karoubi Envelope C’ is also triangulated or exact respectively [Balmer and Schlichting 2001] and [ Buhler 2010]. In this talk we will talk about a unification of these two results, that is to say we will show that if C is an extriangulated category then, so is C’. We will do so by constructing an explicit extriangulated structure on C', which was not originally given in the triangulated and exact cases. However, it will turn out that our explicit description of the extriangulated structure is equivalent to the implicit description given in the triangulated and exact cases. Consequently, we will show that C^, the weak idempotent completion of C is an extension closed subcategory of the Karoubi envelope C', and hence is also an extriangulated category.

Anya Nordskova, "Derived Picard groups of representationfinite symmetric algebras"
I will talk about my recent work on computing the derived Picard groups of symmetric representationfinite algebras of type D. What we obtain is an explicit description via generators and relations. In particular, we prove that these groups are generated by spherical twists along a collection of 0spherical objects, the shift and, roughly speaking, outer automorphisms. One of the key ingredients in the proof is the faithfulness of braid group actions by spherical twists along ADE configurations of 0spherical objects. Another part of the strategy is based on the fact that symmetric representationfinite algebras are tiltingconnected. To apply this result we in particular develop a combinatorialgeometric model for silting mutations in type D, generalising the classical concepts of Brauer trees and Bauer moves. If time permits, I will discuss possible directions in which the result might be extended to a more general categorical context.

Yadira Valdivieso, "CalderoChapoton functions for orbifolds and snake graphs"
Caldero–Chapoton functions define a relation between indecomposable objects of a module category and elements in a cluster algebra. For some families of cluster algebras, for example, the acyclic ones, CalderoChapoton functions form a basis of the algebras. In this talk, we describe CalderoChapoton functions for Jacobian algebras associated with orbifolds with orbifolds points of order three using snake graphs. This is joint work with Esther Banaian.

Alexandra Zvonareva, "tstructures and thick subcategories of discrete cluster categories"
In my talk, I will try to explain the classification of tstructures and thick subcategories in discrete cluster categories C(Z) of type A, introduced by Igusa and Todorov. These cluster categories have a nice geometric model, which is quite useful for this classification. It turns out that both tstructures and thick subcategories can be classified by certain variations of noncrossing partitions. Moreover, the set of all tstructures on C(Z) is a lattice under inclusion of aisles, with meet given by their intersection. The coaisle of a given tstructure in C(Z) can be constructed from its aisle using an analogue of the Kreweras complement for noncrossing partitions, this operation can be easily interpreted on the geometric model. Approximation triangles of objects in C(Z) can also be constructed geometrically. This is based on a joint work with Sira Gratz.
Recordings
Osamu Iyama
Singularity categories and cluster tilting
Anya Nordskova
Derived Picard groups of representationfinite symmetric algebras
Peter Jørgensen
The index with respect to a rigid subcategory of a triangulated category
Jeremy Brightbill
The Nstable Category
Aaron Chan
Categorification of curves and quasitriangulations on unpunctured nonorientable marked surfaces
Haibo Jin
Cutting surfaces and recollements of gentle algebras
Alexandra Zvonareva
tstructures and thick subcategories of discrete cluster categories
Ana Garcia Elsener
Extensions in Jacobian algebras via punctured skein relations
Alex Dugas
Semisimplehomology resolutions of modules
Karin Baur
Cluster structures for Grassmannians
Francesca Fedele
Generalised cluster categories from nCalabiYau triples
Yadira ValdiviesoDíaz
CalderoChapoton functions for orbifolds and snake graphs
Dixy Msapato
The Karoubi envelope of an extriangulated category
Steffen König
Almost selfinjective algebras
Organisation
The meeting will be held online and details sent to registered participants closer to the meeting.
The organisers are Raquel Coelho Simões, Jan Grabowski and David Pauksztello.
We are grateful to the European Union’s Horizon 2020 research and innovation programme for funding under grant agreement No. 838706.