Trivial Extension Algebras Research Articles

On Lie algebras associated with representation-finite algebras

Let Λ be a representation-finite C-algebra which has Hall polynomials, with the universal cover Λ˜ which is a locally bounded directed C-algebra. In this paper we prove that the Z-Lie algebra L(Λ) associated with Λ which is defined by Riedtmann in [17] and the Z-Lie algebra K(Λ) associated with Λ which is defined by Ringel in [19] are isomorphic. As an application we show that if Λ is a representation-finite (generalized) cluster-tilted algebra or representation-finite trivial extension algebra, then K(Λ)≅L(Λ).

Periodicity of self-injective algebras of polynomial growth

Let A be an indecomposable representation-infinite tame finite-dimensional algebra of polynomial growth over an algebraically closed field. We prove that A is a periodic algebra with respect to action of the bimodule syzygy operator if and only if A is Morita equivalent to a socle deformation of an orbit algebra Bˆ/G where Bˆ is the repetitive category of a tubular algebra B and G is an admissible infinite cyclic automorphism group of Bˆ. The main contribution in the paper is to prove the sufficiency part of this equivalence. It is known that every orbit algebra Bˆ/G of a tubular algebra B admits a presentation as an orbit algebra T(B)(r)/H of an r-fold trivial extension algebra T(B)(r) of B with respect to free action of a finite cyclic automorphism group H of T(B)(r). A significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self-injective algebras of polynomial growth and showing that all of them are periodic algebras. Then we deduce the periodicity of all algebras socle equivalent to the orbit algebras Bˆ/G=T(B)(r)/H of tubular algebras B from the periodicity of these ten exceptional algebras, using invariance of periodicity for finite Galois coverings and derived equivalences of algebras.

Jordan Derivations on Trivial Extension Algebras

Jordan Derivations on Trivial Extension Algebras

Combinatorial dimensions: Indecomposability on certain local finite-dimensional trivial extension algebras

We study problems related to indecomposability of modules over certain local finite-dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorified versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.

Homotopy Categories, Leavitt Path Algebras, and Gorenstein Projective Modules

For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite-dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra of a von Neumann regular algebra by an invertible bimodule. A related, but different, result for the homotopy category of acyclic complexes of projective modules is given. Restricting these equivalences to compact objects, we obtain various descriptions of the singularity category of a finite-dimensional algebra with radical square zero, which contain previous results.

Notes on piecewise-Koszul algebras

The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul algebras with arbitrary “jump-degree”.

Generating relations for Grothendieck groups of trivial extension algebras

In this paper we describe explicitly the generating relations for the Grothendieck groups of trivial extension algebras of tame hereditary algebras.

Hochschild Homology Invariants of Külshammer Type of Derived Categories

For a perfect field k of characteristic p > 0 and for a finite dimensional symmetric k-algebra A Külshammer studied a sequence of ideals of the centre of A using the p-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If A is symmetric, then the dual to the Külshammer ideal structure was generalised to higher Hochschild homology in earlier work [23]. In the present article we follow this program and propose an analogue of the dual to the Külshammer ideal structure on the degree m Hochschild homology theory also to not necessarily symmetric algebras.

Graded self-injective algebras “are” trivial extensions

For a positively graded artin algebra A = ⊕ n ⩾ 0 A n we introduce its Beilinson algebra b ( A ) . We prove that if A is well-graded self-injective, then the category of graded A-modules is equivalent to the category of graded modules over the trivial extension algebra T ( b ( A ) ) . Consequently, there is a full exact embedding from the bounded derived category of b ( A ) into the stable category of graded modules over A; it is an equivalence if and only if the 0-th component algebra A 0 has finite global dimension.

Tilting equivalences: From hereditary algebras to symmetric groups

We study Koszul duality for finite dimensional hereditary algebras, and various generalisations to trivial extension algebras, to Schur algebras, to doubles of Schur bialgebras, and to deformations of doubles of Schur bialgebras. We describe applications to the modular representation theory of symmetric groups.

Quotient triangulated categories

For a self-orthogonal module T, the relation between the quotient triangulated category D b(A)/K b(addT) and the stable category of the Frobenius category of T-Cohen-Macaulay modules is investigated. In particular, for a Gorenstein algebra, we get a relative version of the description of the singularity category due to Happel. Also, the derived category of a Gorenstein algebra is explicitly given, inside the stable category of the graded module category of the corresponding trivial extension algebra, via Happel’s functor \(F: D^b(A) \longrightarrow T(A)^{\mathbb{Z}}\mbox{-}\underline{\rm mod}\).

A generalization of trivial extension algebras

In the paper a generalization of the known construction of trivial extension algebras is introduced. For the new constructed algebras one can indicate a construction of their Galois coverings. As an application of these constructions one can obtain a nice Galois coverings for the enveloping algebras of trivial extension algebras of triangular algebras.

Periodic Algebras which are Almost Koszul

The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such ‘almost Koszul’ algebras is developed and other examples are given.

Some Hall Polynomials for Representation-Finite Trivial Extension Algebras

Letkbe a finite field and assume that Λ is a finite dimensional associativek-algebra with 1. Denote by modΛ the category of all finitely generated (right) Λ-modules and by indΛ the full subcategory in which every object is a representative of the isoclass of an indecomposable (right) Λ-module. We are interested in the existance of the Hall polynomial ϕMNLfor anL,M,N∈modΛ (for the definition, see977113or Section 1 below). In case Λ is directed,7has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by4. It has been conjectured in8that any representation-finitek-algebra has Hall polynomials. In this investigation, we shall show that if Λ is a representation-finite trivial extension algebra, then, for anyL,M,N∈modΛ withNindecomposable, Λ has the Hall polynomials ϕMLNand ϕMNL. Using these Hall polynomials, we can naturally structure the free abelian group with a basis indΛ, denoted byK(modΛ), into a Lie algebra and the universal enveloping algebra ofK(modΛ)⊗ZQis just H(Λ)1⊗ZQ, where H(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below).

