Skip to main content Accessibility help

Classification of Symmetric Special Biserial Algebras With At Most One Non-Uniserial Indecomposable Projective

  • Nicole Snashall (a1) and Rachel Taillefer (a2)


We consider a natural generalization of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the weakly symmetric algebras of Euclidean type n, as studied by Bocian et al., as well as some algebras of dihedral type.



Hide All
1.Al-Nofayee, S., Equivalences of derived categories for self-injective algebras, J. Alg. 313(2 (2007), 897904.
2.Alperin, J. L., Local representation theory: modular representations as an introduction to the local representation theory of finite groups, Cambridge Studies in Advanced Mathematics, Volume 11 (Cambridge University Press, 1986).
3.Antipov, M. A. and Generalov, A. I., The Yoneda algebras of symmetric special biserial algebras are finitely generated, St Petersburg Math. J. 17(3 (2006), 377392.
4.Benson, D., Representations and cohomology, I: basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics, Volume 30 (Cambridge University Press, 1991).
5.Bocian, R. and Skowroński, A., Weakly symmetric algebras of Euclidean type, J. Reine Angew. Math. 580 (2005), 157199.
6.Bocian, R., Holm, T. and Skowroński, A., Derived equivalence classification of weakly symmetric algebras of Euclidean type, J. Pure Appl. Alg. 191(1–2) (2004), 4374.
7.Erdmann, K., Blocks of tame representation type and related algebras, Springer Lecture Notes in Mathematics, Volume 1428 (Springer, 1990).
8.Gabriel, P. and Riedtmann, Ch., Group representations without groups, Comment. Math. Helv. 54(2 (1979), 240287.
9.Green, E. L. and Snashall, N., Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group, Forum Math. 16(1 (2004), 1736.
10.Happel, D., Hochschild cohomology of finite-dimensional algebras, in Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, Lecture Notes in Mathematics, Volume 1404, pp. 108126 (Springer, 1989).
11.Holm, T., Derived equivalent tame blocks, J. Alg. 194 (1997), 178200.
12.Holm, T., Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. Alg. 211(1 (1999), 159205.
13.Holm, T. and Zimmermann, A., Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type, J. Alg. 320(9 (2008), 34253437.
14.Kauer, M., Derived equivalence of graph algebras, in Trends in the representation theory of finite dimensional algebras, Contemporary Mathematics, Volume 229, pp. 201213 (American Mathematical Society, Providence, RI, 1998).
15.Keller, B. and Vossieck, D., Sous les catégories dérivées, C. R. Acad. Sci. Paris I 305(6 (1987), 225228.
16.Koenig, S., Liu, Y. and Zhou, G., Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture, Trans. Am. Math. Soc. 364(1 (2012), 195232.
17.Linckelmann, M., A derived equivalence for blocks with dihedral defect groups, J. Alg. 164 (1994), 244255.
18.Liu, Y., Summands of stable equivalences of Morita type, Commun. Alg. 36 (2008), 37783782.
19.Liu, Y., Zhou, G. and Zimmermann, A., Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture, Math. Z. 270(3–4) (2012), 759781.
20.Membrillo-Hernández, F. H., Brauer tree algebras and derived equivalence, J. Pure Appl. Alg. 114(3 (1997), 231258.
21.Pogorzaly, Z., Algebras stably equivalent to self-injective special biserial algebras, Commun. Alg. 22(4 (1994), 11271160.
22.Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Alg. 61(3 (1989), 303317.
23.Rickard, J., Derived equivalences as derived functors, J. Lond. Math. Soc. 43 (1991), 3748.
24.Roggenkamp, K. W., Biserial algebras and graphs, in Algebras and modules II, Conference Proceedings, Canadian Mathematical Society, Volume 24, pp. 481496 (American Mathematical Society, Providence, RI, 1998).
25.Skowroński, A., Selfinjective algebras: finite and tame type, in Trends in representation theory of algebras and related topics, Contemporary Mathematics, Volume 406, pp. 169238 (American Mathematical Society, Providence, RI, 2006).
26.Snashall, N. and Taillefer, R., The Hochschild cohomology ring of a class of special biserial algebras, J. Alg. Appl. 9(1 (2010), 73122.
27.Wald, B. and Waschbüsch, J., Tame biserial algebras, J. Alg. 95(2 (1985), 480500.
28.Xi, C., Stable equivalences of adjoint type, Forum Math. 20(1 (2008), 8197.
29.Zimmermann, A., Invariance of generalized Reynolds ideals under derived equivalences, Math. Proc. R. Irish Acad. A 107(1 (2007), 19.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed