Derived Equivalence Classification Research Articles

Quipu Quivers and Nakayama Algebras with Almost Separate Relations

A Nakayama algebra with almost separate relations is one where the overlap between any pair of relations is at most one arrow. In this paper we give a derived equivalence between such Nakayama algebras and path algebras of quivers of a special form known as quipu quivers. Furthermore, we show how this derived equivalence can be used to produce a complete classification of linear Nakayama algebras with almost separate relations. As an application, we include a list of the derived equivalence classes of all Nakayama algebras of length ≤8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le 8$$\\end{document} with almost separate relations.

τ-Tilting finiteness of two-point algebras II

In this paper, we explain a strategy on [Formula: see text]-vectors to discover some new minimal [Formula: see text]-tilting infinite two-point algebras. Consequently, the [Formula: see text]-tilting finiteness of various two-point monomial algebras, including all radical cube zero cases, could be determined. Moreover, we find that the derived equivalence class of the Kronecker algebra contains only itself and its opposite algebra.

Maximal Tori in HH1 and the Fundamental Group

Abstract We investigate maximal tori in the Hochschild cohomology Lie algebra ${\operatorname {HH}}^1(A)$ of a finite dimensional algebra $A$, and their connection with the fundamental groups associated to presentations of $A$. We prove that every maximal torus in ${\operatorname {HH}}^1(A)$ arises as the dual of some fundamental group of $A$, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.

A complete derived invariant for gentle algebras via winding numbers and Arf invariants

Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections of the associated surface. We associate to each gentle algebra a line field on the corresponding surface and prove that the derived equivalence class of the algebra is completely determined by the homotopy class of the line field up to homeomorphism of the surface. Then, based on winding numbers and the Arf invariant of a certain quadratic form over {mathbb {Z}}_2, we translate this to a numerical complete derived invariant for gentle algebras.

On representation-finite gendo-symmetric algebras with only one non-injective projective module

Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig [15,16] addressed some behaviour of the endomorphism algebra of a generator over a symmetric algebra, which they called gendo-symmetric algebra. Continuing this line of works, we classify in this article the representation-finite gendo-symmetric algebras that have precisely one isomorphism class of indecomposable non-injective projective module. We also determine their almost ν-stable derived equivalence classes in the sense of Wei Hu and Changchang Xi [21]. It turns out that a representative can be chosen as the quotient of a representation-finite symmetric algebra by the socle of a certain indecomposable projective module.

Derived equivalence classification of Brauer graph algebras

We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. Motivated by the connection between Brauer graph algebras and gentle algebras we consider A∞-trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary. This construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences between Brauer graph algebras with the same derived invariants. As part of the proof we provide an interpretation of derived invariants of Brauer graph algebras as orbit invariants of line fields under the action of the mapping class group.

The tilting theory of contraction algebras

To every minimal model of a complete local isolated cDV singularity Donovan–Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions, obtaining a faithful group action on the bounded derived category. Further, we determine precisely those standard equivalences which are induced by two-term tilting complexes and show that any standard equivalence between contraction algebras (up to algebra isomorphism) can be viewed as the composition of our constructed functors. Thus, for a contraction algebra, we obtain a complete picture of its derived equivalence class and, in particular, of its derived autoequivalence group.

On the finiteness of the derived equivalence classes of some stable endomorphism rings

We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction f :X rightarrow {mathrm{Spec};},R, where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra A_{{text {con}}} known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of A_{text {con}} and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.

Equivalence and dissimilarity of ecosystem states

Measuring (dis)similarity between ecosystem states is a key theme in ecology. Much of community and ecosystem ecology is devoted to searching for patterns in ecosystem similarity from an external observer's viewpoint, using variables such as species abundances, measures of diversity and complexity. However, from the point of view of organisms in the ecosystem, proportional population growth rates are the only relevant aspect of ecosystem state, because natural selection acts on groups of organisms with different proportional population growth rates. We therefore argue that two ecosystem states are equivalent if and only if, for each species they contain, the proportional population growth rate does not differ between the states. Based on this result, we develop species-level and aggregated summary measures of ecosystem state and discuss their ecological meaning. We illustrate our approach using a long-term dataset on the plankton community from the Central European Lake Constance. We show that the first three principal components of proportional population growth rates describe most of the variation in ecosystem state in Lake Constance. We strongly recommend using proportional population growth rates and the derived equivalence classes for comparative ecosystem studies. This opens up new perspectives on important existing topics such as alternative stable ecosystem states, community assembly, and the processes generating regularities in ecosystems.

Deformations of complexes for finite dimensional algebras

Let k be a field and let Λ be a finite dimensional k-algebra. We prove that every bounded complex V• of finitely generated Λ-modules has a well-defined versal deformation ring R(Λ,V•) which is a complete local commutative Noetherian k-algebra with residue field k. We also prove that nice two-sided tilting complexes between Λ and another finite dimensional k-algebra Γ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.

Classifying tilting complexes over preprojective algebras of Dynkin type

We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.

Dualities and derived equivalences for category $$\mathcal{O}$$ ℒ

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category $$\mathcal{O}$$ associated to a reductive finite-dimensional Lie algebra. In particular, we find that, contrary to the original category $$\mathcal{O}$$ and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category $$\mathcal{O}$$ as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category $$\mathcal{O}$$ . We subsequently classify all derived equivalence classes of blocks of category $$\mathcal{O}$$ in type A which preserve the Koszul grading.

Derived Equivalence Classification of the Gentle Two-Cycle Algebras

We complete a derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobinski–Malicki.

Derived invariants for surface algebras

In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler [11]. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A cut can be regarded as a grading on the Jacobian algebra of the quiver with potential (Q,W) associated with the triangulation.Fixing a set ϵ of generators of the fundamental group of the surface π1(S), we associate to any cut d a weight wϵ(d)∈Z2g+b, where g is the genus of S and b the number of boundary components. The main result of the paper asserts that the derived equivalence class of the surface algebra is determined by the corresponding weight wϵ(d) up to homeomorphism of the surface. Surface algebras are gentle and of global dimension ≤2, and any surface algebras coming from the same surface (S,M) are cluster equivalent, in the sense of [2]. To prove that the weight is a derived invariant we strongly use results about cluster equivalent algebras from [2].Furthermore we also show that for surface algebras the invariant defined for gentle algebras by Avella-Alaminos and Geiss in [6] is determined by the weight.

Derived equivalence classification of weakly symmetric algebras of domestic type

Derived equivalence classification of weakly symmetric algebras of domestic type

Morita equivalence classes of $2$-blocks of defect three

We give a complete description of the Morita equivalence classes of blocks with elementary abelian defect groups of order $8$ and of the derived equivalences between them. A consequence is the verification of Brou\'e's abelian defect group conjecture for these blocks. It also completes the classification of Morita and derived equivalence classes of $2$-blocks of defect at most three defined over a suitable field.

Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D

We provide a far reaching derived equivalence classification of the cluster-tilted algebras of Dynkin type D and suggest standard forms for the derived equivalence classes. We believe that the classification is complete, but some subtle questions remain open. We introduce another notion of equivalence called good mutation equivalence which is slightly stronger than derived equivalence but is algorithmically more tractable, and give a complete classification together with standard forms.

Algebras of acyclic cluster type: Tree type and type Ã

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

Algebras of acyclic cluster type: Tree type and typeÃ

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

APR Tilting Modules and Graded Quivers with Potential

We study the quiver with relations of the endomorphism algebra of an APR tilting module. We give an explicit description of the quiver with relations by graded quivers with potential (QPs) and mutations. The result also implies that mutations of QPs with algebraic cuts induce tilting mutations between the associated truncated Jacobian algebras. Consequently, mutations of QPs provide a rich source of derived equivalence classes of algebras. As an application, we give a sufficient condition of QPs such that Derksen-Weyman-Zelevinsky's question has a positive answer.