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”, Comm. Algebra, Vol. 26, (1998), no. 3, pp. 749–758. http://dx.doi.org/10.1080/00927879808826161 [6] A. Jaworska and Z. Pogorzały: “On trivial extensions of 2-fundamental algebras”, Comm. Algebra, Vol. 34, (2006), no. 11, pp. 3935–3947. http://dx.doi.org/10.1080/00927870600862748 [7] Z. Pogorzały and M. Sufranek: “Starting and ending components of the Auslander-Reiten quivers of a class of special biserial algebras”, Colloq. Math., Vol. 99, (2004), no. 1, pp. 111–144. [8] A. Skowroński: “Generalized standard Auslander-Reiten components”, J. Math. Soc. Japan, Vol. 46

., Vol. 274, (1982), pp. 399–443. http://dx.doi.org/10.2307/1999116 [10] F. Huard: “Tilted gentle algebras”, Comm. Alg., Vol. 26(1), (1998), pp. 63–72. [11] F. Huard and Sh. Liu: “Tilted special biserial algebras”, J. Algebra, Vol. 217, (1999), pp. 679–700. http://dx.doi.org/10.1006/jabr.1998.7828 [12] F. Huard and Sh. Liu: “Tilted string algebras”, J. Pure Appl. Algebra, Vol. 153, (2000), pp. 151–164. http://dx.doi.org/10.1016/S0022-4049(99)00101-2 [13] Z. Pogorzały and M. Sufranek: “Starting and ending components of the Auslander-Reiten quivers of a class of special

quivers is based on the almost split sequences, by our result we observe that the Auslander–Reiten quiver Γ 𝒳 ¯ {\Gamma_{\underline{\mathcal{X}}}} of mod ⁢ - ⁢ 𝒳 ¯ {{\mathrm{{mod\mbox{-}}}}\underline{\mathcal{X}}} can be considered as a valued full subquiver of Γ 𝒮 𝒳 ⁢ ( Λ ) {\Gamma_{\mathcal{S}_{\mathcal{X}}(\Lambda)}} such that contains all vertices in Γ 𝒮 𝒳 ⁢ ( Λ ) {\Gamma_{\mathcal{S}_{\mathcal{X}}(\Lambda)}} except those of vertices corresponding to the isomorphism classes of the indecomposable objects having of the form ( X ⁢ → 1 ⁢ X ) {(X\overset{1

Abstract

In continuation of our earlier work [2] we describe the indecomposable representations and the Auslander-Reiten quivers of a family of vector space categories playing an important role in the study of domestic finite dimensional algebras over an algebraically closed field. The main results of the paper are applied in our paper [3] where we exhibit a wide class of almost sincere domestic simply connected algebras of arbitrary large finite global dimensions and describe their Auslander-Reiten quivers.

Abstract

In the paper, we introduce a wide class of domestic finite dimensional algebras over an algebraically closed field which are obtained from the hereditary algebras of Euclidean type , n≥1, by iterated one-point extensions by two-ray modules. We prove that these algebras are domestic and their Auslander-Reiten quivers admit infinitely many nonperiodic connected components with infinitely many orbits with respect to the action of the Auslander-Reiten translation. Moreover, we exhibit a wide class of almost sincere domestic simply connected algebras of large global dimensions.

radical, Compositio Math., 1991, 77(3), 313–333 [23] Lenzing H., Skowroński A., Quasi-tilted algebras of canonical type, Colloq. Math., 1996, 71(2), 161–181 [24] Liu S.-P., Almost split sequences for nonregular modules, Fund. Math., 1993, 143(2), 183–190 [25] Liu S.-P., Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc., 1993, 47(3), 405–416 http://dx.doi.org/10.1112/jlms/s2-47.3.405 [26] Liu S.-P., Tilted algebras and generalized standard Auslander-Reiten components, Arch. Math. (Basel), 1993, 61(1), 12–19 http://dx.doi.org/10.1007/BF01258050

://dx.doi.org/10.1016/0021-8693(91)90215-T [9] M. Kleiner, A. Skowroński and D. Zacharia, On endomorphism algebras with small homological dimensions, J. Math. Soc. Japan 54 (2002), 621–648. [10] H. Lenzing and A. Skowroński, Quasi-tilted algebras of canonical type, Colloq. Math. 71 (1996), 161–181. [11] S. Liu, Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc. 47 (1993), 405–416. [12] L. Peng and J. Xiao, On the number of D Tr-orbits containing

Abstract

The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.

Abstract

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category 𝖢 with the following properties.

On the one hand, the d-cluster tilting subcategories of 𝖢 have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of 𝖢 which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d - 1, we show a weakly d-cluster tilting subcategory 𝖳 which has an indecomposable object with precisely ℓ mutations.

The category 𝖢 is the algebraic triangulated category generated by a (d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A .

Abstract

The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver consisting only of stable tubes has been shown to be the class of selfinjective algebras of tubular type, that is, the orbit algebras $$ \hat B$$ /G of the repetitive algebras $$ \hat B$$ of tubular algebras B with respect to the actions of admissible groups G of automorphisms of $$ \hat B$$ . The aim of the paper is to describe the determinants of the Cartan matrices of selfinjective algebras of tubular type and derive some consequences.