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Holm, Thorsten; Skowroński, Andrzej

Derived equivalence classification of symmetric algebras of domestic type. (English) Zbl 1108.18007

J. Math. Soc. Japan 58, No. 4, 1133-1149 (2006).
The authors obtain a complete classification of derived equivalence classes of domestic symmetric algebras. In previous work, Bocian and the second author have classified symmetric algebras of domestic type up to Morita equivalence. Furthermore, Bocian and the present authors obtained a derived equivalence classification up to the problem whether a standard (representation-infinite) domestic symmetric algebra can be derived equivalent to a non-standard one. Recall that “standard” means that there is a simply connected Galois covering. The authors use generalized Reynolds ideals [see B. Külshammer in several papers, J. Algebra, 1981–1985)] to show that derived equivalence between standard and non-standard domestic symmetric algebras does not occur. In the representation-finite case, this leads to a simplification of H. Asashiba’s classification of derived equivalence classes of self-injective algebras [J. Algebra 214, 182–221 (1999; Zbl 0949.16013)].
Reviewer: Wolfgang Rump (Stuttgart)

Cited in 10 Documents

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
16D50 Injective modules, self-injective associative rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras

Citations:

Zbl 0949.16013
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\textit{T. Holm} and \textit{A. Skowroński}, J. Math. Soc. Japan 58, No. 4, 1133--1149 (2006; Zbl 1108.18007)
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References:

[1] H. Asashiba, The derived equivalence classification of representation-finite selfinjective algebras, J. Algebra, 214 (1999), 182-221. · Zbl 0949.16013
[2] I. Assem, J. Nehring and A. Skowroński, Domestic trivial extensions of simply connected algebras, Tsukuba J. Math., 13 (1989), 31-72. · Zbl 0686.16011
[3] I. Assem, D. Simson and A. Skowroński, Elements of Representation Theory of Associative Algebras, Vol.,1: Techniques of Representation Theory, London Math. Soc. Student Texts, 65 , Cambridge Univ. Press, 2006. · Zbl 1092.16001
[4] I. Assem and A. Skowroński, Iterated tilted algebras of type \(\widetilde{\bm{A}}_n\), Math. Z., 195 (1987), 269-290. · Zbl 0601.16022
[5] M. Auslander, I. Reiten and S. O. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., 36 , Cambridge Univ. Press, 1995. · Zbl 0834.16001
[6] R. Bocian, T. Holm and A. Skowroński, Derived equivalence classification of weakly symmetric algebras of Euclidean type, J. Pure Appl. Algebra, 191 (2004), 43-74. · Zbl 1063.16015
[7] R. Bocian, T. Holm and A. Skowroński, Derived equivalence classification of nonstandard algebras of domestic type, Comm. Algebra, · Zbl 1168.16004
[8] R. Bocian and A. Skowroński, Symmetric special biserial algebras of Euclidean type, Colloq. Math., 96 (2003), 121-148. · Zbl 1068.16015
[9] R. Bocian and A. Skowroński, Weakly symmetric algebras of Euclidean type, J. Reine Angew. Math., 580 (2005), 157-199. · Zbl 1099.16002
[10] R. Bocian and A. Skowroński, Socle deformations of selfinjective algebras of Euclidean type, Comm. Algebra, · Zbl 1186.16007
[11] O. Bretscher, C. Läser and C. Riedtmann, Selfinjective and simply connected algebras, Manuscr. Math., 36 (1981), 253-307. · Zbl 0478.16024
[12] M. Broué, Isométrie parfaites, types de blocs, catégories dérivées, Astérisque, 181-182 (1990), 61-92. · Zbl 0704.20010
[13] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv., 62 (1987), 339-389. · Zbl 0626.16008
[14] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc. Lecture Note Series, 119 , Cambridge Univ. Press, 1988. · Zbl 0635.16017
[15] L. Héthelyi, E. Horváth, B. Külshammer and J. Murray, Central ideals and Cartan invariants of symmetric algebras, J. Algebra, 293 (2005), 243-260. · Zbl 1088.16025
[16] T. Holm, Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. Algebra, 211 (1999), 159-205. · Zbl 0932.16009
[17] D. Hughes and J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc., 46 (1983), 347-364. · Zbl 0488.16021
[18] S. König and A. Zimmermann, Derived equivalences for group rings, Lecture Notes in Math., 1685 , Springer, 1998. · Zbl 0898.16002
[19] B. Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Algebra I, II, III, IV, J. Algebra, 72 (1981), 1-7; J. Algebra, 75 (1982), 59-69; J. Algebra, 88 (1984), 279-291; J. Algebra, 93 (1985), 310-323. · Zbl 0488.16010
[20] B. Külshammer, Group-theoretical descriptions of ring theoretical invariants of group algebras, Progress in Math., 95 (1991), 425-441. · Zbl 0747.20002
[21] J. Rickard, Morita theory for derived categories, J. London Math. Soc., 39 (1989), 436-456. · Zbl 0642.16034
[22] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra, 61 (1989), 303-317. · Zbl 0685.16016
[23] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc., 43 (1991), 37-48. · Zbl 0683.16030
[24] C. Riedtmann, Representation-finite selfinjective algebras of class \(\bm{D}_n\), Compositio Math., 49 (1983), 231-282. · Zbl 0514.16019
[25] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., 1099 , Springer-Verlag, 1984. · Zbl 0546.16013
[26] A. Skowroński, Selfinjective algebras of polynomial growth, Math. Ann., 285 (1989), 177-199. · Zbl 0653.16021
[27] A. Skowroński, Selfinjective algebras: finite and tame type, In: Trends in Representation Theory of Algebras and Related Topics, Contemp. Math., 406 (2006), 169-238. · Zbl 1129.16013
[28] A. Skowroński, Classification of selfinjective algebras of domestic type, Preprint, Toruń (2005).
[29] H. Tachikawa, Representations of trivial extensions of hereditary algebras, In: Representation Theory II, Lecture Notes in Math., 832 , Springer-Verlag, 1980, 579-599. · Zbl 0451.16019
[30] J. Waschbüsch, Symmetrische Algebren vom endlichen Modultyp, J. Reine Angew. Math., 321 (1981), 78-98. · Zbl 0441.16027
[31] K. Yamagata, Frobenius algebras, In: Handbook of Algebra, 1 , Elsevier Science B. V., 1996, 841-887. · Zbl 0879.16008
[32] A. Zimmermann, Invariance of generalized Reynolds ideals under derived equivalences.. · Zbl 1131.16003
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