Iterated tilted algebras of type $$\tilde {\mathbb{A}}_n $$

Derived Equivalence Classification of the Gentle Two-Cycle Algebras

“…Gentle algebras were introduced by Assem and Skowroński [6] in their study of the algebras derived equivalent to the hereditary algebras of Euclidean typeÃ. Namely, they have proved that the algebras derived equivalent to the hereditary algebras of Euclidean typeà are precisely the gentle one-cycle algebras which satisfy the clock condition.…”

Algebras with cycle-finite Galois coverings

“…The class of cycle-finite algebras is wide and contains the algebras of finite type, the tame tilted algebras [Ke], the tame double tilted algebras [RS1], the tame generalized double tilted algebras [RS2], the tame quasi-tilted algebras [LS], [Sk10], the tame generalized multicoil algebras [MS], and the strongly simply connected algebras of polynomial growth [Sk8]. It has also been proved in [AS1], [AS2], [AS3] that the class of algebras A for which the derived category D b (mod A) of bounded complexes over mod A is cycle-finite coincides with the class of algebras which are tilting-cotilting equivalent to the hereditary algebras of Dynkin type, hereditary algebras of Euclidean type, or tubular algebras (in the sense of [Ri]), and consequently these algebras A are also cycle-finite. We also mention that by a result due to Peng and Xiao [PX] and the second named author [Sk4], the Auslander-Reiten quiver of any algebra A has at most finitely many D Trorbits containing directing modules (not lying on a cycle in mod A).…”

A classification of symmetric algebras of strictly canonical type

“…The representation theory of the trivial extension algebras has been extensively developed (see [2], [3], [5]- [8], [14], [16], [17], [22], [25], [26], [28]- [30], [34]- [36], [38]- [40] for some research in this direction).…”