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Biserial algebras via subalgebras and the path algebra of $D_4$

Julian Külshammer

Published in Journal of Algebra, 2011

We give two new criteria for a basic algebra to be biserial. The first one states that an algebra is biserial iff all subalgebras of the form $eAe$ where $e$ is supported by at most $4$ vertices are biserial. The second one gives some condition on modules that must not exist for a biserial algebra. These modules have properties similar to the module with dimension vector for the path algebra of the quiver $D_4$. Both criteria generalize criteria for an algebra to be Nakayama. They rely on the description of a basic biserial algebra in terms of quiver and relations given by R. Vila-Freyer and W. Crawley-Boevey.

Representation type of Frobenius–Lusztig kernels

Julian Külshammer

Published in The Quarterly Journal of Mathematics, 2013

In this article, we show that almost all blocks of all Frobenius–Lusztig kernels are of wild representation type extending results of Feldvoss and Witherspoon, who proved this result for the principal block of the zeroth Frobenius–Lusztig kernel. Furthermore, we verify the conjecture that there are infinitely many Auslander–Reiten components for a finite-dimensional algebra of infinite representation type for selfinjective algebras whose cohomology satisfies certain finiteness conditions.

Auslander–Reiten theory of Frobenius–Lusztig kernels

Julian Külshammer

Published in Bulletin of the London Mathematical Society, 2013

In this paper, we show that the tree class of a component of the stable Auslander–Reiten quiver of a Frobenius–Lusztig kernel is one of the three infinite Dynkin diagrams. For the special case of the small quantum group, we show that the periodic components are homogeneous tubes and that the non-periodic components have shape $\mathbb{Z}[\mathbb{A}_\infty]$ if the component contains a module for the infinite-dimensional quantum group.

Auslander–Reiten theory of small half quantum groups

Julian Külshammer

Published in Proceedings of the American Mathematical Society, 2014

For the small half quantum groups $u_\zeta(\mathfrak{b})$ and $u_\zeta(\mathfrak{n})$ we show that the components of the stable Auslander-Reiten quiver containing gradable modules are of the form $\mathbb{Z}[\mathbb{A}_\infty]$.

Quasi-hereditary algebras, exact Borel subalgebras, A_∞-algebras and boxes

Steffen Koenig, Julian Külshammer, and Sergiy Ovsienko

Published in Advances in Mathematics, 2014

Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category $\mathcal{O}$. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category $\mathcal{F}(\Delta)$ of modules with standard (Verma, Weyl, …) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg algebra associated with the $A_\infty$-structure on $\operatorname{Ext}^*(\Delta,\Delta)$. Its underlying algebra is an exact Borel subalgebra.

In the bocs seat: Quasi-hereditary algebras and representation type

Julian Külshammer

Published in Representation theory--current trends and perspectives (EMS Series of Congress Reports), 2017

This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation type of the category filtered by standard modules for a quasi-hereditary algebra. In this direction, joint work with Thiel is presented showing that the subcategory of modules filtered by Weyl modules for tame Schur algebras is of finite representation type. The paper also includes a new proof for the classification of quasi-hereditary algebras with two simple modules, a result originally obtained by Membrillo-Hernández.

Pro-species of algebras I: Basic Properties

Julian Külshammer

Published in Algebras and Representation Theory, 2017

In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß, Leclerc, and Schröer. In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories.

Nakayama-type phenomena in higher Auslander–Reiten theory

Gustavo Jasso and Julian Külshammer

Published in Contemporary Mathematics, 2018

Ringel duality as an instance of Koszul duality

Agnieszka Bodzenta and Julian Külshammer

Published in Journal of Algebra, 2018

In joint work S. Koenig, S. Ovsienko and the second author showed that every quasi-hereditary algebra is Morita equivalent to the right algebra, i.e. the opposite algebra of the left dual, of a coring. Let $A$ be an associative algebra and $V$ an $A$-coring whose right algebra $R$ is quasi-hereditary. In this paper, we give a combinatorial description of an associative algebra $B$ and a $B$-coring $W$ whose right algebra is the Ringel dual of $R$. We apply our results in small examples to obtain restrictions on the $A_\infty$-structure of the Ext-algebra of standard modules over a class of quasi-hereditary algebras related to birational morphisms of smooth surfaces.

