From iterated tilted algebras to cluster tilted algebras

Sonia Trepode

Joint work with M. Barot, E. Fernandez, M. I. Platzeck and I. Pratti

We study the connection between cluster tilted algebras and iterated tilted algebras. The latter coincide with the algebras derived equivalent to hereditary algebras.

The notion of relation-extension of a finite dimensional k-algebra, introduced by Assem, Brüstle and Schiffler plays an important role in this work. It is known that relation extension of a tilted algebra is a cluster tilted algebra, and any cluster tilted algebra is obtained in this way.

We prove that an iterated tilted algebra B of global dimension at most two gives rise to a unique cluster tilted algebra C which is a split extension of B, and maps onto the relation- extension R(B) of B. Though this map φ is not always an isomorphism, it induces an isomorphism between the corresponding quivers. We characterize when φ is an isomorphism in the Dynkin case.

An iterated tilted algebra B is the endomorphism ring of a tilting complex T in the derived category of a hereditary algebra. The strategy of our proof consists in proving that T induces a tilting object in the cluster category, in case gl dim B ≤ 2.

For an algebra C = kQ/I we define an admissible cut C as the quotient C/Δ, where Δ is an ideal of C generated by exactly one arrow of each chordless cycle of C. In the Dynkin case we characterize iterated tilted algebras of global dimension at most 2 as admissible cuts of cluster tilted algebras.

The interplay between iterated tilted and cluster tilted algebras is very useful in the classification of cluster tilted algebras an in the study of their relations. It also gives new insight on tilted an iterated tilted algebras.