Tilted String Algebras

Tilted algebras, that is endomorphism algebras of tilting modules over a hereditary algebra, have been one of the main objects of study in representation theory of algebras since their introduction by Happel and Ringel [10]. As a generalization, Happel, Reiten and Smalø studied endomorphism algebras of tilting objects of a hereditary abelian category which they call quasi-tilted algebras [9]. The latter has attracted a lot of attention of recent investigations. So far all complete characterizations of tilted or quasi-tilted algebras are module-theoretical [9, 10]. On the other hand, Gabriel’s theorem says that a finite-dimensional algebra over an algebraically closed field is determined, up to Morita equivalence, by its bound quiver [6] . It is then natural and interesting to characterize tilted or quasi-tilted algebras in terms of their bound quiver. This has been done for tilted algebras of type An, Ãn and for tame concealed algebras [1, 11, 15]. As the problem in general seems very difficult, if not impossible, we shall consider it for string algebras, that is monomial biserial algebras [3, 5]. As results we shall find some simple combinatorial criteria for a string algebra to be tilted or quasi-tilted. As a consequence, this will enable one to construct a lot of new examples of tilted algebras. Finally we shall determine all quasi-tilted string algebras which are not tilted.