Auslander-Reiten Quiver of Nakayama Algebra Nn-2,n (original) (raw)
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Auslander Reiten Quiver of Nakayama Algebra type Dynkin Graph An
2013
In this paper we will determine Auslander Reiten quiver of Nakayama algebra with quiver type Dynkin graph A n for all natural number n ≥ 2. The ARquiver is a visualization of module category of finite dimensional algebras. From the AR-quiver of an algebra A we may know all the isomorphism classes of indecomposable modules in mod A and the homomorphism between them. Once we get the general shape of the AR-quiver of this algebra, we will use it to compute a tilting module of this algebra.
A characterization of finite Auslander—Reiten quivers
Journal of Algebra, 1984
Let II be an indecomposable artin algebra of finite representation type, R the center of A, k = R/rad R. Let r,, be the associated Auslander-Reiten quiver. Then r,, is finite k-modulated translation quiver and it is unique up to isomorphism of modulated translation quivers (Section 2). The main results of this paper are:
On the Postprojective Partitions and Components of the Auslander-Reiten Quivers
Communications in Algebra, 2016
In this paper we shall investigate further the connections between the postprojective partition of an algebra and its Auslander-Reiten quiver. Auslander-Smalø introduced, in [3], the notion of postprojective partition and modules (under the name of preprojective). The connection between such a partition and the structure of the Auslander-Reiten quiver has been investigated in several papers such as [1, 3, 6, 7, 8, 12]. The purpose of this paper is to follow such investigations. We introduce the notion of P-discrete component of the Auslander-Reiten quiver Γ A as follows. Let {P i } of indA with i ∈ N ∞ = N ∪ {∞} be the postprojective partition of A (recall the definition below). A component Γ of Γ A is P-discrete if for each i ≥ 0 and each M ∈ Γ ∩ P i , we have that tr P i+1 (M) = tr P∞ (M), where tr C (M) denotes the trace of the set of modules C in M (see Section 1 below). Theorem 2.3. Let A be a representation-infinite Artin algebra. If Γ is a P-discrete connected component of Γ A then there is no arrow M → N in Γ with M ∈ P i and N ∈ P j such that i + 1 < j < ∞. Also, using the notion of left degree of a morphism (introduced by Liu in [10]), we prove the following result.
Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers
2019
We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.
Right n-Nakayama Algebras and their Representations
Algebras and Representation Theory, 2019
In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra Λ is representation-finite if and only if Λ is right n-Nakayama for some positive integer n. We classify hereditary right n-Nakayama algebras. We also define right n-coNakayama algebras and show that an artin algebra Λ is right n-coNakayama if and only if Λ is left n-Nakayama. We then study right 2-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right 2-Nakayama algebra. We end by classifying finite dimensional right 2-Nakayama algebras in terms of their quivers with relations. 1. Introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as an R-module. Given an artin algebra Λ, it is a quite natural question to ask for the classification of all the indecomposable finitely generated right Λ-modules. Only for few classes of algebras such a classification is known, one of the first such class were the Nakayama algebras. A Nakayama algebra Λ is an algebra such that the indecomposable projective right Λ-modules as well as the indecomposable injective right Λ-modules are uniserial. This then implies that all the indecomposable right Λmodules are uniserial. Nakayama algebras were studied by Tadasi Nakayama who called them generalized uniserial rings [9, 10]. A right Λ-module M is called uniserial if it has a unique composition series. Uniserial modules are the simplest indecomposables and this makes it interesting to understand their role in the category mod(Λ) of finitely generated right Λ-modules. An artin algebra Λ is said to be representation-finite, provided there are only finitely many isomorphism classes of indecomposable right Λ-modules. In representation theory, representation-finite algebras are of particular importance since in this case one has a complete combinatorial description of the module category in terms of the Auslander-Reiten quiver. The class of Nakayama algebras is one of the fundamental classes of representation-finite algebras whose representation theory is completely understood. In this paper we introduce the notion of n-factor serial modules. We say that a nonuniserial right Λ-module M of length l is n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. In some sense, n is an invariant that measures how far M is from being uniserial. We say that an artin algebra Λ is right n-Nakayama if every finitely generated indecomposable right Λ-module is i-factor serial for some 1 i n