On the tameness of trivial extension algebras

For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if $T_A$ is a tilting module and $B=End T_A$, then T(A) is tame if and only if

Auslander-Reiten triangles in derived categories of finite-dimensional algebras

Let A A be a finite-dimensional algebra. The category b m o d A bmod A of finitely generated left A A -modules canonically embeds into the derived category D b ( A ) {D^b}\left ( A \right ) of bounded complexes over b m o d A bmod A and the stable category mod _ Z T ( A ) {\underline {\bmod } ^\mathbb {Z}}T\left ( A \right ) of Z \mathbb {Z} -graded modules over the trivial extension algebra of A A by the minimal injective cogenerator. This embedding can be extended to a full and faithful functor from D b ( A ) {D^b}\left ( A \right ) to mod _ Z T ( A ) \underline {\bmod }^{\mathbb {Z}}T\left ( A \right ) . Using the concept of Auslander-Reiten triangles it is shown that both categories are equivalent only if A A has finite global dimension.

Quadratic forms and iterated tilted algebras

Given a finite quiver A without oriented cycles, an algebra A is called an iterated tilted algebra of type A [2] if there exists a sequence of algebras A = A,, A 1) . ..) A,,,, where A, is the path algebra of A, and a sequence of tilting modules T>, (0 , and every indecomposable A ,-module satisfies either Hom,J T’, M) = 0 or Exta,( T’, M) = 0. If m < 1, A is called a tilted algebra of type A [25]. The representation theory of iterated tilted algebras was proved to be closely related to that of a class of symmetric algebras, namely, the trivial extension algebras; see [3,4, 18, 27, 291. Recently, they were also shown to arise naturally in the study of the derived category of a finite dimensional algebra; see [23, 24, 71. Iterated tilted algebras of type A, where the underlying graph of A is a Dynkin diagram, were studied in [ 1, 2, 8, 223, and the iterated tilted algebras of Euclidean type A,,, (m 3 1) were classified in [S]. 55 0021-8693/90 $3.00

Iterated tilted algebras induced from coverings of trivial extensions of hereditary algebras

Recently the relations between tiltingtheory and trivialextension algebras are deeply studied. Let A and B be basic connected artin algebras over a commutative artin ring C. In [6] Tachikawa and Wakamatsu showed that the existence of stably equivalence between categories over the trivialextension algebras T(A)=A kDA and T(B)=Bt<DB under the assumption that there is a tiltingmodule TA with B=End(TA). In case C is a field,Hughes and Waschblisch proved that if T(B) is representation-finiteof Cartan class A, then there exists a tiltedalgebra A of Dynkin type A such that T(B)^T(A) [4]. Assem, Happel and Roldan showed that, for an algebra B over an algebraically closed field, T(B) is representation-finiteiff B is an iterated tiltedalgebra of Dynkin type [1]. However in case T(B) is not of finiterepresentation type the condition T{B) = T{A) with A hereditary does not forces B to be an iterated tiltedalgebra. Let's consider the covering A of T(A) [4]. The author proved that the condition A^B implies T(A) = T(B) and that the converse holds if T{A) is representation-finite[5]. In this paper, we prove that the condition B = A with A hereditary implies that B is an iterated algebra obtained from A. It is to be noted that in case A is not necessary representation-finite. Moreover, the proof of our theorem shows that such an algebra B is obtained by at most 3m times processes tilting from A, where m is the number of non-isomorphic primitive idempotents of A.

Tilting functors and stable equivalences for selfinjective algebras

(1) Exta (T, T) = 0, (2) Ext: (T, ) = 0, (3) There is a short exact sequence 0 --+ A, + T> + T” + 0, with T’, T” being direct sums of summands of T. Then putting B= End T, we call (B, T, A) and Hom,(T, ): mod-A -+ mod-B a tilting triple and a tilting functor, respectively. Tilting functors have been introduced by Brenner and Butler [6] as a generalization of the Bernstein-Gelfand-Ponomarev’s reflection functors [S]. They and Happel and Ringel [S] have proved that we have (usually nonhereditary) torsion theories (F-, F) in mod-A and (X, g) in mod-B, where F = {Xe mod-A (Extj, (T, X) = 0} and X= { YEmod-BI YOB T=O), and the tilting functor and Ext: (T, -) give category-equivalences between F and g and between F and X, respectively. These equivalences give us, however, no information about indecomposable A-modules and B-modules which do not belong to the above subcategories F-, F”, !Z, and CV. The purpose of this paper is to point out that there is a method to enlarge our view by which we can supply the lack of information. Let us consider trivial extension algebras R = A K DA and S = B K DB of A and B by DA and DB respectively, where ,DA, = Hom,(A, k) and BDB, = Hom,(B, k) are injective cogenerator bimodules. In this case R and S are selfinjective (more precisely symmetric) algebras and mod-A and mod-B are naturally embedded into projectively stable categories m-R and m-S. Then our main theorem states that for any tilting triple

Isomorphisms between coverings of trivial extension algebras

Isomorphisms between coverings of trivial extension algebras