On minimal non-tilted algebras

Introduced in the early 80’s by Happel-Ringel [4], the class of tilted algebras has played an important role in the development of the representation theory of algebras. However, it is not always easy to identify a tilted algebra looking, for instance, at its ordinary quiver. On the other hand, the tilted algebras behave nicely with respect to their semi-convex subcategories, that is, if A is a tilted algebra and B is a semi-convex subcategory of A, then also B is tilted. Recall that, given an algebra A, a subcategory B of A is semi-convex provided there is a sequence B = Bs, · · · , B0 = A such that Bi = Ci[M ′ i ] (respectively, Bi = [M ′ i ]Ci) is a one-point (co-)extension of a convex subcategory Ci of Bi−1 = Ci[M ′ i ⊕ M ′′ i ] (respectively Bi−1 = [M ′ i ⊕M ′′ i ]Ci) by a Ci-module M ′ i , possibly M ′′ i = 0. Clearly, convex subcategories are semi-convex. By looking at the minimal non-tilted algebras, we hope to get some insight of how the tilted algebras are built up. By minimal non-tilted we mean a triangular algebra which is not tilted but any of its semi-convex subcategory is tilted. In this talk, we will discuss this problem and give several results of classification of classes of minimal non-tilted algebras. The results discussed here are joint work with J. A. de la Pena (UNAM-Mexico) and S. Trepode (Universidad de Mar del Plata Argentina) and are contained in [1, 2, 3].