Derived equivalences for tame weakly symmetric algebras having only periodic modules.(English)Zbl 1050.16010
Let \(K\) be an algebraically closed field. One of the main results of the paper is a classification, up to derived equivalence, of weakly symmetric finite-dimensional \(K\)-algebras \(A\) that have tame representation type, admit simply connected Galois coverings and all of whose indecomposable modules of finite dimension are periodic. A set of pairwise nonisomorphic representatives of the derived equivalence classes of these algebras is presented.
Reviewer: Daniel Simson (Toruń)
MSC:
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
Keywords:
Auslander-Reiten quivers;periodic modules;Galois coverings;simply connected algebras;weakly symmetric algebras;derived equivalences;Cartan matricesCite
Full Text:DOI
References:
| [1] | Białkowski, J.; Skowroński, A., Selfinjective algebras of tubular type, Colloq. Math., 94, 175-194 (2002) ·Zbl 1023.16015 |
| [2] | J. Białkowski, A. Skowroński, On tame weakly symmetric algebras having only periodic modules, Arch. Math., in press; J. Białkowski, A. Skowroński, On tame weakly symmetric algebras having only periodic modules, Arch. Math., in press ·Zbl 1063.16020 |
| [3] | Drozd, Yu. A., Tame and wild matrix problems, (Representation Theory II. Representation Theory II, Lecture Notes in Math., 832 (1980), Springer), 242-258 ·Zbl 0454.16014 |
| [4] | Erdmann, K., Algebras and quaternion defect groups I, Math. Ann., 281, 545-560 (1988) ·Zbl 0655.16011 |
| [5] | Erdmann, K., Algebras and quaternion defect groups II, Math. Ann., 281, 561-582 (1988) ·Zbl 0655.16011 |
| [6] | Happel, D., On the derived category of a finite-dimensional algebra, Comment. Math. Helv., 62, 339-389 (1987) ·Zbl 0626.16008 |
| [7] | Happel, D., Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser., 119 (1988), Cambridge University Press ·Zbl 0635.16017 |
| [8] | Happel, D.; Ringel, C. M., The derived category of a tubular algebra, (Representation Theory I, Finite Dimensional Algebras. Representation Theory I, Finite Dimensional Algebras, Lecture Notes in Math., 1177 (1986), Springer), 156-180 ·Zbl 0626.16011 |
| [9] | Happel, D.; Vossieck, D., Minimal algebras of infinite representation type with preprojective component, Manuscripta Math., 42, 221-243 (1983) ·Zbl 0516.16023 |
| [10] | Holm, T., Derived equivalent tame blocks, J. Algebra, 194, 178-200 (1997) ·Zbl 0878.20007 |
| [11] | Holm, T., Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. Algebra, 211, 159-205 (1999) ·Zbl 0932.16009 |
| [12] | Krause, H., Stable equivalence preserves representation type, Comment. Math. Helv., 72, 266-284 (1997) ·Zbl 0901.16008 |
| [13] | Nehring, J.; Skowroński, A., Polynomial growth trivial extensions of simply connected algebras, Fund. Math., 132, 117-134 (1989) ·Zbl 0677.16008 |
| [14] | Ohnuki, Y.; Takeda, K.; Yamagata, K., Automorphisms of repetitive algebras, J. Algebra, 232, 708-724 (2000) ·Zbl 1035.16010 |
| [15] | Pogorzały, Z.; Skowroński, A., Symmetric algebras stably equivalent to the trivial extensions of tubular algebras, J. Algebra, 181, 95-111 (1996) ·Zbl 0867.16005 |
| [16] | Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Algebra, 61, 303-317 (1989) ·Zbl 0685.16016 |
| [17] | Rickard, J., Morita theory for derived categories, J. London Math. Soc., 39, 436-456 (1989) ·Zbl 0642.16034 |
| [18] | Rickard, J., Derived equivalences as derived functors, J. London Math. Soc., 43, 37-48 (1991) ·Zbl 0683.16030 |
| [19] | Ringel, C. M., Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., 1099 (1984), Springer: Springer Berlin ·Zbl 0546.16013 |
| [20] | Skowroński, A., Selfinjective algebras of polynomial growth, Math. Ann., 285, 177-199 (1989) ·Zbl 0653.16021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
