arXiv subjects

Christoffer Söderberg

Publications and source records attributed to Christoffer Söderberg.

4 recordsLinked to original sources

arXiv2026

Mutating Species with Potentials and Cluster Tilting Objects

Buan, Iyama, Reiten and Smith proved that cluster-tilting objects in triangulated 2-Calabi--Yau categories are closely connected with mutation of quivers with potentials over an algebraically closed field. We prove a more general statement where instead of working with quivers with potentials we consider species with potential over a perfect field. We describe the $3$-preprojective algebra of the tensor product of two tensor algebras of acyclic species using a species with potential. In the case when the Jacobian algebra of a species with potential is self-injective, we provide a description of the Nakayama automorphism of a particular case of mutation of the species with potential where you mutate along orbits of the Nakayama permutation, which preserves self-injectivity. For certain types of Jacobian algebras of species with potentials, we prove that they lie in the scope of the derived Auslander-Iyama correspondence due to Jasso-Muro. Mutating along orbits of the Nakayama permutation stays within this setting, yielding a rich source of examples. All $2$-representation finite $l$-homogeneous algebras that are constructed using certain species with potential and mutations of such species with potentials are considered.

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arXiv2025

Jacobian Algebras of Species with Potentials and 2-Representation Finite Algebras

We study $2$-representation finite $\mathbb{K}$-algebras obtained from tensor products of tensor algebras of species. In earlier work we computed the higher preprojective algebra of said algebras to be given as Jacobian algebras of certain species with potential $(S, W)$, which are self-injective and finite dimensional. Truncating these Jacobian algebras yields a rich source of $2$-representation finite $\mathbb{K}$-algebras. Under suitable assumptions, we prove that the set of all cuts of $(S, W)$ is transitive under successive cut-mutations. Furthermore, we show that cuts and cut-mutation correspond to truncated Jacobian algebras and $2$-APR tilting, respectively. Consequently, under certain assumptions, all truncated Jacobian algebras are related to each other via $2$-APR tilting. We produce various new examples of $2$-representation finite $\mathbb{K}$-algebras.

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arXiv2023

Category O for Takiff sl_2

We investigate various ways to define an analogue of BGG category $\mathcal{O}$ for the non-semi-simple Takiff extension of the Lie algebra $\mathfrak{sl}_2$. We describe Gabriel quivers for blocks of these analogues of category $\mathcal{O}$ and prove extension fullness of one of them in the category of all modules.

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arXiv2022

Preprojective algebras of $d$-representation finite species with relations

In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We generalize this result to the species case. We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by a mapping cone of a certain chain map. We also study a higher dimensional analogue of representation finite hereditary algebras called $d$-representation finite algebras. One source of $d$-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.

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