Representation-finite Algebras Research Articles

Degenerations for modules over representation-finite algebras

Let A A be a representation-finite algebra. We show that a finite dimensional A A -module M M degenerates to another A A -module N N if and only if the inequalities dim K ⁡ H o m A ( M , X ) ≤ dim K ⁡ H o m A ( N , X ) \dim _{K} Hom_{A}(M,X)\leq \dim _{K} Hom_{A}(N,X) hold for all A A -modules X X . We prove also that if Ext A 1 ⁡ ( X , X ) = 0 \operatorname {Ext}_{A}^{1}(X,X)=0 for any indecomposable A A -module X X , then any degeneration of A A -modules is given by a chain of short exact sequences.

The Δ-Filtered Modules Without Self-Extensions for the Auslander Algebra of k[ T ]/〈 T n 〉

It is well known that the Auslander algebra of any representation finite algebra is quasi-hereditary. We consider the Auslander algebra A n of k[T]/〈 n 〉 (here, k is a field, T a variable and n a natural number). We determine all Δ-filtered A n -modules without self-extensions. They can be described purely combinatorially. Given any Δ-filtered module N, we show that there is (up to isomorphism) a unique Δ-filtered module M without self-extensions which has the same dimension vector. In the case where k is an infinite field, N is a degeneration of this module M. In particular, we see that in this case, the set of Δ-filtered modules with a fixed dimension vector is the closure of an open orbit (thus irreducible). As observed by Hille and Rohrle, the problem of describing all Δ-filtered A n -modules is the same as that of describing the conjugacy classes of elements in the unipotent radical of a parabolic subgroup P of GL(m, k) under the action of P, thus we recover Richardson's dense orbit theorem in this instance.

Hammocks and the Nazarova-Roiter Algorithm

Hammocks have been considered by Brenner [1], who gave a numerical criterion for a finite translation quiver to be the Auslander–Reiten quiver of some representation-finite algebra. Ringel and Vossieck [11] gave a combinatorial definition of left hammocks which generalised the concept of hammocks in the sense of Brenner, as a translation quiver H and an additive function h on H (called the hammock function) satisfying some conditions. They showed that a thin left hammock with finitely many projective vertices is just the preprojective component of the Auslander–Reiten quiver of the category of S-spaces, where S is a finite partially ordered set (abbreviated as ‘poset’). An important role in the representation theory of posets is played by two differentiation algorithms. One of the algorithms was developed by Nazarova and Roiter [8], and it reduces a poset S with a maximal element a to a new poset S′=a∂S. The second algorithm was developed by Zavadskij [13], and it reduces a poset S with a suitable pair (a, b) of elements a, b to a new poset S′=∂(a,b)S. The main purpose of this paper is to construct new left hammocks from a given one, and to show the relationship between these new left hammocks and the Nazarova–Roiter algorithm. In a later paper [5], we discuss the relationship between hammocks and the Zavadskij algorithm.

Non-split extensions over representation-finite hereditary algebras

Let be the path algebra of a quiver with underlying graph An or Dn. For all pairs of indecomposable Λ-modules M and N is computed and a basis consisting of short exact sequences 0 → M → Y → N → 0 is constructed.

Degenerations for Modules over Representation-Finite Biserial Algebras

Degenerations for Modules over Representation-Finite Biserial Algebras

Some Hall polynomials for representation-finite selfinjective algebra with modpΛ without short chains

Some Hall polynomials for representation-finite selfinjective algebra with modpΛ without short chains

Strongly simply connected Auslander algebras

Letkbe an algebraically closed field. By an algebra is meant an associative finite dimensionalk-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

Hammocks and the Algorithms of Zavadskii

w x Hammocks have been considered by Brenner 3 in order to give a numerical criterion for a finite translation quiver to be the Auslander] Reiten quiver of some representation-finite algebra. Ringel and Vossieck w x 13 gave a combinatorial definition of left hammocks, which generalizes the concept of hammocks, in the sense of Brenner, as a translation quiver Ž . H and an additive function h on H called the hammock function satisfying some conditions. They also showed that a thin left hammock with finitely many projective vertices is just the preprojective component of the Auslander]Reiten quiver of the category of S , where S is a finite Ž . partially ordered sets abbreviated poset . An important role of posets in representation theory is played by two differentiation algorithms. One of w x the algorithms is due to Nazarova and Roiter 9 and it reduces a poset S with a maximal element a g S to a new poset S X s ­ S with same a w x representation type. The second algorithm is due to Zavadskii 15 and it Ž . reduces a poset S with a suitable pair a, b of elements a, b to a new poset S X s ­ S with same representation type. Zavadskii ’s algorithm Ža, b. is successfully used to give new proofs for characterizing posets of finite w x w x type 5 and for characterizing posets of wild type 10 in studying posets of w x w x finite growth 15 . In the paper 7 , we discussed the relationship between hammocks and the algorithm of Nazarova and Roiter. The main purpose of the present paper is to construct some new left hammocks from a given one, and to show the relationship between these new left hammocks and the algorithm of Zavadskii . In Section 2, we recall some basic definitions and facts. Let H be a thin Ž . left hammock with hammock function h , let p a a projective vertex of H Ž . H different from the source, and let q b an injective vertex of H different

