Indecomposable Objects Research Articles

Primitively generated Hall algebras

In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for non-hereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.

Injective objects of monomorphism categories

For an acyclic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q,A) and prove that Mon(Q,A) has enough injective objects.

String and Band Complexes over Certain Algebra of Dihedral Type

We give a combinatorial description of a family of indecomposable objects in the triangulated category of perfect complexes \(\mathcal {K}^{b}(\mathcal {P}_{\Lambda })\), where Λ is a certain algebra of dihedral type (as introduced by K. Erdmann) with three simple modules. We then discuss the shape of the corresponding components of the Auslander-Reiten quiver of \(\mathcal {K}^{b}(\mathcal {P}_{\Lambda })\) containing these objects.

Categorical Realizations of Quivers

We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers (in the sense of [9]), and bilinear pairings as special cases. In this general context, we prove the cancelation from direct sums, and show the existence and uniqueness of decomposition into a sum of indecomposable objects, provided certain assumptions hold. This yields similar results for the special cases just mentioned. Using similar ideas, we also prove a version of Fitting's Lemma for natural transformations between functors.

Gorenstein homological theory for differential modules

We show that a differential module is Gorenstein projective (injective, respectively) if and only if its underlying module is Gorenstein projective (injective, respectively). We then relate the Ringel–Zhang theorem on differential modules to the Avramov–Buchweitz–Iyengar notion of projective class of differential modules and prove that for a ring R there is a bijective correspondence between projectively stable objects of split differential modules of projective class not more than 1 and R-modules of projective dimension not more than 1, and this is given by the homology functor H and stable syzygy functor ΩD. The correspondence sends indecomposable objects to indecomposable objects. In particular, we obtain that for a hereditary ring R there is a bijective correspondence between objects of the projectively stable category of Gorenstein projective differential modules and the category of all R-modules given by the homology functor and the stable syzygy functor. This gives an extended version of the Ringel–Zhang theorem.

Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

AbstractThe (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

Krull–Schmidt categories and projective covers

Krull–Schmidt categories are additive categories such that each object decomposes into a finite direct sum of indecomposable objects having local endomorphism rings. We provide a self-contained introduction which is based on the concept of a projective cover.

Atoms in the p-localization of Stable Homotopy Category

We study p-localizations, where p is an odd prime, of the full subcategories $$ {\mathcal{S}}^n $$ of stable homotopy category formed by CW-complexes with cells in n successive dimensions. Using the technique of triangulated categories and matrix problems, we classify the atoms (indecomposable objects) in $$ {\mathcal{S}}_p^n $$ for n ≤ 4(p − 1) and show that, for n > 4(p − 1), this classification is wild in a sense of the representation theory.

Reduced representations of rooted trees

Let k be a field and Q a quiver. The category of finite dimensional representations of Q will be denoted by repk(Q). We can define the pointwise tensor product on the category repk(Q), i.e. for two representations N, M the representation N⊗M is defined at a vertex x∈Q0 as (N⊗M)x=Nx⊗Mx and for an arrow α∈Q1 we define (N⊗M)α=Nα⊗Mα. This tensor product has been investigated in recent work of Herschend and Kinser [1–3]. In [3] Kinser investigated the representation ring of rooted trees. A rooted tree (Q,σ) is a quiver Q, whose underlying graph is tree and which has a unique sink σ. The representation ring of Q is the free abelian group generated by the isomorphism classes of indecomposable objects in repk(Q). It is endowed with a multiplication induced by the pointwise tensor product. See [3] for details. For these rooted trees, Kinser constructed reduced representations, which are defined inductively. These reduced representations then lead to a set of orthogonal primitive idempotents in the representation ring of Q. On the other hand, there is an inductive construction of so-called radiation modules for a tree with arbitrary orientation in [5]. In the special case of a rooted tree quiver, this construction results in the reduced representations introduced by Kinser. The idea used in the construction of the radiation modules then allows us to prove a new characterization of reduced representations which isTheoremLet Q be a rooted tree quiver. An indecomposable representationV∈repk(Q)is reduced if and only if V is a direct summand ofV⊗V.This theorem then allows us to prove further results for the special case, where Q is a rooted tree quiver, whose underlying graph is of Dynkin type. In that case V being reduced is also equivalent to the condition that the vector space of V at the sink σ is one dimensional.

From submodule categories to preprojective algebras

Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\). Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\); the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\). Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\), thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.

Cotorsion pairs in the cluster category of a marked surface

We study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surface and the indecomposable objects in the associated cluster category, we prove that the dimension of extension space of two indecomposable objects in the cluster categories equals to the intersection number of the corresponding curves. By using this result, we prove that there are no non-trivial t-structures in the cluster categories when the surface is connected. Based on this result, we give a classification of cotorsion pairs in these categories. Moreover we define the notion of paintings of a marked surface without punctures and their rotations. They are a geometric model of cotorsion pairs and of their mutations respectively.