Toupie algebra, some examples of laura algebras
arXiv (Cornell University), 2010
We consider the bound quiver algebras whose ordinary quiver is that of a canonical algebra. We determine which of those algebras are hereditary, tilted, quasitilted, weakly shod or laura algebras. 1. Introduction. The introduction of quasitilted algebra in the early 90's by Happel, Ringel and Smalø give rise to the study of two classes of modules named the right and the left part of the module category. In [10], they proved that an algebra is quasitilted if and only if all indecomposable projective modules lie in the left part (and equivalently, that all indecomposable injective modules lie in the right part). Moreover, in this case all indecomposables modules must lie in one of those part. During the last two decades, the structure of quasitilted algebras and their module categories has been thoroughly studied and, by a consequence, the knowledge of the structure of the left and right part of the module category of an arbitrary algebra also. Those works leads to various generalisations of classes of algebras. In this work, we will study examples of the following subclasses of algebras, each of which is properly contained in the following one: Hereditary algebras, Tilted algebras, Quasitilted algebras, Weakly shod algebras, and Laura algebras. Let k be a commutative field. A bound quiver algebra A = kQ/I is called a toupie algebra if Q is a non linear quiver with a unique source 0, a unique sink ∞ and for any other vertex x in Q, there is exactly one arrow starting at x and exactly one arrow ending at x. Paths going from 0 to ∞ are called branches of Q. Remark that those quiver correspond with those of canonical algebras introduce in 1984 by Ringel in [15] or are of type A n. Our goal is to determine to which of the above classes a given toupie algebra belongs by looking at its bound quiver. The motivation for this classification of toupie algebra came from a result of Assem and Coelho [2] saying that, for the above classes, for any idempotent e of A, the algebra eAe belongs to the same classes of algebras than A, extending [8], [11] and [13]. Observe that, for any algebra A, there always exist an idempotent e such that eAe is a toupie algebra.
Journal of Algebra, 2003
Let ∆ be a Dynkin diagram and k an algebraically closed field. Let A be an iterated tilted finitedimensional k-algebra of type ∆ and denote by its repetitive algebra. We approach the problem of finding a combinatorial algorithm giving the embedding of the vertices of the Auslander-Reiten quiver Γ A of A in the Auslander-Reiten quiver Γ (mod(Â)) Z∆ of the stable category mod(Â). Let T be a trivial extension of finite representation type and Cartan class ∆. Assume that we know the vertices of Z∆ corresponding to the radicals of the indecomposable projective T -modules. We determine the embedding of Γ A in Z∆ for any algebra A such that T (A) T .
Journal of Algebra, 2003
Let ∆ be a Dynkin diagram and k an algebraically closed field. Let A be an iterated tilted finitedimensional k-algebra of type ∆ and denote by its repetitive algebra. We approach the problem of finding a combinatorial algorithm giving the embedding of the vertices of the Auslander-Reiten quiver Γ A of A in the Auslander-Reiten quiver Γ (mod(Â)) Z∆ of the stable category mod(Â). Let T be a trivial extension of finite representation type and Cartan class ∆. Assume that we know the vertices of Z∆ corresponding to the radicals of the indecomposable projective T-modules. We determine the embedding of Γ A in Z∆ for any algebra A such that T (A) T .
On the classification of higher Auslander algebras for Nakayama algebras
Journal of Algebra
We give new improved bounds for the dominant dimension of Nakayama algebras and use those bounds to give a classification of Nakayama algebras with n simple modules that are higher Auslander algebras with global dimension at least n. The classification is then used to extend the results on the inequality for the global dimension of Nakayama algebras obtained in [MM].