arXiv subjects

Bernhard Keller

Publications and source records attributed to Bernhard Keller.

At least 19 recordsLinked to original sources

arXiv2025

A Higgs category for the cluster variety of triples of flags

The cluster variety of triples of flags (associated with a split simple Lie group of Dynkin type Delta) plays a key role in higher Teichmuller theory as developed by Fock-Goncharov, Jiarui Fei, Ian Le, ... and Goncharov-Shen. We refer to it as the basic triangle associated with Delta. In this paper, for simply laced Delta, we construct and study a Higgs category (in the sense of Yilin Wu) which we expect to categorify the basic triangle. This category is a certain exact dg category (in the sense of Xiaofa Chen) which is Frobenius and stably 2-Calabi-Yau. We show that it has indeed the expected cyclic group symmetry and that its derived category has the expected braid group symmetry. A key ingredient in our construction is a conjecture by Merlin Christ, whose proof occupies most of this paper. The proof is based on a new description of the Higgs category in terms of Gorenstein projective dg modules. Our techniques are in the spirit of Orlov in his work on triangulated categories of graded B-branes.

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arXiv2025

Dg enhanced orbit categories and applications

Our aim in this paper is to prove two results related to the three constructions of cluster categories: as orbit categories, as singularity categories and as cosingularity categories. In the first part of the paper, we prove the universal property of pretriangulated orbit categories of dg categories first stated by the second-named author in 2005. We deduce that the passage to an orbit category commutes with suitable dg quotients. We apply these results to study collapsing of grading for (higher) cluster categories constructed from bigraded Calabi-Yau completions as introduced by Ikeda-Qiu. The second part of the paper is motivated by the construction of cluster categories as (co)singularity categories. We show that, for any dg algebra $A$, its perfect derived category can be realized in two ways: firstly, as an (enlarged) cluster category of a certain differential bigraded algebra, generalizing a result of Ikeda-Qiu, and secondly as a (shrunk) singularity category of another differential bigraded algebra, generalizing a result of Happel following Hanihara. We relate these two descriptions using a version of relative Koszul duality.

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arXiv2024

$g$-vectors and $DT$-$F$-polynomials for Grassmannians

We review $\mathrm{Hom}$-infinite Frobenius categorification of cluster algebras with coefficients and use it to give two applications of Jensen--King--Su's Frobenius categorification of the Grassmannian: 1) we determine the $g$-vectors of the Pl\"ucker coordinates with respect to the triangular initial seed and 2) we express the $F$-polynomials associated with the Donaldson--Thomas transformation in terms of $3$-dimensional Young diagrams thus providing a new proof for a theorem of Daping Weng.

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arXiv2024

On Amiot's conjecture

In a survey paper in 2011, Amiot proposed a conjectural characterisation of the cluster categories which were conceived in the mid 2000s to lift the combinatorics of Fomin-Zelevinsky's cluster algebras to the categorical level. This paper is devoted to a proof of (a variant of) her conjecture. More generally, cluster categories admit higher-dimensional and relative variants, the so-called Higgs categories recently introduced by Wu. We also prove higher-dimensional and relative variants of the conjecture.

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arXiv2024

The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence

We provide an outline of the proof of the Donovan--Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds. The proof relies on results of August, of Hua and the second-named author, Wemyss, and on the Derived Auslander--Iyama Correspondence -- a recent result by the first- and third-named authors.

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arXiv2024

Calabi-Yau structures on Drinfeld quotients and Amiot's conjecture

In 2009, Claire Amiot gave a construction of Calabi-Yau structures on Verdier quotients. We sketch how to lift it to the dg setting. We use this construction as an important step in an outline of the proof of her conjecture on the structure of 2-Calabi-Yau triangulated categories with a cluster-tilting object.

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arXiv2023

Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces

Cluster algebras *with coefficients* are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, ... . The approach of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow to categorify such cluster algebras. In previous work, the third-named author has constructed Higgs categories and relative cluster categories in the relative Jacobi-finite setting (arXiv:2109.03707). Higgs categories generalize the Frobenius categories used by Geiss-Leclerc-Schröer. In this article, we construct the Higgs category and the relative cluster category in the relative Jacobi-infinite setting under suitable hypotheses. These cover for example the case of Jensen-King-Su's Grassmannian cluster category. As in the relative Jacobi-finite case, the Higgs category is no longer exact but still extriangulated in the sense of Nakaoka-Palu. We also construct a cluster character refining Plamondon's. In the appendix, Chris Fraser and the second-named author categorify quasi-cluster morphisms using Frobenius categories. A recent application of this result is due to Matthew Pressland, who uses it to prove a conjecture by Muller-Speyer.

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arXiv2023

The valuation pairing on an upper cluster algebra

It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed $t$: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in $t$ and another element not divisible by the cluster variables in $t$. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, $d$-vectors, $F$-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of $d$-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its $F$-polynomial; the $F$-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.

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arXiv2023

An introduction to relative Calabi-Yau structures

These are notes taken by the second author for a series of three lectures by the first author on absolute and relative Calabi-Yau completions and Calabi-Yau structures given at the workshop of the International Conference on Representations of Algebras which was held online in November 2020. Such structures are relevant for (higher) representation theory as well as for the categorification of cluster algebras with coefficients. After a quick reminder on dg categories and their Hochschild and cyclic homologies, we present examples of absolute and relative Calabi-Yau completions (in the sense of Yeung). In many examples, these are related to higher preprojective algebras in the sense of Iyama-Oppermann. We conclude with the definition of relative (left and right) Calabi-Yau structures after Brav-Dyckerhoff.

