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

My research area is the representation theory of quivers and finite dimensional algebras, specifically of gentle and preprojective algebras. I am studying their module varieties and Ringel-Hall algebras with a focus on connections to quiver representations over $\mathbb{F}_1$ and cluster algebras.

Publications and Preprints

2. Scalar extensions of quiver representations over $\mathbb{F}_1$. Preprint, submitted. arXiv:2403.04597
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.

3. 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 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