Syzygy categories over Iwanaga-Gorenstein algebras

• Khrystyna Serhiyenko (University of Kentucky)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal Syzygies are a fundamental topic of study in both commutative and non-commutative algebra. They are defined as submodules of projectives and appear as kernels of projective resolutions that are used to approximate every module. It is then natural to study the category of all syzygies, particularly when the entire module category cannot be explicitly understood. In this series of talks, we will explore syzygy categories over Iwanaga-Gorenstein algebras of Gorenstein parameter 1, particularly focusing on the class of 2-Calabi-Yau tilted algebras. In this case, it is known that the syzygy category is triangulated and equivalent to the singularity category of an algebra as well as the category of maximal Cohen-Macauley modules. It is an open question to characterize algebras of finite CM type. As a first step in this direction, we will investigate the behavior of syzygy categories when passing from an algebra A to its quotient A/Ae_iA by an idempotent e_i, which corresponds to deleting a vertex from the quiver of A. Consequently, we describe various equivalent characterizations for these two algebras to have equivalent syzygy categories. We will then focus on the special class of Cohen-Macauley finite algebras, called dimer tree algebras and their skew group algebras, and provide a geometric model for their syzygy categories in terms of their associated checkerboard polygons.

Koszul algebras in representation theory

• Johanne Haugland (NTNU)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal Koszul algebras were first defined by Priddy in 1970. These algebras have been extensively studied and arise naturally in various fields of mathematics, such as algebraic geometry, noncommutative geometry, topology and number theory. In this lecture series, we give an introduction to Koszul algebras and the role they play in representation theory. Many central classes of algebras in representation theory turn out to be Koszul. Examples include hereditary algebras, gentle algebras, quadratic monomial algebras, polynomial algebras and exterior algebras, as well as certain preprojective algebras and trivial extensions. A main reason for the importance of Koszul algebras is their duality theory, as studied in the influential paper "Koszul duality patterns in representation theory" by Beilinson, Ginzburg and Soergel. To any Koszul algebra, there is an associated Koszul dual algebra. A key topic in the mini-course is the Koszul duality equivalence and how it reflects the strong connection between a Koszul algebra and its dual. As a motivating example, we look at how Koszul duality manifests in the geometric model for gentle algebras due to Opper, Plamondon and Schroll. A core perspective in the lecture series is the notion of higher Koszul algebras, or n-T-Koszul algebras, which yields a natural connection to Iyama's higher Auslander–Reiten theory. This framework builds on a generalization of T-Koszul algebras, as introduced by Madsen and Green, Reiten and Solberg. We discuss a higher version of classical Koszul duality and sketch some applications for n-hereditary algebras. This part of the lecture series builds on joint work with Mads H. Sandøy, who is also responsible for the exercise classes in the course.

Regular Sequences in Triangulated Categories

• Antonia Kekkou (University of Utah)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal Regular sequences are a fundamental tool in commutative algebra. In this talk, we introduce regular sequences in R-linear triangulated categories, where R is a graded-commutative ring, and give some examples. As an application of this definition, we show that the length of regular sequences provides lower bounds on levels. This is joint work with Janina C. Letz and Marc Stephan.

Representations of lattices, Auslander regularity, and Coxeter permutations

• Hugh Thomas (Université du Québec à Montréal)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal In this minicourse, I will be focusing on the representation theory of lattices (that is to say, of their incidence algebras). This turns out to be an interesting setting in which to explore Auslander regularity and related notions, in part because they make contact with certain already studied notions within lattice theory, which I will also discuss. This minicourse assumes no prior knowledge of lattice theory, and the representation theory background required is fairly minimal. I will explain the definition of Auslander regularity, and then present the result of Iyama and Marczinzik showing that a lattice is Auslander regular if and only if it is distributive. I will also explain how Auslander regularity (of incidence algebras of lattices) can be read off from the Coxeter matrix. This leads naturally to the notion of Coxeter permutation, and the rowmotion operation on distributive lattices. Finally, I will discuss the extension of rowmotion to semidistributive lattices, the way this manifests in terms of the Coxeter matrix, and how this can also be understood in representation-theoretic terms.

Exercise Class

• Khrystyna Serhiyenko (University of Kentucky)

Room: Endenicher Allee 60/0-006 - Seminarraum There will also be exercises in rooms N0.007 and N0.008.

