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...lassification of iterated tilted algebras of type A and type A in [5] and [7] respectively. They now play an important role in many areas of mathematics: in algebra, they occur in cluster theory as Jacobian algebras associated to surface triangulations [6, 29, 38], and in recent advances in invariant theory [23]. In addition to their widespread appearance in representation theory and algebra, gentle algebras also are instrumental in geometric contexts. For example, their singularity category [15] – which measures how far an algebra or variety is away from being nonsingular – was described in [36]. They feature prominently in the programme to understand singularities of nodal curves [16, 17, 18] and similar combinatorics occur in an algebraic approach to mirror symmetry based on dimer models [10]. From now on let Λ be a gentle algebra and Db(Λ) be its bounded derived category with shift functor Σ. The indecomposable objects in Db(Λ) have been classified [8] in terms of string combinatorics: namely they are given in terms of homotopy strings and homotopy bands; see also [16] for a similar approach in the context of nodal algebras. Note that the terminology originates in [9]. Using the H...

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...xample, to cite but a few of the earlier results, the structure of the indecomposable representations [42,29,50], almost split sequences [19], maps between indecomposable representations [22,37], and the structure of the Auslander–Reiten quiver [24]. Recently there has been renewed interested in this class of algebras. On the one hand this interest stems from its connecting with cluster theory. In [5] the authors show that the Jacobian algebras of surface cluster algebras are gentle algebras, and hence a subclass of special biserial algebras. This class has been extensively studied since, see [21,35] for examples of the most recent results. On the other hand with the introduction of τ -tilting and silting theory [2,3], there has been a renewed interested in special biserial algebras and symmetric special biserial algebras, in particular, see [1,38,51,52]. For self-injective algebras, Brauer tree and Brauer graph algebras have been useful in the classification of group algebras and blocks of group algebras of finite and tame representation type [13,16,32,23] and the derived equivalence classification of self-injective algebras of tame representation type, see for example in [4,47] and the ...

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... is triangulated equivalent to the singularity category of the cluster-tilted algebra of Dynkin type. For cluster-tilted algebras of type A, since they are gentle algebras, it was settled by Kalck in =-=[Ka]-=-. We use the good mutation defined in [BHL2] to give a direct proof, see Corollary 3.6. For type D, use the description of its quiver by Geng and Peng, Vatne [GP, Va], we get some selfinjective algebr...