arXiv subjects

Darius Dramburg

Publications and source records attributed to Darius Dramburg.

7 recordsLinked to original sources

arXiv2026

Isomorphisms of graded semiconnected algebras

Let $A_\bullet$ and $B_\bullet$ be nonnegatively graded algebras, finitely generated in degrees less than $2$ and with semisimple base rings $A_0$ and $B_0$. We prove that if $A \simeq B$ as ungraded algebras, then $A_\bullet \simeq B_\bullet$ as graded algebras. This generalises a theorem of Bell and Zhang arXiv:1509.08812 in the connected case $A_0 = k = B_0$, and the path-algebra version when $A_0$ and $B_0$ are $k$-elementary due to Gaddis arXiv:1712.01650. We also discuss some related open problems for graded algebras.

math.RA↗

arXiv2026

$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part I: Extensions of abelian groups

Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. The resulting skew-group algebra $R \ast G$ is $3$-Calabi-Yau, and can sometimes be endowed with the structure of a $3$-preprojective algebra. However, not every such $R \ast G$ admits such a structure. The finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ are classified into types (A) to (L). We consider the groups $G$ of types (C) and (D) and determine for each such group whether the algebra $R \ast G$ admits a $3$-preprojective cut, that is a $3$-preprojective structure arising from a grading of the McKay quiver of $G$. We show that the algebra $R \ast G$ admits a $3$-preprojective cut if and only if $9 \mid |G|$. Our proof is constructive and yields a description of the involved $2$-representation infinite algebras. This is based on the semi-direct decomposition $G \simeq N \rtimes K$ for an abelian group $N$, and we show that the existence of a $3$-preprojective structure on $R \ast G$ is essentially determined by the existence of one on $R \ast N$. This provides new classes of $2$-representation infinite algebras, and we discuss some $2$-Auslander-Platzeck-Reiten tilts. Along the way, we give a detailed description of the involved groups and their McKay quivers by iteratively applying skew-group constructions.

math.RT↗

arXiv2026

$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part II: Central extensions and exceptionals

Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. We continue our investigation from arXiv:2505.10683 [math.RT] into when the resulting skew-group algebra $R \ast G$ is a $3$-preprojective algebra of a $2$-representation infinite algebra, defined by a so-called cut. We consider the subgroups arising from $\operatorname{GL}_2(\mathbb{C}) \hookrightarrow \operatorname{SL}_3(\mathbb{C})$, called type (B), as well as the exceptional subgroups, called types (E) -- (L). For groups of type (B), we show that a $3$-preprojective cut exists on $R \ast G$ if and only if $G$ is not isomorphic to a subgroup of $\operatorname{SL}_2(\mathbb{C})$ or $\operatorname{PSL}_2(\mathbb{C})$. For groups $G$ of the remaining types (E) -- (L), every $R \ast G$ admits a $3$-preprojective cut, except for type (H) and (I). To prove our results for type (B), we explore how the notion of isoclinism interacts with the shape of McKay quivers. We compute the McKay quivers in detail, using a knitting-style heuristic. For the exceptional subgroups, we compute the McKay quivers directly, as well as cuts, and we discuss how this task can be done algorithmically. This provides many new examples of $2$-representation infinite algebras, and together with arXiv:2401.10720 [math.RT], arXiv:2505.10683 [math.RT] completes the classification of finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ for which $R \ast G$ is a $3$-preprojective algebra.

math.RT↗

arXiv2025

On the Orbifold origin of Higher Form Symmetries in Geometric Engineering

In this work we explore the relation between orbifold singularities and higher form symmetries. Using the geometric engineering dictionary, we argue that the discrete higher symmetries of 5d SCFTs constructed from M-theory on a non-compact Calabi-Yau threefold can be related to a quantum symmetry of the associated BPS quiver. Through un-orbifolding the quantum symmetry we obtain a new theory without higher form symmetry, providing a notion of ''minimality'' for a theory. This procedure is carried out via algebraic manipulations of the BPS/McKay quiver describing the crepant resolution of the singular geometry. This technique can also be reverted and thus, starting from any ''minimal'' theory, one can orbifold it and generate new theories with the desired higher form symmetries. We test our technology on classes of 5d SCFTs that arise from M-theory geometric engineering on Calabi-Yau threefolds that are non-toric non-complete intersections, which have historically been challenging to tackle.

hep-th↗

arXiv2025

On compatibility of Koszul- and higher preprojective gradings

We investigate compatibility of gradings for an almost Koszul or Koszul algebra $R$ that is also the higher preprojective algebra $Π_{n+1}(A)$ of an $n$-hereditary algebra $A$. For an $n$-representation finite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with an almost Koszul grading. For an acyclic basic $n$-representation infinite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with a Koszul grading. From this we deduce that a higher preprojective grading of an (almost) Koszul algebra $R = Π_{n+1}(A)$ is, in both cases, isomorphic to a cut of the (almost) Koszul grading. Up to a further assumption on the tops of the degree $0$ subalgebras for the different gradings, we also show a similar result without the basic assumption in the $n$-representation infinite case. As an application, we show that $n$-APR tilting preserves the property of being Koszul for $n$-representation infinite algebras.

math.RT↗

arXiv2024

A classification of $n$-representation infinite algebras of type Ã

We classify $n$-representation infinite algebras $Λ$ of type Ã. This type is defined by requiring that $Λ$ has higher preprojective algebra $Π_{n+1}(Λ) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq \operatorname{SL}_{n+1}(k)$ is finite abelian. For the classification, we group these algebras according to a more refined type, and give a combinatorial characterisation of these types. This is based on so-called height functions, which generalise the height function of a perfect matching in a Dimer model. In terms of toric geometry and McKay correspondence, the types form a lattice simplex of junior elements of $G$. We show that all algebras of the same type are related by iterated $n$-APR tilting, and hence are derived equivalent. By disallowing certain tilts, we turn this set into a finite distributive lattice, and we construct its maximal and minimal elements.

math.RT↗

arXiv2024

The 3-Preprojective Algebras Of Type $Ã$

Let $G \leq \operatorname{SL}_{n+1}(\mathbb{C})$ act on $R = \mathbb{C}[X_1, \ldots, X_{n+1}]$ by change of variables. Then, the skew-group algebra $R \ast G$ is bimodule $(n+1)$-Calabi-Yau. Under certain circumstances, the algebra admits a locally finite-dimensional grading of Gorenstein parameter $1$, in which case it is the $(n+1)$-preprojective algebra of its $n$-representation infinite degree $0$ piece, as defined by Herschend, Iyama and Oppermann. If the group $G$ is abelian, the $(n+1)$-preprojective algebra is said to be of type $Ã$. For a given group $G$, it is not obvious whether $R \ast G$ admits such a grading making it into an $(n+1)$-preprojective algebra. We study the case when $n=2$ and $G$ is abelian. We give an explicit classification of groups such that $R \ast G$ is $3$-preprojective by constructing such gradings. This is possible as long as $G$ is not a subgroup of $\operatorname{SL}_2(\mathbb{C})$ and not $C_2 \times C_2$. For a fixed $G$, the algebra $R \ast G$ admits different $3$-preprojective gradings, so we associate a type to a grading and classify all types. Then we show that gradings of the same type are related by a certain kind of mutation. This gives a classification of $2$-representation infinite algebras of type $Ã$. The involved quivers are those arising from hexagonal dimer models on the torus, and the gradings we consider correspond to perfect matchings on the dimer, or equivalently to periodic lozenge tilings of the plane. Consequently, we classify these tilings up to flips, which correspond to the mutation we consider.

math.RT↗