Köln Algebra and Representation Theory Seminar

Abstract: In this talk we introduce homological notions -- such as finite global dimension and gorensteinness -- for compactly generated triangulated categories.  We observe that the latter generalise classical notions from homological algebra and study those same attributes for other triangulated categories of interest in representation theory, such as the homotopy category of injectives of an Artin algebra or the derived category of a dg algebra. This is based on joint work with C. Psaroudakis and J. Vitória.

Abstract: In this talk, we study the notion of Gorenstein projective dg modules over connective dg algebras, recently introduced by Zhenhui Ding in his ongoing Ph.D thesis. Motivated by Yilin Wu’s Higgs category in cluster theory, we discuss the Frobenius Morita type theorem in the context of exact dg categories. We extend known equivalence between Higgs categories concentrated in degree zero and Gorenstein projective modules over the boundary algebra, to the case of proper relative Ginzburg dg algebras with non vanishing cohomology in multiple degrees. Additionally, we will investigate an important class of examples  given by the relative Ginzburg dg algebras associated with principally framed acyclic quivers. This is based on joint work with Zhenhui Ding.

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Abstract: We study singularity categories of Gorenstein algebras. There are many such algebras whose singularity categories are known to be Calabi-Yau, for example, preprojective algebras, their quotients by tilting ideals, cluster tilted algebras, and so on. A lift of the Calabi-Yau property of a triangulated category to its dg enhancement is called a Calabi-Yau structure. In the talk we will explain a systematic construction of Calabi-Yau structures on dg singularity categories. This is based on a joint work with Bernhard Keller.

Abstract: Among various types of algebras originating in differential geometry, those known as Lie-Yamaguti algebras still remain somewhat mysterious algebraically. Last year, A.Das introduced their 'associative' version, the Yamaguti algebras, which are supposed to serve as envelopes of Lie-Yamaguti algebras. I'll report on joint work with Frédéric Chapoton, which exhibits relationships between this algebraic structure, combinatorics of noncrossing partitions, and the Lie algebra sl_2.

Abstract: In arXiv:2105.13334, Gyenge, Koppensteiner and Logvinenko constructed a 2-categorification of the Heisenberg algebra of any (possibly noncommutative) smooth projective variety, and decategorified it via Grothendieck group. In this talk, I will first give an overview of this 2-categorification and then explain how to decategorify it via the Hochschild homology HH_*, instead. Roughly, we extend the decategorification map from a lattice in HH_0 to the whole Hochschild homology. Computing the decategorified action of the Heisenberg algebra explicitly, we relate it to both the original Grojnowski and Nakajima’s Heisenberg algebra action on the cohomology of Hilbert schemes of points on a surface and its K-theoretic generalisation to by Segal and Wang.

Abstract: For a wide range of subcategories of the module category of a finitely generated algebra, we show a variant of the inductive step of the second Brauer-Thrall conjecture. That is, if there are infinitely many non-isomorphic indecomposable modules of the same finite dimension in the subcategory, then there are infinitely many dimensions that each admit infinitely non-isomorphic indecomposable modules in the subcategory. This also implies a variant of the first Brauer-Thrall conjecture in this context. The subcategories in question are the constructible subcategories, which are those that consist of all modules that vanish on a finitely presented functor. A key ingredient of the proof is a new connection between the Ziegler spectrum and schemes of finite dimensional modules that allows for a geometric approach. An important step is to find a suitable curve inside a constructible subset of the scheme. This result is contributed by Andres Fernandez Herrero. 

Abstract: In this talk we introduce homological notions -- such as finite global dimension and gorensteinness -- for compactly generated triangulated categories.  We observe that the latter generalise classical notions from homological algebra and study those same attributes for other triangulated categories of interest in representation theory, such as the homotopy category of injectives of an Artin algebra or the derived category of a dg algebra. This is based on joint work with C. Psaroudakis and J. Vitória.

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Abstract: It is known that partially wrapped Fukaya categories of marked surfaces are closely related to derived categories of gentle algebras. We discuss how this relation changes when the surface has nodal singularities. The main algebraic objects are pinched gentle algebras, which arise from localizing graded gentle algebras at certain spherical band objects. We explain how one can associate a minimal A_\infty-category to a nodal surface, giving a minimal model for the formal localization of the topological Fukaya category. We will also discuss some examples coming from simple pinching operations on marked surfaces. This is based on a joint work with Severin Barmeier, Pierre Bodin, and Sibylle Schroll.

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