Exercise Class

• Johanne Haugland (NTNU)

Room: Endenicher Allee 60/0-006 - Seminarraum There will also be exercises in rooms N0.007 and N0.008.

Homological properties of category O

• Volodymyr Mazorchuk (Uppsala University)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal In this series of lectures I will try to describe various homological properties of BGG category O and their relation to the Kazhdan-Lusztig combinatorics of Hecke algebras. The plan is as follows: the first lecture will be about Hecke algebras and Kazhdan-Lusztig combinatorics. The second lecture will be about basics of category O and its connection with Hecke algebras. Finally, the third lecture will focus on various homological properties, in particular, on projective dimension of structural modules and Auslander regularity. The excercise classes will treat small rank cases in detail, including sl_2 (type A_1) and sl_3 (type A_2).

Syzygy categories over Iwanaga-Gorenstein algebras

• Khrystyna Serhiyenko (University of Kentucky)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

A Caldero-Chapoton map for the derived category of gentle algebras

• Azzurra Ciliberti (Ruhr Universität Bochum)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal The derived category of a gentle algebra A can be described via the geometry of surface dissections, where indecomposable complexes of A-modules correspond to possibly infinite graded arcs on the surface, and morphisms between them are encoded by crossings between the associated arcs. Moreover, their mapping cones are given by the resolution of these crossings. We introduce a Caldero-Chapoton map in this setting. Specifically, we associate a Laurent polynomial to every finite indecomposable complex, and show that skein relations hold whenever the corresponding arcs cross in the interior. For each complex, its Caldero-Chapoton map specializes to the corresponding element of the Grothendieck group. Furthermore, if A is hereditary, the algebra generated by these functions is an ordinary cluster algebra. This is joint work in progress with Esther Banaian, Ilaria Di Dedda, Khrystyna Serhiyenko, Yadira Valdivieso-Díaz and Kayla Wright.

Koszul algebras in representation theory

• Johanne Haugland (NTNU)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Exercise Class

• Volodymyr Mazorchuk (Uppsala University)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal There will also be exercises in room N0.007.

Exercise Class

• Hugh Thomas (Université du Québec à Montréal)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal There will also be exercises in room N0.007.

Mini Talks Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Open Problem Sessions Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Representations of lattices, Auslander regularity, and Coxeter permutations

• Hugh Thomas (Université du Québec à Montréal)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Homological properties of category O

• Volodymyr Mazorchuk (Uppsala University)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Categorification of the quantum electrical algebra

• Jonas Nehme (University of Bonn)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We introduce the quantum electrical algebra, which is a quantum version of the electrical Lie algebra governing electrical networks. We will then provide a categorification of these algebras via electrical KLR algebras mimicking the classical construction of categorification of the positive half of quantum groups. In addition, this electrical KLR algebra bears close connections to the representation theory of the periplectic Lie superalgebra. (jt. with Catharina Stroppel)

Syzygy categories over Iwanaga-Gorenstein algebras

• Khrystyna Serhiyenko (University of Kentucky)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Exercise Class

• Khrystyna Serhiyenko (University of Kentucky)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal There will also be exercises in room 1.008.

Exercise Class

• Volodymyr Mazorchuk (Uppsala University)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal There will also be exercises in room 1.008.

Koszul algebras in representation theory

• Johanne Haugland (NTNU)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Representations of lattices, Auslander regularity, and Coxeter permutations

• Hugh Thomas (Université du Québec à Montréal)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

On extended hearts of t-structures

• Marvin Plogmann (Universität zu Köln)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal For a finite-dimensional algebra A, the category mod(A) can be recovered inside the bounded derived category of A as the full subcategory of objects whose cohomology is concentrated in degree 0. A natural enlargement is obtained by considering, for a fixed n>0, those objects whose cohomology is concentrated in degrees 1-n,...,0. This subcategory is called the n-extended heart of the standard t-structure. For n=1, one recovers the abelian category mod(A), while for n>1 the extended heart is no longer abelian. Nevertheless, it retains many features reminiscent of module categories; for example, it admits Auslander--Reiten sequences. I will explain why extended hearts are useful for studying finite-dimensional algebras and their derived analogues.

Homological properties of category O

• Volodymyr Mazorchuk (Uppsala University)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Exercise Class

• Johanne Haugland (NTNU)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal There will also be exercises in room 0.006.

Exercise Class

• Hugh Thomas (Université du Québec à Montréal)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal There will also be exercises in room 0.006.