Representation-infinite Algebras Research Articles

MINIMAL (-)TILTING INFINITE ALGEBRAS

AbstractMotivated by a new conjecture on the behavior of bricks, we start a systematic study of minimal $\tau $ -tilting infinite (min- $\tau $ -infinite, for short) algebras. In particular, we treat min- $\tau $ -infinite algebras as a modern counterpart of minimal representation-infinite algebras and show some of the fundamental similarities and differences between these families. We then relate our studies to the classical tilting theory and observe that this modern approach can provide fresh impetus to the study of some old problems. We further show that in order to verify the conjecture, it is sufficient to treat those min- $\tau $ -infinite algebras where almost all bricks are faithful. Finally, we also prove that minimal extending bricks have open orbits, and consequently obtain a simple proof of the brick analogue of the first Brauer–Thrall conjecture, recently shown by Schroll and Treffinger using some different techniques.

τ-tilting finiteness of non-distributive algebras and their module varieties

We treat the τ-tilting finiteness of minimal representation-infinite algebras and particularly the non-distributive ones. Building upon the new results of Bongartz, we fully determine which algebras in this family are τ-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the minimal representation-infinite biserial algebras. Consequently, we obtain nontrivial explicit conditions for τ-tilting infiniteness of a large family of algebras. This also produces concrete families of “minimal τ-tilting infinite algebras”, recently studied in our work.We further use our results to establish a conjectural connection between the τ-tilting theory and two geometric notions (the dense orbit property and Schur-representation finiteness) introduced by Chindris, Kinser and Weyman while studying module varieties. We verify the conjectures for the algebras studied in this note: For the minimal representation-infinite algebras which are non-distributive or biserial, if Λ has the dense orbit property, then Λ is τ-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is τ-tilting finite. This gives a categorical interpretation of Schur-representation-finiteness over this family of algebras.

Yoneda algebras and their singularity categories

For a finite dimensional algebra Λ $\Lambda$ of finite representation type and an additive generator M $M$ for mod Λ $\operatorname{mod}\Lambda$ , we investigate the properties of the Yoneda algebra Γ = ⨁ i ⩾ 0 Ext Λ i ( M , M ) $\Gamma =\bigoplus _{i \geqslant 0}\operatorname{Ext}_\Lambda ^i(M,M)$ . We show that Γ $\Gamma$ is graded coherent and Gorenstein of self-injective dimension at most 1, and the graded singularity category D sg Z ( Γ ) $\mathrm{D_{sg}^\mathbb {Z}}(\Gamma )$ of Γ $\Gamma$ is triangle equivalent to the derived category of the stable Auslander algebra of Λ $\Lambda$ . These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category Y $\mathcal {Y}$ of Λ $\Lambda$ as the additive closure of the shifts of the Λ $\Lambda$ -modules in the derived category D b ( mod Λ ) $\mathrm{D^b}(\operatorname{mod}\Lambda )$ . We show that Y $\mathcal {Y}$ is coherent and Gorenstein of self-injective dimension at most 1, and the singularity category of Y $\mathcal {Y}$ is triangle equivalent to the derived category D b ( mod ( mod ̲ Λ ) ) $\mathrm{D^b}(\operatorname{mod}(\operatorname{\underline{\operatorname{mod}}}\Lambda ))$ of the stable category mod ̲ Λ $\operatorname{\underline{\operatorname{mod}}}\Lambda$ . To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an f-category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.

Hochschild cohomology for periodic algebras of polynomial growth

We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.

On Simple Connectedness of Minimal Representation-Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

Regular modules over 2-dimensional quantum Beilinson algebras of Type $$S$$ S

In the study of a finite dimensional hereditary algebra of infinite representation type, understanding regular modules is essential. Recently, Herschend, Iyama and Oppermann introduced the notions of \(d\)-representation infinite algebra and \(d\)-regular module, extending the above notions to finite dimensional algebras of global dimension \(d\ge 1\). Since the Beilinson algebras of AS-regular algebras of dimension \(d+1\) are typical examples of \(d\)-representation infinite algebras, the purpose of this paper is to study the behavior of \(d\)-regular modules over such algebras. In particular, we will show that the isomorphism classes of simple 2-regular modules over a 2-representation tame quantum Beilinson algebra of Type \(S\) are parameterized by \({\mathbb {P}}^{2}\).