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

Construction of representation-finite selfinjective artin algebras of classB n andC n

In this paper, we discuss the representation-finite selfinjective artin algebras of classB n andC n and obtain the following main results:

A characterization of representation-finite algebras

Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented c

The kiev algorithm for bocses applies directly to representation-finite algebras

The kiev algorithm for bocses applies directly to representation-finite algebras

On Omnipresent Modules in Simply Connected Algebras

Journal of the London Mathematical SocietyVolume s2-36, Issue 3 p. 385-392 Notes and papers On Omnipresent Modules in Simply Connected Algebras J. A. de la Peña, J. A. de la Peña Instituto de Matematicas, Area de la Investigacion Cientifica, Circuito Exterior, Ciudad Universitaria, Mexico DF CP 04510Search for more papers by this author J. A. de la Peña, J. A. de la Peña Instituto de Matematicas, Area de la Investigacion Cientifica, Circuito Exterior, Ciudad Universitaria, Mexico DF CP 04510Search for more papers by this author First published: December 1987 https://doi.org/10.1112/jlms/s2-36.3.385Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes2-36, Issue3December 1987Pages 385-392 RelatedInformation

Quotients of representation-finite algebras

(1987). Quotients of representation-finite algebras. Communications in Algebra: Vol. 15, No. 1-2, pp. 279-307.

Stable equivalence of representation-finite trivial extension algebras

Stable equivalence of representation-finite trivial extension algebras

Tame biserial algebras

A similar result is suspected if we drop the property “representationfinite” in classes 1 and 2, and we will start an investigation of biserial algebras of infinite representation type. The key to the result for representation-finite algebras was to observe that each representation-finite biserial algebra is special [13] (see also the basic notations below), which means that it can be presented by a quiver with “nice” relations. Special algebras play an important role in the modular representation theory of finite groups. Namely, each representation-finite block of a group algebra and some of the tame blocks (they occur only in characteristic 2) are special [7, 10,4]. Moreover in the complex representation theory of the Lorentz group the so-called Harish-Chandra modules are defined over (tame) special algebras [6]. In this paper we show that /l(A)62 for any special algebra A and that special algebras of infinite representation type are tame. Our paper consists of four parts. First we prove that each special algebra is a factor of a special symmetric algebra (Theorem 1.5). For the latter the methods of Gelfand and Ponomarev used in the classification of the indecomposable Harish-Chandra modules of the Lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras (Proposition 2.3) presented in Section 2. From this list we 480 0021~8693/85 $3.00

Representation-finite algebras and multiplicative bases

Representation-finite algebras and multiplicative bases

On representation-finite algebras whose Auslander-Reiten quiver contains a stable complete slice

On representation-finite algebras whose Auslander-Reiten quiver contains a stable complete slice

Critical simply connected algebras

It is well-known that simply connected algebras are uniquely determined by a graded tree.Reversely,each graded tree gives rise to a not necessarily representation-finite algebra. We call an algebra critical provided it is not representation-finite, but any proper convex full subalgebra is.All critical algebras arising from graded trees are classified.

Automorphisms of representation finite algebras

Coverings in representation theory have been successfully used by: BongartzGabriel [2], Gabriel [3], Green [4], Riedtman [6, 7]. Of particular importance is the study of the relations between the coverings of the Auslander-Reiten quiver and the ordinary quiver. In this direction P. Gabriel posed, in the Third International Conference on Representations of Algebras, Puebla M6xico, 1980, the following conjecture: "Let g be an automorphism of an algebra A of finite representation type. For each right module M, denote by M g the A-module with underlying group M and scalar multiplication (m, 2 )~mgl (2 ) . If Sgq~S for each simple S, then M g q~ M for each indecomposable M". The aim of this note is to give a short proof of this statement. The conjecture is not true for algebras of infinite representation type. Con-