Lukas Bonfert

I'm a PhD student of Catharina Stroppel, funded by the Max Planck Institute for Mathematics.

Contact

Email: bonfert[symbol in every email address]mpim-bonn.mpg.de

Research Interests

I am interested in representation theory, particularly that of Lie (super)algebras and related topics such as perverse sheaves or Koszul duality. My PhD project is (mostly) about silting theory, which means that I study the relation between weight structures (aka co-$t$-structures) and $t$-structures on triangulated categories such as derived categories of finite-dimensional algebras or dg algebras. I'm also thinking about perverse $t$-structures on flag varieties, which have deep connections to the representation theory of Lie algebras. Furthermore, I spent some time thinking about combinatorial ways to understand the structure of Lie superalgebras.

Publications and Preprints

3. Serre functor and $\mathbb{P}$-objects for perverse sheaves on $\mathbb{P}^n$ (joint with Alessio Cipriani). Preprint. arXiv:2506.06051
Abstract

We show that the inverse Serre functor for the constructible derived category $\mathbf{D}^\mathrm{b}_\mathrm{c}(\mathbb{P}^n)$ is given by the $\mathbb{P}$-twist at the simple perverse sheaf corresponding to the open stratum. Moreover, we show that all indecomposable perverse sheaves on $\mathbb{P}^n$ are $\mathbb{P}$-like objects, and explicitly construct morphisms spanning their total endomorphism spaces.

4. Derived projective covers and Koszul duality of simple-minded and silting collections. To appear in Applied Categorical Structures. arXiv:2309.00554
Abstract

We introduce derived projective covers and explain how they are related to the notion of enough derived projectives. This provides an if-and-only if criterion for when derived projective covers form a silting collection. We prove moreover a Koszul duality result for silting and simple-minded collections.

5. The Weyl Groupoids of $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(r|2n)$ (joint with Jonas Nehme). Journal of Algebra 641 (2024), pp. 795–822. arXiv:2305.04751
Abstract

We provide a convenient formulation of the definition of Cartan graphs and Weyl groupoids introduced by Heckenberger and Schneider, and construct Cartan graphs for regular symmetrizable contragredient Lie superalgebras. For $\mathfrak{sl}(m|n)$, $\mathfrak{osp}(2m+1|2n)$ and $\mathfrak{osp}(2m|2n)$ we explicitly describe the Cartan graph in terms of partitions and determine the relations between the generators of their Weyl groupoids.

Teaching

• Tutor for Foundations of Representation Theory, lecture course by Prof. Schröer, summer term 2023
• Tutor for Representation Theory II: Representation Theory, Geometry and Combinatorics, lecture course by Prof. Stroppel, winter term 2022/23
• Tutor for Algebra I, lecture course by Prof. Blomer, summer term 2022
• Tutor for Einführung in die Algebra, lecture course by Prof. Blomer, winter term 2021/22
• Tutor for Representation Theory II: Auslander–Reiten Theory, lecture course by Prof. Schröer, summer term 2021
• Tutor for Advanced Algebra I: Lie Algebras and their Representations, lecture course by Dr. Heidersdorf, winter term 2020/21
• Tutor for Algebra II: Modular Forms, lecture course by Prof. Blomer, winter term 2019/20
• Tutor for Einführung in die komplexe Analysis, lecture course by Prof. Bödigheimer, summer term 2019
• Tutor for Lineare Algebra I, lecture course by Prof. Huybrechts, winter term 2018/19
• Tutor for Analysis II, lecture course by Prof. Koch, summer term 2018
• Tutor for the Programmierkurs C++, preparatory course by Dr. Brenner, 09/2018
• Tutor for the Programmierkurs C, preparatory course by Johannes Holke and Clelia Albrecht, 09/2017