Tame Hereditary Algebras Research Articles

A generic classification of locally free representations of affine GLS algebras

Throughout, let K be an algebraically closed field of characteristic 0. We provide a generic classification of locally free representations of Geiß-Leclerc-Schröer's algebras HK(C,D,Ω) associated to affine Cartan matrices C with minimal symmetrizer D and acyclic orientation Ω. Affine GLS algebras are “smooth” degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several “tame” phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type BC˜1 we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a 1-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular τ-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free H-module is isomorphic to a direct sum of τ-rigid modules and modules from our 1-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types and we obtain such a decomposition by folding the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are E-tame in the sense of Derksen-Fei and Asai-Iyama.

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.

Hall polynomials for tame quivers with automorphism

We prove that Hall polynomial exists for each triple of decomposition sequences associated with tame quivers with automorphism (or, equivalently, for all tame hereditary algebras over finite fields). This generalizes the results of Hubery (2010) [18] and Deng and Ruan (2017) [9].

Minimal silting modules and ring extensions

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behavior of minimality under ring extensions. We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism. Similar results are obtained for commutative rings of small homological dimension.

Higher representation infinite algebras from McKay quivers of metacyclic groups

For each prime number s we introduce examples of - and s-representation infinite algebras in the sense of Herschend, Iyama and Oppermann, which arise from skew group algebras of some metacyclic groups embedded in and . For this purpose, we give a description of the McKay quiver with a superpotential of such groups. Moreover, we show that for s = 2 our examples correspond to the classical tame hereditary algebras of type .

The No Gap Conjecture for tame hereditary algebras

The “No Gap Conjecture” of Brüstle–Dupont–Pérotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be “polygonally deformed” into each other. We prove this stronger conjecture for all tame hereditary algebras over any field, equivalently, for any acyclic tame skew-symmetrizable exchange matrix.

Rigid and Schurian modules over cluster-tilted algebras of tame type

We give an example of a cluster-tilted algebra $$\Lambda $$ with quiver Q, such that the associated cluster algebra $$\mathcal {A}(Q)$$ has a denominator vector which is not the dimension vector of any indecomposable $$\Lambda $$ -module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra $$\Lambda $$ , we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid $$\Lambda $$ -modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid $$\Lambda $$ -modules in this case.

Stratifying systems over hereditary algebras

This paper deals with stratifying systems over hereditary algebras. In the case of tame hereditary algebras we obtain a bound for the size of the stratifying systems composed only by regular modules and we conclude that stratifying systems cannot be complete. For wild hereditary algebras, with more than two vertices, we show that there exists a complete stratifying system whose elements are regular modules. In any other case, we conclude that there are no stratifying system consisting of regular modules.

N-representation infinite algebras

From the viewpoint of higher dimensional Auslander–Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n-regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext1-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras.Applying Minamotoʼs theory on Fano algebras in non-commutative algebraic geometry, we describe the category of n-regular modules in terms of the corresponding preprojective algebra. Then we introduce n-representation tame algebras, and show that the category of n-regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an n-representation tame algebra is at least n+2.

PURITY RELATIVE TO CLASSES OF FINITELY PRESENTED MODULES

We investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different. Then we consider purities over finite-dimensional algebras, giving a general description of the relative pure-injectives which we make completely explicit in the case of tame hereditary algebras.

Extensions of tame algebras and finite group schemes of domestic representation type

Abstract Let k be an algebraically closed field. Given an extension A : B of finite-dimensional k-algebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These results are then used to show that principal blocks of finite group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of domestic representation type and the underlying group schemes are closely related to binary polyhedral group schemes.

Tilting modules over tame hereditary algebras

Abstract. We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form where is a union of tubes, and denotes the universal localization of R at in the sense of Schofield and Crawley-Boevey. Here is a direct sum of the Prüfer modules corresponding to the tubes in . Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class consists of all modules without indecomposable preprojective summands. Over an arbitrary tame hereditary algebra, T can have finite dimensional summands, but the infinite dimensional part of T is still built up from universal localizations, Prüfer modules and (localizations of) the Lukas tilting module. We also recover the classification of the infinite dimensional cotilting R-modules due to Buan and Krause.

The Gabriel-Roiter submodules of simple homogeneous modules

Let Λ \Lambda be a connected tame hereditary algebra over an algebraically closed field. We show that if Λ = k Q \Lambda =kQ is of type A ~ n \widetilde {\mathbb {A}}_n , D ~ n \widetilde {\mathbb {D}}_n , E ~ 6 \widetilde {\mathbb {E}}_6 or E ~ 7 \widetilde {\mathbb {E}}_7 , then every Gabriel-Roiter submodule of a quasi-simple module of rank 1 1 (i.e. a simple homogeneous module) has defect − 1 -1 . In particular, any Gabriel-Roiter submodule of a simple homogeneous module yields a Kronecker pair, and thus induces a full exact embedding of the category mod ⁡ k A ~ 1 \operatorname {mod} k\widetilde {\mathbb {A}}_1 into mod ⁡ Λ \operatorname {mod}\Lambda , where A ~ 1 \widetilde {\mathbb {A}}_1 is the Kronecker quiver. Consequently, we obtain that all quasi-simple modules are Gabriel-Roiter factor modules.

Generating relations for Grothendieck groups of trivial extension algebras

In this paper we describe explicitly the generating relations for the Grothendieck groups of trivial extension algebras of tame hereditary algebras.

Large tilting modules and representation type

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in Angeleri Hugel, L., Herbera, D., Trlifaj, J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010) for tame hereditary algebras.

Baer and Mittag-Leffler modules over tame hereditary algebras

We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if \({{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}\) for all torsion modules T, and M is Mittag-Leffler in case the canonical map \({M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)}\) is injective where \({\{Q_i\}_{i\in I}}\) are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.

Comparison of Auslander–Reiten Theory and Gabriel–Roiter Measure Approach to the Module Categories of Tame Hereditary Algebras

Let Λ be a tame hereditary algebra over an algebraically closed field, i.e., Λ = kQ with Q a quiver of type , , , , or . Two different kinds of partitions of the module category can be obtained by using Auslander–Reiten theory, and on the other hand, Gabriel–Roiter measure approach. We compare these two kinds of partitions and see how the modules are rearranged according to Gabriel–Roiter measure. We also show that the Gabriel–Roiter submodules can be used to build orthogonal exceptional pairs for indecomposable preprojective Λ-modules when Λ is of type and .

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group $\mathcal{G}$ is largely determined by a linearly reductive subgroup scheme $\hat{\mathcal{G}}$ of SL(2), with the McKay quiver of $\hat{\mathcal{G}}$ relative to its standard module being the Gabriel quiver of the principal block $\mathcal{B}_0(\mathcal{G})$ . The graphs underlying these quivers are extended Dynkin diagrams of type $\tilde{A}, \tilde{D}$ or $\tilde{E}$ , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.

Infinite Dimensional Representations of Canonical Algebras

AbstractThe aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to themore general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Prüfer modules, and how other modules are determined by these modules.

On the degree of irreducible morphisms

We study the degree of irreducible morphisms in generalized standard convex components of the Auslander–Reiten quiver of an artin algebra with the property that paths with the same origin and end vertices have equal length. We call the components with this last property components with length. In particular, we give two criteria to determine wether the degree of such an irreducible morphism f is finite or infinite. One of them is given in terms of the compositions of f with non-zero maps between modules in the component. The other states that the left degree of an irreducible map f is finite if and only if Ker f belongs to the component. We apply our results to irreducible morphisms over artin algebras of finite representation type and over tame hereditary algebras.