Quantum Grothendieck rings and derived Hall algebras

Abstract Let 𝔤 = 𝔫 ⊕ 𝔥 ⊕ 𝔫- be a simple Lie algebra over ℂ of type A, D, E, and let Uq (L𝔤) be the associated quantum loop algebra. Following Nakajima [Ann. of Math. (2) 160 (2004), 1057–1097], Varagnolo–Vasserot [Studies in memory of Issai Schur, Birkhäuser-Verlag, Basel (2002), 345–365], and the first author [Adv. Math. 187 (2004), 1–52], we study a t-deformation 𝒦 t of the Grothendieck ring of a tensor category 𝒞ℤ of finite-dimensional Uq (L𝔤)-modules. We obtain a presentation of 𝒦 t by generators and relations. Let Q be a Dynkin quiver of the same type as 𝔤. Let DH(Q) be the derived Hall algebra of the bounded derived category Db (mod(F Q)) over a finite field F, introduced by Toën [Duke Math. J. 135 (2006), 587–615]. Our presentation shows that the specialization of 𝒦 t at t = |F|1/2 is isomorphic to DH(Q). Under this isomorphism, the classes of fundamental Uq (L𝔤)-modules are mapped to scalar multiples of the classes of indecomposable objects in DH(Q). Our presentation of 𝒦 t is deduced from the preliminary study of a tensor subcategory 𝒞 Q of 𝒞ℤ analogous to the heart mod(F Q) of the triangulated category Db (mod(F Q)). We show that the t-deformed Grothendieck ring 𝒦 t,Q of 𝒞 Q is isomorphic to the positive part of the quantum enveloping algebra of 𝔤, and that the basis of classes of simple objects of 𝒦 t,Q corresponds to the dual of Lusztig's canonical basis. The proof relies on the algebraic characterizations of these bases, but we also give a geometric approach in the last section. It follows that for every orientation Q of the Dynkin diagram, the category 𝒞 Q gives a new categorification of the coordinate ring ℂ[N] of a unipotent group N with Lie algebra 𝔫, together with its dual canonical basis.

Indecomposables live in all smaller lengths

We show that there are no gaps in the lengths of the indecomposable objects in an abelian k k -linear category over a field k k provided all simples are absolutely simple. To derive this natural result we prove that any distributive minimal representation-infinite k k -category is isomorphic to the linearization of the associated ray category which is shown to have an interval-finite universal cover with a free fundamental group so that the well-known theory of representation-finite algebras applies.

Cluster tilting vs. weak cluster tilting in Dynkin type A infinity

Abstract This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category 𝖢 with the following properties. On the one hand, the d-cluster tilting subcategories of 𝖢 have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of 𝖢 which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d - 1, we show a weakly d-cluster tilting subcategory 𝖳 ℓ ${\mathsf {T}_{\ell }}$ which has an indecomposable object with precisely ℓ mutations. The category 𝖢 is the algebraic triangulated category generated by a (d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A ∞.

A Module-Theoretic Interpretation of Schiffler's Expansion Formula

We give a module-theoretic interpretation of Schiffler's expansion formula which is defined combinatorially in terms of complete (Γ, γ)-paths in order to get the expansion of the cluster variables in the cluster algebra of a marked surface (S, M). Based on the geometric description of the indecomposable objects of the cluster category of the marked surface (S, M), we show the coincidence of Schiffler–Thomas’ expansion formula and the cluster character defined by Palu.

Deligne’s category \underline{𝑅𝑒}𝑝(𝐺𝐿_{𝛿}) and representations of general linear supergroups

We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric Schur polynomials, and give a method for decomposing their tensor products. Along the way, we describe indecomposable objects in Re _ p ⁡ ( G L δ ) \underline {\operatorname {Re}}\!\operatorname {p}(GL_\delta ) and explain how to decompose their tensor products.

A Caldero–Chapoton map for infinite clusters

We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. The map is not necessarily defined on all objects of the triangulated category, but we show that it is a (weak) cluster map in the sense of Buan-Iyama-Reiten-Scott. As a corollary, it induces a surjection from the set of exceptional objects which can be reached from the cluster tilting subcategory to the set of cluster variables of an associated cluster algebra. Along the way, we study the interaction between Calabi-Yau reduction, cluster structures, and the Caldero-Chapoton map. We apply our results to the cluster category D of Dynkin type A infinity which has a rich supply of cluster tilting subcategories with infinitely many indecomposable objects. We show an example of a cluster map which cannot be extended to all of D. The case of D also permits us to illuminate results by Assem-Reutenauer-Smith on SL_2-tilings of the plane.

The relative singularity category of a non-commutative resolution of singularities

In this article, we study a triangulated category associated with a non-commutative resolution of singularities. In particular, we give a complete description of this category in the case of a curve with nodal singularities, classifying its indecomposable objects and computing its Auslander–Reiten quiver and K-group.

A geometric model of tube categories

We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented quiver of type A∞∞.

Categorification of a frieze pattern determinant

Broline, Crowe and Isaacs have computed the determinant of a matrix associated to a Conway–Coxeter frieze pattern. We generalise their result to the corresponding frieze pattern of cluster variables arising from the Fomin–Zelevinsky cluster algebra of type A. We give a representation-theoretic interpretation of this result in terms of certain configurations of indecomposable objects in the root category of type A.