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Colloquium Mathematicum

1993 | 66 | 2 | 265-281
Tytuł artykułu

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

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Języki publikacji
EN
Abstrakty
EN
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].
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
265-281
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-04-14
poprawiono
1993-04-15
Twórcy
autor
• Institute of Mathematics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Bibliografia
• [1] I. Assem and A. Skowroński, Iterated tilted algebras of type $Ã_n$, Math. Z. 195 (1987), 269-290.
• [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.
• [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.
• [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.
• [9] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443.
• [10] J. Nehring, Polynomial growth trivial extensions of non-simply connected algebras, Bull. Polish Acad. Sci. Math. 36 (1988), 441-445.
• [11] Z. Pogorzały, Algebras stably equivalent to selfinjective special biserial algebras, preprint.
• [12] Z. Pogorzały and A. Skowroński, Selfinjective biserial standard algebras, J. Algebra 138 (1991), 491-504.
• [13] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303-317.
• [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.
• [16] A. Skowroński and J. Waschbüsch, Representation-finite biserial algebras, J. Reine Angew. Math. 345 (1983), 172-181.
• [17] H. Tachikawa and T. Wakamatsu, Tilting functors and stable equivalences for self-injective algebras, J. Algebra 109 (1987), 138-165.
• [18] B. Wald and J. Waschbüsch, Tame biserial algebras, J. Algebra 95 (1985), 480-500.
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