Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$
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 . The Tachikawa-Wakamatsu result  also reveals the importance of these problems in the study of tilting equivalent algebras (compare with ). 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 . We shall also use freely information on Auslander-Reiten sequences which can be found in .
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