Nakayama Algebra 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.

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.

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.

The role of gentle algebras in higher homological algebra

Abstract We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra Λ contains a d-cluster tilting subcategory for some d ≥ 2 {d\geq 2} , then Λ is a radical square zero Nakayama algebra. This gives a complete classification of weakly d-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if 𝒟 b ⁡ ( Λ ) {\operatorname{\mathcal{D}}^{b}(\Lambda)} contains a d-cluster tilting subcategory that is closed under [ d ] {[d]} , then Λ is derived equivalent to an algebra of Dynkin type A. Furthermore, our approach gives a geometric characterization of all d-cluster tilting subcategories of 𝒟 b ⁡ ( Λ ) {\operatorname{\mathcal{D}}^{b}(\Lambda)} that are closed under [ d ] {[d]} .

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.

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.

Derived tame Nakayama algebras

We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.

Intersections, sums, and the Jordan-Hölder property for exact categories

We investigate how the concepts of intersection and sums of subobjects carry to exact categories. We obtain a new characterisation of quasi-abelian categories in terms of admitting admissible intersections in the sense of [23] . There are also many alternative characterisations of abelian categories as those that additionally admit admissible sums and in terms of properties of admissible morphisms . We then define a generalised notion of intersection and sum which every exact category admits. Using these new notions, we define and study classes of exact categories that satisfy the Jordan-Hölder property for exact categories, namely the Diamond exact categories and Artin-Wedderburn exact categories. By explicitly describing all exact structures on A = rep Λ for a Nakayama algebra Λ we characterise all Artin-Wedderburn exact structures on A and show that these are precisely the exact structures with the Jordan-Hölder property.

The finitistic dimension of a Nakayama algebra

Let A be an artin algebra, Gélinas has introduced an interesting upper bound for the finitistic dimension fin-pro A of A, namely the delooping level del A. We assert that fin-proA=delA for any Nakayama algebra A. This yields also a new proof that the finitistic dimension of A and its opposite algebra are equal, as shown recently by Sen. If S is a simple module, let e(S) be the minimum of the projective dimension of S and of its injective envelope (one of these numbers has to be finite); and e⁎(S) the minimum of the injective dimension of S and of its projective cover. Then the finitistic dimension of A is the maximum of the numbers e(S), as well as the maximum of the numbers e⁎(S).Using suitable syzygy modules, we construct a permutation h of the simple modules S such that e⁎(h(S))=e(S). In particular, this shows for z∈N that the number of simple modules S with e(S)=z is equal to the number of simple modules S′ with e⁎(S′)=z.

Tilting modules over Auslander algebras of Nakayama algebras with radical cube zero

Let [Formula: see text] be the Auslander algebra of a finite-dimensional basic connected Nakayama algebra [Formula: see text] with radical cube zero and [Formula: see text] simple modules. Then the cardinality [Formula: see text] of the set consisting of isomorphism classes of basic tilting [Formula: see text]-modules is [Formula: see text]

Classifying tilting modules over the Auslander algebras of radical square zero Nakayama algebras

Let [Formula: see text] be a radical square zero Nakayama algebra with [Formula: see text] simple modules and let [Formula: see text] be the Auslander algebra of [Formula: see text]. Then every indecomposable direct summand of a tilting [Formula: see text]-module is either simple or projective. Moreover, if [Formula: see text] is self-injective, then the number of tilting [Formula: see text]-modules is [Formula: see text]; otherwise, the number of tilting [Formula: see text]-modules is [Formula: see text].

Resolution quiver and cyclic homology criteria for Nakayama algebras

If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [17], Shen gave such a characterization using Ringel's resolution quiver. In [11], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example that came up in our attempt to generalize this comparison result to arbitrary monomial relation algebras [8].

Cell structure of bimodules over radical square zero Nakayama algebras

In this paper, we describe the combinatorics of the cell structure of the tensor category of bimodules over a radical square zero Nakayama algebra. This accounts to an explicit description of left, right and two-sided cells.

Higher Nakayama algebras I: Construction

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 ZA∞ and the tubes.

Monomial algebras of dominant dimension at least two

Let A be a finite dimensional algebra having the double centralizer property with respect to a minimal faithful projective-injective left module Af for some idempotent f. We prove that in this case A is a monomial algebra if and only if A is a Nakayama algebra given by a quiver with admissible relations.

On bounds of homological dimensions in Nakayama algebras

Let A A be a Nakayama algebra with n n simple modules and a simple module S S of even projective dimension. Choose m m minimal such that a simple A A -module with projective dimension 2 m 2m exists. Then we show that the global dimension of A A is bounded by n + m − 1 n+m-1 . This gives a combined generalisation of results of Gustafson [J. Algebra 97 (1985), pp. 14–16] and Madsen [Projective dimensions and Nakayama algebras, Amer. Math. Soc., Providence, RI, 2005]. In [Comm. Algebra 22 (1994), pp. 1271–1280], Brown proved that the global dimension of quasi-hereditary Nakayama algebras with n n simple modules is bounded by n n . Using our result on the bounds of global dimensions of Nakayama algebras, we give a short new proof of this result and generalise Brown’s result from quasi-hereditary to standardly stratified Nakayama algebras, where the global dimension is replaced with the finitistic dimension.

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring R(Λ,V) and we describe R(Λ,V) explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.

STRONGLY QUASI-HEREDITARY ALGEBRAS AND REJECTIVE SUBCATEGORIES

Ringel’s right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline–Parshall–Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

ABSTRACTLet T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule HomK(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ℕ-graded second Hochschild homology group HH2(A) in the sense of Sköldberg.