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arXiv2023

Relative Calabi-Yau structures and ice quivers with potential

In 2015, Van den Bergh showed that complete $3$-Calabi-Yau algebras over an algebraically closed field of characteristic $0$ are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields. The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism. For example, every ice quiver with potential gives rise to a Ginzburg morphism. We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism. In particular, in dimension $3$ and over an algebraically closed ground field of characteristic $0$, it is given by an ice quiver with potential. Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a noncommutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry.

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arXiv2023

The dg Leavitt algebra, singular Yoneda category and singularity category

For any finite dimensional algebra $Λ$ given by a quiver with relations, we prove that its dg singularity category is quasi-equivalent to the perfect dg derived category of a dg Leavitt path algebra. The result might be viewed as a deformed version of the known description of the dg singularity category of a radical-square-zero algebra in terms of a Leavitt path algebra with trivial differential. The above result is achieved in two steps. We first introduce the singular Yoneda dg category of $Λ$, which is quasi-equivalent to the dg singularity category of $Λ$. The construction of this new dg category follows from a general operation for dg categories, namely an explicit dg localization inverting a natural transformation from the identity functor to a dg endofunctor. This localization turns out to be quasi-equivalent to a dg quotient category. Secondly, we prove that the endomorphism algebra of the quotient of $Λ$ modulo its Jacobson radical in the singular Yoneda dg category is isomorphic to the dg Leavitt path algebra. The appendix is devoted to an alternative proof of the result using Koszul-Moore duality and derived localizations.

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arXiv2023

Cluster categories and rational curves

We study rational curves on smooth complex Calabi--Yau threefolds via noncommutative algebra. By the general theory of derived noncommutative deformations due to Efimov, Lunts and Orlov, the structure sheaf of a rational curve in a smooth CY 3-fold $Y$ is pro-represented by a nonpositively graded dg algebra $Γ$. The curve is called nc rigid if $H^0Γ$ is finite dimensional. When $C$ is contractible, $H^0Γ$ is isomorphic to the contraction algebra defined by Donovan and Wemyss. More generally, one can show that there exists a $Γ$ pro-representing the (derived) multi-pointed deformation (defined by Kawamata) of a collection of rational curves $C_1,\ldots,C_t$ so that ${\mathrm{dim}}({\rm{Hom}}_Y({\mathcal{O}}_{C_i},{\mathcal{O}}_{C_j}))=δ_{ij}$. The collection is called nc rigid if $H^0Γ$ is finite dimensional. We prove that $Γ$ is a homologically smooth bimodule 3CY algebra. As a consequence, we define a (2CY) cluster category ${\mathcal{C}}_Γ$ for such a collection of rational curves in $Y$. It has finite-dimensional morphism spaces iff the collection is nc rigid. When $\bigcup_{i=1}^tC_i$ is (formally) contractible by a morphism $\hat{Y}\to \hat{X}$, ${\mathcal{C}}_Γ$ is equivalent to the singularity category of $\hat{X}$ and thus categorifies the contraction algebra of Donovan and Wemyss. The Calabi-Yau structure on $Y$ determines a canonical class $[w]$ (defined up to right equivalence) in the zeroth Hochschild homology of $H^0Γ$. Using our previous work on the noncommutative Mather--Yau theorem and singular Hochschild cohomology, we prove that the singularities underlying a 3-dimensional smooth flopping contraction are classified by the derived equivalence class of the pair $(H^0Γ, [w])$. We also give a new necessary condition for contractibility of rational curves in terms of $Γ$.

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arXiv2023

A refined multiplication formula for cluster characters

We obtain a multiplication formula for cluster characters on (stably) 2-Calabi-Yau (Frobenius or) triangulated categories. This formula generalizes those known for arbitrary pairs of objects and for Auslander-Reiten triangles. As an application, we show that for cluster algebras of acyclic types, specialization of a cluster variable to 1 sends all cluster variables to elements of a cluster algebra of smaller rank. We also obtain application to the reduction of friezes of acyclic type.

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arXiv2022

The Derived Auslander-Iyama Correspondence

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how -- combined with crucial results of August and Hua-Keller -- our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds.

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arXiv2022

On Leclerc's conjectural cluster structures for open Richardson varieties

In 2016, Leclerc constructed conjectural cluster structures on open Richardson varieties using representations of preprojective algebras. A variant with more explicit seeds was obtained by Ménard in his thesis. We show that Ménard's seeds do yield *upper* cluster algebra structures on open Richardson varieties and discuss the problems that remain in order to prove that they are cluster algebra structures.

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arXiv2020

A survey on maximal green sequences

Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic bases. We survey recent progress on their existence and properties and give a representation-theoretic proof of Greg Muller's theorem stating that full subquivers inherit maximal green sequences. In the appendix, Laurent Demonet describes maximal chains of torsion classes in terms of bricks generalizing a theorem by Igusa.

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arXiv2020

Singular Hochschild cohomology via the singularity category

We show that for a noetherian algebra $A$ whose bounded dg derived category is smooth, the singular Hochschild cohomology (=Tate--Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category of $A$. The existence of such an isomorphism is suggested by recent work of Zhengfang Wang.

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arXiv2020

Tame algebras have dense $\mathbf{g}$-vector fans

The $\mathbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\mathbf{g}$-vectors of its $2$-term presilting objects. We prove that the $\mathbf{g}$-vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster $\mathbf{g}$-vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster scattering diagrams only differ by wall-crossing functions on the walls contained in a separating hyperplane. The appendix is devoted to the construction of truncated twist functors and their adjoints.

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