Nakayama Algebras Research Articles

Nakayama Algebras of Small Homological Dimension and Pattern Avoiding Permutations

Nakayama Algebras of Small Homological Dimension and Pattern Avoiding Permutations

Rigidity dimensions of self-injective Nakayama algebras

Rigidity dimension is a new homological dimension which is intended to measure the quality of the best resolution of an algebra. In this paper, we determine the rigidity dimensions of self-injective Nakayama algebras An,m with n simple modules and the Loewy length m⩾n.

A functorial approach to monomorphism categories II: Indecomposables

AbstractWe investigate the (separated) monomorphism category of a quiver over an Artin algebra . We show that there exists an epivalence (called representation equivalence in the terminology of Auslander) from to , where is the category of finitely generated ‐modules and and denote the respective injectively stable categories. Furthermore, if has at least one arrow, then we show that this is an equivalence if and only if is hereditary. In general, the epivalence induces a bijection between indecomposable objects in and noninjective indecomposable objects in , and we show that the generalized Mimo‐construction, an explicit minimal right approximation into , gives an inverse to this bijection. We apply these results to describe the indecomposables in the monomorphism category of radical square zero Nakayama algebras, and to give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length 3 with the same residue field. The main tool to prove these results is the language of a free monad of an exact endofunctor on an arbitrary abelian category. This allows us to avoid the technical combinatorics arising from quiver representations. The setup also specializes to more general settings, such as representations of modulations. In particular, we obtain new results on the singularity category of the algebras that 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 algebras that were introduced by Lu and Wang to realize groups via semiderived Hall algebras.

Non-piecewise hereditary Nakayama algebras

Happel and Seidel gave a classification of piecewise hereditary Nakayama algebras, where the relations are given by some power of the radical.Here we explore what happens for general relations. We develop techniques for showing that a given algebra is not piecewise hereditary, illustrating them on numerous mid-sized examples. Then we observe cases where the property of being non-piecewise hereditary can be extended to other (larger) Nakayama algebras. While a complete classification remains elusive, we are able to identify two types of patterns of relations preventing piecewise heredity, indicating that for large quivers many Nakayama algebras are non-piecewise hereditary.

Hochschild homology dimension and a class of Hochschild extension algebras of truncated quiver algebras

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite.

WHEN IS THE SILTING-DISCRETENESS INHERITED?

Abstract We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.

Selfextensions of modules over group algebras

Let KG be a group algebra with G a finite group and K a field and M an indecomposable KG-module. We pose the question, whether ExtKG1(M,M)≠0 implies that ExtKGi(M,M)≠0 for all i≥1. We give a positive answer in several important special cases such as for periodic groups and give a positive answer also for all Nakayama algebras, which allows us to improve a classical result of Gustafson. We then specialise the question to the case where the module M is simple, where we obtain a positive answer also for all tame blocks of group algebras. For simple modules M, the appendix provides a Magma program that gives strong evidence for a positive answer to this question for groups of small order.

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.

Classification results for n-hereditary monomial algebras

We classify n-hereditary monomial algebras in three natural contexts: First, we give a classification of the n-hereditary truncated path algebras. We show that they are exactly the n-representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the n-hereditary quadratic monomial algebras. In the case n=2, we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating kA3⊗kkA3, where A3 is the Dynkin quiver of type A with bipartite orientation. In the case n≥3, we show that the only n-representation finite algebras are the n-representation-finite Nakayama algebras with quadratic relations.

When does the Auslander–Reiten translation operate linearly on the Grothendieck group? – Part I

For a hereditary, finite-dimensional algebra A the Coxeter transformation extends the action of the Auslander–Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the category of finitely generated A-modules. It is natural to ask whether other algebras admit a similar linear extension. We show that this is indeed the case for all Nakayama algebras. Conversely, we show that finite-dimensional algebras with non-acyclic and connected quiver admitting such a linear extension are already cyclic Nakayama algebras.