On Mild Contours in Ray Categories

We generalize and refine the structure and disjointness theorems for non-deep contours obtained in the fundamental article ‘Multiplicative bases and representation-finite algebras’. In particular we show that these contours do not occur in minimal representation-infinite algebras.

ON THE GEOMETRY OF ORBIT CLOSURES FOR REPRESENTATION-INFINITE ALGEBRAS

AbstractFor the Kronecker algebra, Zwara found in [14] an example of a module whose orbit closure is neither unibranch nor Cohen-Macaulay. In this paper, we explain how to extend this example to all representation-infinite algebras with a preprojective component.

The number of the Gabriel–Roiter measures admitting no direct predecessors over a wild quiver

A famous result by Drozd says that a finite-dimensional representation-infinite algebra is of either tame or wild representation type. But one has to make assumption on the ground field. The Gabriel–Roiter measure might be an alternative approach to extend these concepts of tame and wild to arbitrary Artin algebras. In particular, the infiniteness of the number of GR segments, i.e. sequences of Gabriel–Roiter measures which are closed under direct predecessors and successors, might relate to the wildness of Artin algebras. As the first step, we are going to study the wild quiver with three vertices, labeled by 1, 2 and 3, and one arrow from 1 to 2 and two arrows from 2 to 3. The Gabriel–Roiter submodules of the indecomposable preprojective modules and quasi-simple modules τ − i M , i ≥ 0 are described, where M is a Kronecker module and τ = D T r is the Auslander–Reiten translation. Based on these calculations, the existence of infinitely many GR segments will be shown. Moreover, it will be proved that there are infinitely many Gabriel–Roiter measures admitting no direct predecessors.

Lower bounds for Auslander's representation dimension

The representation dimension is an invariant introduced by Auslander to measure how far a representation infinite algebra is from being representation finite. In 2005, Rouquier gave the first examples of algebras of representation dimension greater than three. Here we give the first general method for establishing lower bounds for the representation dimension of given algebras or families of algebras. The classes of algebras for which we explicitly apply this method include (but do not restrict to) most of the previous examples of algebras of large representation dimension, for some of which the lower bound is improved to the correct value

Moduli Spaces for Representations of Concealed-Canonical Algebras

Continuing M. Domokos and H. Lenzing [ J. Algebra 228 (2000), 738–762], we investigate the relative invariants and moduli spaces introduced in A. D. King [ Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530] which are related to a separating family of stable tubes in the module category of a concealed-canonical algebra. The class of concealed-canonical algebras consists of representation-infinite algebras of tame or wild representation type and contains in particular the representation theory of extended Dynkin quivers. We present a uniform characteristic-free approach avoiding case-by-case analysis. We introduce the notion of admissible weight for such algebras; these are the weights (i.e., elements of the dual of the Grothendieck group of the module category) attached to the infinite moduli spaces for families of modules from the separating subcategory. By using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories. We construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras. It turns out that for any concealed-canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space. Moreover, any such moduli space can be naturally identified with the moduli space belonging to a special weight, called the rank. As a consequence we show that in the case of a tame concealed algebra, any infinite moduli space for families of modules is a projective space, and all fields of rational invariants on irreducible components of representation spaces are purely transcendental.

Derived dynkin extensions

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri]. We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions. Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments.

On the category of modules of second kind for Galois coverings

Let F: R → R/G be a Galois covering and $mod_1(R/G)$ (resp. $mod_2(R/G)$) be a full subcategory of the module category mod (R/G), consisting of all R/G-modules of first (resp. second) kind with respect to F. The structure of the categories $(mod (R/G))/[m

Complete slices and homological properties of tilted algebras

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

Galois coverings of representation-infinite algebras

Galois coverings of representation-infinite algebras