Derived invariance of support varieties

Julian Külshammer, Chrysostomos Psaroudakis, and Øystein Skartsæterhagen

Published in Proceedings of the American Mathematical Society, 2019

The (Fg) condition on Hochschild cohomology as well as the support variety theory are shown to be invariant under derived equivalence.

Higher Nakayama algebras I: Construction

Gustavo Jasso and Julian Külshammer

Published in Advances in Mathematics, 2019

We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander–Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional algebra $A^{(d)}$ having a distinguished $d$-cluster-tilting $A^{(d)}$-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of $A$. We also construct higher dimensional analogues of the mesh category of type $\mathbb{Z}\mathbb{A}_\infty$ and the tubes.

From quasi-hereditary algebras with exact Borel subalgebras to directed bocses

Tomasz Brzeziński, Steffen Koenig, and Julian Külshammer

Published in Bulletin of the London Mathematical Society, 2020

Up to Morita equivalence, every quasi-hereditary algebra is the dual algebra of a directed bocs or a coring. From the bocs, an exact Borel subalgebra is obtained. In this paper a characterisation of exact Borel subalgebras arising in this way is given.

A functorial approach to monomorphism categories of species I

Nan Gao, Julian Külshammer, Sondre Kvamme, and Chrysostomos Psaroudakis

Published in Communications in Contemporary Mathematics, 2022

We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalized species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit computations as in the case of Ringel and Schmidmeier.

Uniqueness of exact Borel subalgebras and bocses

Julian Külshammer and Vanessa Miemietz

Preprint, 2023

Together with Koenig and Ovsienko, the first author showed that every quasi-hereditary algebra can be obtained as the (left or right) dual of a directed bocs. In this monograph, we prove that if one additionally assumes that the bocs is basic, a notion we define, then this bocs is unique up to isomorphism. This should be seen as a generalisation of the statement that the basic algebra of an arbitrary associative algebra is unique up to isomorphism. The proof associates to a given presentation of the bocs an $A_\infty$-structure on the $\operatorname{Ext}$-algebra of the standard modules of the corresponding quasi-hereditary algebra. Uniqueness then follows from an application of Kadeishvili's theorem.

Towards bound quivers for exact categories

Julian Külshammer

Preprint, 2023

A functorial approach to monomorphism categories of species II: Indecomposables

Nan Gao, Julian Külshammer, Sondre Kvamme, and Chrysostomos Psaroudakis

Published in Proceedings of the London Mathematical Society, 2024

We investigate the (separated) monomorphism category $\operatorname{mono}(Q,Λ)$ of a quiver $Q$ over an Artin algebra $\Lambda$. We construct an epivalence from $\overline{\operatorname{mono}}(Q,\Lambda)$ to $\operatorname{rep}(Q,\overline{\operatorname{mod}} Λ)$, where $\operatorname{mod}\Lambda$ is the category of finitely generated modules and $\overline{\operatorname{mod}}\Lambda$ and $\overline{\operatorname{mono}}(Q,\Lambda)$ denote the respective injectively stable categories. Furthermore, if $Q$ has at least one arrow, then we show that this is an equivalence if and only if $\Lambda$ is hereditary. In general, it induces a bijection between indecomposable objects in $\operatorname{rep}(Q,\overline{\operatorname{mod}}\Lambda)$ and non-injective indecomposable objects in $\operatorname{mono}(Q,\Lambda)$. We show that the generalized $\operatorname{Mimo}$-construction, an explicit minimal right approximation into $\operatorname{mono}(Q,\Lambda)$, gives an inverse to this bijection. Using this, we describe the indecomposables in the monomorphism category of a radical-square-zero Nakayama algebra, and give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length 3 with the same residue field. These results are proved using free monads on an abelian category, in order to avoid the technical combinatorics arising from quiver representations. The setup also specializes to representations of modulations. In particular, we obtain new results on the singularity category of the algebras H which were introduced by Geiss, Leclerc, and Schröer in order to extend their results relating cluster algebras and Lusztig’s semicanonical basis to symmetrizable Cartan matrices. We also recover results on the ιquivers algebras which were introduced by Lu and Wang to realize ιquantum groups via semi-derived Hall algebras.

Julian Külshammer

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