Hereditary Algebras Research Articles

Semistable torsion classes and canonical decompositions in Grothendieck groups

AbstractWe study two classes of torsion classes that generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen–Fei. More strongly, for ‐tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. We also answer a question by Derksen–Fei negatively by giving examples of algebras that do not satisfy the ray condition. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type .

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.

Wide coreflective subcategories and torsion pairs

We revisit a construction of wide subcategories going back to work of Ingalls and Thomas. To a torsion pair in the category R−mod of finitely presented modules over a left artinian ring R, we assign two wide subcategories in the category R−Mod of all R-modules and describe them explicitly in terms of an associated cosilting module. It turns out that these subcategories are coreflective, and we address the question of which wide coreflective subcategories can be obtained in this way. Over a tame hereditary algebra, they are precisely the categories which are perpendicular to collections of pure-injective modules.

Tilting Quivers for Hereditary Algebras

Let A be a finite dimensional hereditary algebra over an algebraically closed field k. In this paper, we study the tilting quiver of A from the viewpoint of τ-tilting theory. First, we prove that there exists an isomorphism between the support τ-tilting quiver Q(sτ-tilt A) of A and the tilting quiver Q(tilt A¯) of the duplicated algebra A¯. Then, we give a new method to calculate the number of arrows in the tilting quiver Q(tilt A) when A is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of Q(tilt A) contains only finitely many non-saturated vertices. We provide an example to show that this conjecture does not hold for some algebras whose quivers are wild with at least four vertices.

Exact Borel subalgebras of path algebras of quivers of Dynkin type [formula omitted

Hereditary algebras are quasi-hereditary with respect to any adapted partial order on the indexing set of the isomorphism classes of their simple modules. For any adapted partial order on {1,…,n}, we compute the quiver and relations for the Ext-algebra of standard modules over the path algebra of a uniformly oriented linear quiver with n vertices. Such a path algebra always admits a regular exact Borel subalgebra in the sense of König and we show that there is always a regular exact Borel subalgebra containing the idempotents e1,…,en and find a minimal generating set for it. For a quiver Q and a deconcatenation Q=Q1⊔Q2 of Q at a sink or source v, we describe the Ext-algebra of standard modules over KQ, up to an isomorphism of associative algebras, in terms of that over KQ1 and KQ2. Moreover, we determine necessary and sufficient conditions for KQ to admit a regular exact Borel subalgebra, provided that KQ1 and KQ2 do. We use these results to obtain sufficient and necessary conditions for a path algebra of a linear quiver with arbitrary orientation to admit a regular exact Borel subalgebra.

Morphisms and extensions between bricks over preprojective algebras of type A

The bricks over preprojective algebras of type A are known to be in bijection with certain combinatorial objects called “arcs”. In this paper, we show how one can use arcs to compute bases for the Hom-spaces and first extension spaces between bricks. We then use this description to classify the “weak exceptional sequences” over these algebras. Finally, we explain how our result relates to a similar combinatorial model for the exceptional sequences over hereditary algebras of type A.

Preprojective algebras of d-representation finite species with relations

In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We generalize this result to the species case.We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by the mapping cone of a chain map, which is homogeneous of degree 1 with respect to a certain grading. We also study a higher dimensional analogue of representation finite hereditary algebras called d-representation finite algebras. One source of d-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.

Algebra extensions and derived-discrete algebras

Let be an algebra homomorphism between finite dimensional algebras. We prove that if is split, the derived-discreteness of A implies the derived-discreteness of B; if is separable and the right A-module BA is projective, the converse holds. We prove an analogous statement for piecewise hereditary algebras.

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.

Tilting quivers for BB‐tilted algebras

AbstractLet be a hereditary algebra, be a tilted algebra. We will construct tilting ‐modules from tilting ‐modules and use this result to show how tilting quivers of BB‐tilted algebras can be obtained from those of .

Sections in the bounded derived category of piecewise hereditary algebras

Let [Formula: see text] be a finite-dimensional algebra over an algebraically closed field. We study sections in the Auslander–Reiten quiver of the categories of complexes of fixed size and in the Auslander–Reiten quiver of the bounded derived category. One of the aims of this work is to show where the indecomposable [Formula: see text]-modules can be found in the Auslander–Reiten quiver of the derived category, whenever [Formula: see text] is a piecewise hereditary algebra. We also prove that we can obtain a transjective component of the bounded derived category from a section. As an application of the results of sections, we find a bound on the strong global dimension of some piecewise hereditary algebras of tree type. Moreover, we also determine the strong global dimension of some algebras taking into account their ordinary quivers with relations.

