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Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.Comment: Revised version. To appear in Math.

Abstract. The notion of silting mutation was introduced by Iyama and the author. In this paper we mainly study silting mutation for self-injective algebras and prove that any representation-finite symmetric algebra is tilting-connected. Moreover we give some sufficient conditions for a Bongartz-type Lemma to hold for silting objects.

We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.

We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$ -tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$ -tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$ -tilting-finite algebra is representation-finite.