Representation theory of strongly locally finite quivers

Bautista, R.; Liu, Shiping; Paquette, Charles

doi:10.1112/plms/pds039

Cited by 26 publications

(91 citation statements)

References 35 publications

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“…By König's Lemma, the number of paths between any two pre-fixed vertices is finite. Thus, Q is strongly locally finite, and hence, the representation theory of Q obtained in [2] applies in this case. For each x ∈ Q 0 , let P x , I x , and S x be the indecomposable projective representation, the indecomposable injective representation, and the simple representation at x, respectively, which are defined in a canonical way; see, for example, [2, Section 1].…”

Section: Representations Of Quiversmentioning

confidence: 96%

“…Let rep(Q) denote the category of finite dimensional k-linear representations of Q. This is a Hom-finite hereditary abelian category, having almost split sequences; see [2,18]. The Auslander-Reiten quiver Γ rep(Q) of rep(Q) is chosen to contain the representations P x , I x and S x with x ∈ Q 0 , and its Auslander-Reiten translation is written as τ Q .…”

Section: Representations Of Quiversmentioning

confidence: 99%

“…Recall that Γ rep(Q) has a unique preprojective component P containing all the P x with x ∈ Q 0 , a unique preinjective component I containing all the I x with x ∈ Q 0 , and possibly some other regular components which contain none of the P x and I x with x ∈ Q 0 ; see [2].…”

Section: Representations Of Quiversmentioning

confidence: 99%

“…) has a connecting component C, which is obtained by gluing the preprojective component P of Γ rep(Q) with the shift by −1 of the preinjective component I in such a way that each arrow x → y in Q induces a unique arrow I x [−1] → P y in C; see [2,Section 7], and compare [8]. For convenience, we quote the following result from [2, Section 7].…”

Section: Derived Categoriesmentioning

confidence: 99%

“…Therefore, the canonical orbit category C (Q) of D b (rep(Q)), as mentioned above, is a natural candidate for a cluster category of type Q. By making use of some results obtained in [2], we shall describe the Auslander-Reiten components of C (Q); see (4.1) and prove that the projective representations in rep(Q) generate a cluster-tilting subcategory of C (Q); see (4.4). In case Q is of infinite Dynkin type, every indecomposable object of C(Q) is rigid, and consequently, a cluster-tilting subcategory of C(Q) is simply a maximal rigid subcategory which is functorially finite; see (4.7).…”

Section: Introductionmentioning

confidence: 97%

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Cluster categories of type $${\mathbb {A}}_\infty ^\infty $$ A ∞ ∞ and triangulations of the infinite strip

Liu

Paquette

2016

Math. Z.

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Abstract. We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In particular, its Auslander-Reiten components are explicitly described. When the quiver is of type A∞ or A ∞ ∞ , we show that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure as defined in [3]. When the quiver is of type A ∞ ∞ , we shall give a geometrical description of the cluster structure of the cluster category by using triangulations of the infinite strip in the plane. In particular, we shall show that the cluster-tilting subcategories are precisely given by compact triangulations.

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Section: Representations Of Quiversmentioning

confidence: 96%

Section: Representations Of Quiversmentioning

confidence: 99%

Section: Representations Of Quiversmentioning

confidence: 99%

Section: Derived Categoriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 3 more Smart Citations

Cluster categories of type $${\mathbb {A}}_\infty ^\infty $$ A ∞ ∞ and triangulations of the infinite strip

Liu

Paquette

2016

Math. Z.

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Distinguished Bases of Exceptional Modules

Ringel

2013

Algebras, Quivers and Representations

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An indecomposable representation M of a quiver Q = (Q 0 , Q 1 ) is said to be exceptional provided Ext 1 (M, M ) = 0. And it is called a tree module provided one can choose a set B of bases of the vector spaces M x (x ∈ Q 0 ) such that the coefficient quiver Γ(M, B) is a tree quiver; we call B a tree basis of M . It is known that exceptional modules are tree modules. A tree module usually has many tree bases and the corresponding coefficient quivers may look quite differently. The aim of this note is to introduce a class of indecomposable modules which have a distinguished tree basis, the "radiation modules" (generalizing an inductive construction considered already by Kinser). For a Dynkin quiver, nearly all indecomposable representations turn out to be radiation modules, the only exception is the maximal indecomposable module in case E 8 . Also, the exceptional representations of the generalized Kronecker quivers are given (via the universal cover) by radiation modules. Consequently, with the help of Schofield induction one can display all the exceptional modules of an arbitrary quiver in a nice way.Let Q = (Q 0 , Q 1 ) be a locally finite quiver. We will consider finite-dimensional representations of Q (thus kQ-modules, where kQ is the path algebra of Q). An indecomposable representation M of Q is said to be exceptional provided Ext 1 (M, M ) = 0. And it is called a tree module provided one can choose a set B of bases B x of the vector spaces M x (x ∈ Q 0 ) such that the coefficient quiver Γ(M, B) is a tree quiver; in this case, we call B a tree basis of M . Let me recall the definition of the coefficient quiver Γ(M, B) as introduced in [R3], it is a quiver whose vertices and arrows are labeled by elements of Q 0 and Q 1 , respectively. Its vertex set is the disjoint union of the sets B x , the elements of B x being labeled by x. The arrows of Γ(M, B) are obtained as follows: For an arrow α :It is known [R3] that exceptional modules are tree modules. But even if we know that a module M is a tree module, it often seems to be difficult to find directly a tree basis. The aim of this note is to provide for the exceptional modules an algorithm for obtaining a tree basis. It turns out that we should start by looking at indecomposable representations M with a thin vertex (a vertex x is said to be thin for M provided the vector space M x is one-dimensional). The first modules which we will consider are what we call the radiation modules, see sections 2 and 3. They are inductively defined, in any step one constructs an 1 indecomposable module with a thin vertex. This generalizes a construction introduced by Kinser [K] for rooted tree quivers (using the name "reduced representations").As we will see in section 4, nearly all indecomposable representations of a Dynkin quiver are radiation modules, the only exception is the maximal indecomposable module in case E 8 . The radiation modules which are our main concern are the preprojective and preinjective representations of a tree without leaves with bipartite orientation (...

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Irreducible Morphisms and Locally Finite Dimensional Representations

Paquette

2016

Algebr Represent Theor

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Let A be a Hom-finite additive Krull-Schmidt k-category where k is an algebraically closed field. Let modA denote the category of locally finite dimensional A-modules, that is, the category of covariant functors A → modk. We prove that an irreducible monomorphism in modA has a finitely generated cokernel, and that an irreducible epimorphism in modA has a finitely cogenerated kernel. Using this, we get that an almost split sequence in modA has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results to the study of rep(Q), the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of rep(Q).2000 Mathematics Subject Classification. 16G20, 16G70, 16D90. Key words and phrases. irreducible morphism, almost split sequence, Krull-Schmidt category, locally finite dimensional module, strongly locally finite quiver, Auslander-Reiten quiver.

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Representation theory of strongly locally finite quivers

Cited by 26 publications

References 35 publications

Cluster categories of type $${\mathbb {A}}_\infty ^\infty $$ A ∞ ∞ and triangulations of the infinite strip

Cluster categories of type $${\mathbb {A}}_\infty ^\infty $$ A ∞ ∞ and triangulations of the infinite strip

Distinguished Bases of Exceptional Modules

Irreducible Morphisms and Locally Finite Dimensional Representations

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