On the nilpotency index of the radical of a module category

Let A be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of QA are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of A. Precisely, we study the above mentioned problem whenever A is a monomial or a toupie algebra where their Auslander-Reiten quivers are component with length. We also study the above problem for string algebras. We give concrete formulas to compute such index for many classes of algebras such as toupie algebras, tree algebras and some classes of pullback algebras over the hereditary algebras of type An, Dn and E6. Furthermore, we determine the index taking into account the ordinary quiver of the given algebras.

Rigidity dimensions of Hochschild extensions of hereditary algebras of type D

Rigidity dimension of algebras is a new homological dimension which measures the quality of resolutions of algebras by algebras of finite global dimension and large dominant dimension. In this paper, we calculate the rigidity dimension of the Hochschild extension of a hereditary algebra H of Dynkin type D n ( n ≥ 4 ) by the standard duality H -bimodule. It turns out that this dimension is independent of n and the orientation of D n .

Classifying torsion classes for algebras with radical square zero via sign decomposition

To study the set of torsion classes of a finite dimensional basic algebra over a field, we use a decomposition, called sign-decomposition, parameterized by elements of {±1}n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ϵ∈{±1}n there is a hereditary algebra Aϵ! with radical square zero and a bijection between the set of torsion classes of A associated to ϵ and the set of faithful torsion classes of Aϵ!. Furthermore, this bijection preserves the property of being functorially finite. From a point of view of tilting theory, it implies that there is a bijection between the set of isomorphism classes of basic two-term silting complexes for A associated to ϵ and the set of isomorphism classes of basic tilting Aϵ!-modules. As an application, we prove that the number of two-term tilting complexes over Brauer line algebras (respectively, Brauer cycle algebras) having n edges is (2nn) (respectively, 22n−1 if n is odd, and ∞ if n is even).

On 2-Representation Infinite Algebras Arising From Dimer Models

AbstractThe Jacobian algebra arising from a consistent dimer model is a bimodule 3-Calabi–Yau algebra, and its center is a 3-dimensional Gorenstein toric singularity. A perfect matching (PM) of a dimer model gives the degree, making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a 2-representation infinite algebra that is a generalization of a representation infinite hereditary algebra. Internal PMs, which correspond to toric exceptional divisors on a crepant resolution of a 3-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Combining this characterization with the theorems due to Amiot–Iyama–Reiten, we show that the stable category of graded maximal Cohen–Macaulay modules admits a tilting object for any 3-dimensional Gorenstein toric isolated singularity. We then show that all internal PMs corresponding to the same toric exceptional divisor are transformed into each other using the mutations of PMs, and this induces derived equivalences of 2-representation infinite algebras.

Positive Fuss–Catalan Numbers and Simple-Minded Systems in Negative Calabi–Yau Categories

Abstract We establish a bijection between $d$-simple-minded systems ($d$-SMSs) of $(-d)$-Calabi–Yau cluster category $\mathcal C_{-d}(H)$ and silting objects of ${\mathcal {D}}^{\mathrm {b}}(H)$ contained in ${\mathcal {D}}^{\le 0}\cap {\mathcal {D}}^{\ge 1-d}$ for hereditary algebra $H$ of Dynkin type and $d\ge 1$. We show that the number of $d$-SMSs in $\mathcal C_{-d}(H)$ is the positive Fuss–Catalan number $C_{d}^{+}(W)$ of the corresponding Weyl group $W$, by applying this bijection and Buan–Reiten–Thomas’ and Zhu’s results on Fomin–Reading’s generalized cluster complexes. Our results are based on a refined version of silting-$t$-structure correspondence.

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].

Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

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.

Piecewise hereditary incidence algebras of Dynkin and extended Dynkin type

Let be the incidence algebra of a poset . We study some aspects of incidence algebras that are piecewise hereditary, which we denominate PHI algebras. We describe the quiver with relations of the PHI algebras of Dynkin type. Also we introduce a new family of PHI algebras of extended Dynkin type, which we call ANS family, in reference to Assem, Nehring, and Skowroński. In this description, the important method was the one of cutting sets on trivial extensions. We implemented this method in the computer algebra system GAP/QPA to shows exactly the quivers with relations resulted.