Tilted Special Biserial Algberas

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 [9]. As a generalization, Happel, Reiten and Smalø studied endomorphism algebras of tilting objects of a hereditary abelian category which they call quasi-tilted algebras [8]. We are concerned with the problem of characterizing these algebras in terms of bound quivers. In our previous paper [12], we have found some simple combinatorial criteria to determine if a string algebra is quasi-tilted or tilted or neither. In this paper, we shall consider the same problem for special biserial algebras which are not string algebras. Note that such an algebra is tilted if and only if it is quasi-tilted since there are some indecomposable projective-injective modules [3]. Our strategy is to study the combinatorial interpretation of some behavious of the homological dimensions of the indecomposable modules. This enables us to find first a combinatorial characterization of the special biserial algebras of global dimension at most two, and then some simple necessary and sufficient conditions for a special biserial algebra to be tilted. As one of the applications, this allows one to construct a large class of new examples of tilted algebras.