# Algebras stably equivalent to trivial extensions of hereditary algebras of type ${\xc3}_{n}$

Colloquium Mathematicae (1993)

- Volume: 66, Issue: 2, page 265-281
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topPogorzały, Zygmunt. "Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$." Colloquium Mathematicae 66.2 (1993): 265-281. <http://eudml.org/doc/210248>.

@article{Pogorzały1993,

abstract = {The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting equivalent algebras then their trivial extensions T(A) and T(B) are stably equivalent. Consequently, there is a special need to describe algebras that are stably equivalent to the trivial extensions of tame hereditary algebras. In the paper, there are studied algebras which are stably equivalent to the trivial extensions of hereditary algebras of type $Ã_n$, that is, algebras given by quivers whose underlying graphs are of type $Ã_n$. These algebras are isomorphic to the trivial extensions of very nice algebras (see Theorem 1). Moreover, in view of [1, 8], Theorem 2 shows that every stable equivalence of such algebras is induced in some sense by a derived equivalence of well chosen subalgebras. In our study of stable equivalence, we shall use methods and results from [11]. We shall also use freely information on Auslander-Reiten sequences which can be found in [2].},

author = {Pogorzały, Zygmunt},

journal = {Colloquium Mathematicae},

keywords = {biserial algebras; basic connected hereditary algebras; trivial extensions; minimal injective cogenerators; stable module categories; finitely generated right modules; tilting-cotilting equivalent to hereditary algebras},

language = {eng},

number = {2},

pages = {265-281},

title = {Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$},

url = {http://eudml.org/doc/210248},

volume = {66},

year = {1993},

}

TY - JOUR

AU - Pogorzały, Zygmunt

TI - Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$

JO - Colloquium Mathematicae

PY - 1993

VL - 66

IS - 2

SP - 265

EP - 281

AB - The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting equivalent algebras then their trivial extensions T(A) and T(B) are stably equivalent. Consequently, there is a special need to describe algebras that are stably equivalent to the trivial extensions of tame hereditary algebras. In the paper, there are studied algebras which are stably equivalent to the trivial extensions of hereditary algebras of type $Ã_n$, that is, algebras given by quivers whose underlying graphs are of type $Ã_n$. These algebras are isomorphic to the trivial extensions of very nice algebras (see Theorem 1). Moreover, in view of [1, 8], Theorem 2 shows that every stable equivalence of such algebras is induced in some sense by a derived equivalence of well chosen subalgebras. In our study of stable equivalence, we shall use methods and results from [11]. We shall also use freely information on Auslander-Reiten sequences which can be found in [2].

LA - eng

KW - biserial algebras; basic connected hereditary algebras; trivial extensions; minimal injective cogenerators; stable module categories; finitely generated right modules; tilting-cotilting equivalent to hereditary algebras

UR - http://eudml.org/doc/210248

ER -

## References

top- [1] I. Assem and A. Skowroński, Iterated tilted algebras of type ${\xc3}_{n}$, Math. Z. 195 (1987), 269-290. Zbl0601.16022
- [2] M. Auslander and I. Reiten, Representation theory of artin algebras III, Comm. Algebra 3 (1975), 239-294.
- [3] P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), 311-337. Zbl0628.16019
- [4] Yu. A. Drozd, Tame and wild matrix problems, in: Representations and Quadratic Forms, Kiev, 1979, 39-73 (in Russian).
- [5] K. Erdmann and A. Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), 165-189. Zbl0757.16004
- [6] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, in: Lecture Notes in Math. 831, Springer, 1980, 1-71.
- [7] P. Gabriel, The universal cover of a representation-finite algebra, in: Lecture Notes in Math. 903, Springer, 1981, 68-105.
- [8] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339-389. Zbl0626.16008
- [9] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443. Zbl0503.16024
- [10] J. Nehring, Polynomial growth trivial extensions of non-simply connected algebras, Bull. Polish Acad. Sci. Math. 36 (1988), 441-445. Zbl0777.16008
- [11] Z. Pogorzały, Algebras stably equivalent to selfinjective special biserial algebras, preprint. Zbl0805.16008
- [12] Z. Pogorzały and A. Skowroński, Selfinjective biserial standard algebras, J. Algebra 138 (1991), 491-504. Zbl0808.16019
- [13] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303-317. Zbl0685.16016
- [14] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984.
- [15] A. Skowroński, Group algebras of polynomial growth, Manuscripta Math. 59 (1987), 499-516. Zbl0627.16007
- [16] A. Skowroński and J. Waschbüsch, Representation-finite biserial algebras, J. Reine Angew. Math. 345 (1983), 172-181. Zbl0511.16021
- [17] H. Tachikawa and T. Wakamatsu, Tilting functors and stable equivalences for self-injective algebras, J. Algebra 109 (1987), 138-165. Zbl0616.16012
- [18] B. Wald and J. Waschbüsch, Tame biserial algebras, J. Algebra 95 (1985), 480-500. Zbl0567.16017

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.