Markus Kleinau

I am a second year PhD-student under the supervision of Jan Schröer and a member of the Algebra and Representation Theory group.

Contact

Email:
Office: Endenicher Allee 60 N2.003

Research interests

I am interested in the intersection between the representation theory of finite dimensional algebras and combinatorics. This includes

• Homological properties of representations of incidence algebras,
• Ringel-Hall algebras of gentle algebras,
• Quiver representations over $\mathbb{F}_1$.

Posters

2025: Lattices of torsion classes and 2-cluster categories
2024: Quiver representations over $\mathbb F_1$

Publications and Preprints

3. Pure minimal injective resolutions and perfect modules for lattices Preprint. arXiv:2511.03385
joint with Tal Gottesmann, Viktória Klász, René Marczinzik.
Abstract

In a recent article, Iyama and Marczinzik showed that a lattice is distributive if and only if the incidence algebra is Auslander regular, giving a new connection between homological algebra and lattice theory. In this article we study when a distributive lattice has a pure minimal injective coresolution, a notion first introduced and studied in a work of Ajitabh, Smith and Zhang. We will see that this problem naturally leads to studying when certain antichain modules are perfect modules. We give a classification of perfect antichain modules under the assumption that their canonical antichain resolution is minimal and use this to give a complete classification in lattice theoretic terms of incidence algebras of distributive lattices with pure minimal injective coresolution.

4. Scalar extensions of quiver representations over $\mathbb{F}_1$ arXiv:2403.04597
   Algebr. Represent. Theory 28 (2025), no. 2, 531–548.
Abstract

Let $V$ and $W$ be quiver representations over $\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of $\DeclareMathOperator{\Hom}{Hom} \Hom_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.

5. The Aizenbud-Lapid binary operation for symmetrizable Cartan types Preprint. arXiv:2311.17036
Abstract

Aizenbud and Lapid recently introduced a binary operation on the crystal graph $B(-\infty)$ associated to a symmetric Cartan matrix. We extend their construction to symmetrizable Cartan matrices and strengthen a cancellation property of the binary operation.

Activities

• Talk in Cologne "Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics" 21.10.2025
• Talk in Bonn "Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics" 17.7.2025
• Talk in Trento "Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics" 25.6.2025
• Talk in Bonn "Representation theory over $\mathbb{F}_1$" 23.1.2025
• Research stay in Bochum with Tal Gottesman 14.1.-17.1.2025
• Talk in Versailles "Representation theory over $\mathbb{F}_1$" 11.12.2024
• Research stay in Versailles with Judith Marquardt 11.12-13.12.2024