Nakayama Algebras and Fuchsian Singularities

Nakayama Algebras and Fuchsian Singularities

Infinitesimal semi‐invariant pictures and co‐amalgamation

AbstractThe purpose of this paper is to study the local structure of the semi‐invariant picture of a tame hereditary algebra near the null root. Using a construction that we call co‐amalgamation, we show that this local structure is completely described by the semi‐invariant pictures of a collection of self‐injective Nakayama algebras. We then describe the cones of this local structure using cluster‐like structures that we call support regular clusters. Finally, we show that the local structure is (piecewise linearly) invariant under cluster tilting.

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field , being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.

Homological ideals as integer specializations of some Brauer configuration algebras

UDC 512.5 The homological ideals associated with some Nakayama algebras are characterized and enumerated via integer specializations of some suitable Brauer configuration algebras. In addition, it is shown how the number of these homological ideals can be connected with the process of categorification of Fibonacci numbers defined by Ringel and Fahr.

Simple transitive 2-representations of bimodules over radical square zero Nakayama algebras via localization

We study the classification problem of simple transitive 2-representations of the 2-category of finite-dimensional bimodules over a radical square zero Nakayama algebra. This results in a complete classification of simple transitive 2-representations whose apex is a finitary two-sided cell. We define a notion of localization of 2-representations. We construct previously unknown simple transitive 2-representations as localizations of cell 2-representations. Using the universal property of our construction we prove that any simple transitive 2-representation with finitary apex is equivalent to a localization of a cell 2-representation.

Linear Nakayama algebras which are higher Auslander algebras

An artin algebra A is said to be a higher Auslander algebra provided the global dimension is finite and bounded by the dominant dimension. We say that a linear Nakayama algebra is concave, provided its Kupisch series first increases, then decreases. We are going to classify the concave Nakayama algebras which are higher Auslander algebras. Let us stress that the classification strongly depends on the parity of the global dimension of A.

A new characterization of quasi-hereditary Nakayama algebras and applications

We call a finite dimensional algebra A S-connected if the projective dimensions of the simple A-modules form an interval. We prove that a Nakayama algebra A is S-connected if and only if A is quasi-hereditary. We apply this result to improve an inequality for the global dimension of quasi-hereditary Nakayama algebras due to Brown. We furthermore classify the Nakayama algebras where equality is attained in Brown’s inequality and show that they are enumerated by the even indexed Fibonacci numbers if the algebra is cyclic and by the odd indexed Fibonacci numbers if the algebra is linear.

Monobrick, a uniform approach to torsion-free classes and wide subcategories

For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injection. We call such a set a monobrick. In this paper, we provide a uniform method to study torsion-free classes and wide subcategories via monobricks. We show that monobricks are in bijection with left Schur subcategories, which contains all subcategories closed under extensions, kernels and images, thus unifies torsion-free classes and wide subcategories. Then we show that torsion-free classes bijectively correspond to cofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free) classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories, without using τ-tilting theory. For Nakayama algebras, left Schur subcategories are the same as subcategories closed under extensions, kernels and images, and we show that its number is related to the large Schröder number.

Noncommutative symplectic manifolds for self-injective Nakayama algebras

In the paper, we study noncommutative symplectic manifolds (A,llbracket omega rrbracket ), where A is a self-injective Nakayama algebra. There is given a full description of exact symplectic manifolds for A being the trivial extension of an hereditary Nakayama algebra.

Characterizations of right rejective chains

AbstractIn this paper, we give characterizations of the category of finitely generated projective modules having a right rejective chain. By focusing on the characterizations, we give sufficient conditions for right rejective chains to be total right rejective chains. Moreover, we prove that Nakayama algebras with heredity ideals, locally hereditary algebras and algebras of global dimension at most two satisfy the sufficient conditions. As an application, we show that these algebras are right-strongly quasi-hereditary algebras.