Skip to main content

On Representation-Finite Gendo-Symmetric Biserial Algebras


Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost ν-stable derived equivalences, introduced in Hu and Xi (I. Nagoya Math. J., 200:107–152, 2010), between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.

This is a preview of subscription content, access via your institution.


  1. Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85(3), 633–668 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  2. Auslander, M.: Representation theory of Artin algebras I. Comm. Algebra 1, 117–268 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  3. Auslander, M., Platzeck, M.I., Todorov, G.: Homological theory of idempotent ideals. Trans. Amer. Math. Soc. 332(2), 667–692 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  4. Auslander, M., Reiten, I., Smalo, S.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  5. Asashiba, H.: On a lift of an individual stable equivalence to a standard derived equivalence for representation-finite self-injective algebras. Algebr. Represent. Theory (4) 6, 427–447 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. Boe, B., Nakano, D., Wiesner, E.: Category \(\mathcal {O}\) for the Virasoro algebra: cohomology and Koszulity. Pacific J. Math. 234(1), 1–21 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  7. Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra III: category \(\mathcal {O}\). Represent. Theory 15, 170–243 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. Chen, X.W.: Gorenstein Homological Algebra of Artin Algebras. arXiv:1712.04587 (2017)

  9. Chen, H.X., Koenig, S.: Ortho-symmetric modules, Gorenstein algebras and derived equivalences. Int. Math. Res. Not. IMRN. (2016).

  10. Chen, H.X., Xi, C.C.: Dominant dimensions, derived equivalences and tilting modules. Israel J. Math. (2016).

  11. Demonet, L.: Algebras of partial triangulations. arXiv:1602.01592 (2016)

  12. Dugas, A.: Tilting mutation of weakly symmetric algebras and stable equivalence. Algebr. Represent. Theory 17(3), 863–884 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  13. Enochs, E., Jenda, O.: Relative Homological Algebra. de Gruyter Expositions in Mathematics, 30. Walter de Gruyter & Co., Berlin (2000). xii+ 339

    Book  Google Scholar 

  14. Fang, M., Hu, W., Koenig, S.: Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions. arXiv:1607.03513 (2016)

  15. Fang, M., Koenig, S.: Endomorphism algebras of generators over symmetric algebras. J. Algebra 332, 428–433 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  16. Fang, M., Koenig, S.: Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension. Trans. Amer. Math. Soc. 7, 5037–5055 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Gabriel, P., Riedtmann, C.: Group representations without groups. Comment. Math. Helv. 54(2), 240–287 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  18. Green, E., Schroll, S.: Brauer configuration algebras: A generalization of Brauer graph algebras. arXiv:1508.03617 (2015)

  19. Green, E., Schroll, S.: Almost gentle algebras and their trivial extensions. arXiv:1603.03587 (2016)

  20. Green, J.A.: Walking around the brauer tree. J. Austral. Math. Soc. 17, 197–213 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  21. Happel, D.: On Gorenstein algebras. Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), 389–404, Progr. Math., 95, Birkhäuser, Basel (1991)

  22. Hu, W., Xi, C.C.: Derived equivalences and stable equivalences of Morita type. I. Nagoya Math. J. 200, 107–152 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  23. Hu, W., Xi, C.C.: Derived equivalences and stable equivalences of Morita type II. arXiv:1412.7301 (2014)

  24. Iyama, O.: Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210, 22–50 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  25. Iyama, O.: Cluster tilting for higher Auslander algebras. Adv. Math. 226(1), 1–61 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  26. Kerner, O., Yamagata, K.: Morita algebras. J. Algebra 382, 185–202 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  27. Koenig, S., Slungård, I.H., Xi, C.C.: Double centralizer properties, dominant dimension, and tilting modules. J. Algebra 240(1), 393–412 (2001)

    MathSciNet  Article  Google Scholar 

  28. Külshammer, J.: In the bocs seat: Quasi-hereditary algebras and representation type. To appear in SPP 1388 Conference Proceedings (2016)

  29. Marczinzik, R.: Upper bounds for the dominant dimension of Nakayama and related algebras. J. Algebra 496, 216–241 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  30. Marczinzik, R.: Dominant dimension, standardly stratified algebras and Gorenstein homological algebra preprint (2016)

  31. Marczinzik, R.: On a conjecture about dominant dimensions of algebras. arXiv:1606.00340 (2016)

  32. Marczinzik, R.: Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra preprint (2016)

  33. Martin, P.P.: The structure of the partition algebras. J. Algebra 183, 319–358 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  34. Morita, K.: On algebras for which every faithful representation is its own second commutator. Math. Z. 69, 429–434 (1958)

    MathSciNet  Article  MATH  Google Scholar 

  35. Mueller, B.: The classification of algebras by dominant dimension. Can. J. Math. 20, 398–409 (1968)

    MathSciNet  Article  MATH  Google Scholar 

  36. Perrina, D., Restivo, A.: Words. Chapter 8 in Handbook of Enumerative Combinatorics (Discrete Mathematics and Its Applications) edited by Miklos Bona

  37. Rickard, J.: Derived categories and stable equivalence. Pure Appl. Algebra 61 (3), 303–317 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  38. Rickard, J.: Equivalences of derived categories for symmetric algebras. J. Algebra 257, 460–481 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  39. Seidel, P.: Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics, p viii+ 326. European Mathematical Society (EMS), Zürich (2008)

    Book  MATH  Google Scholar 

  40. Skowronski, A., Waschbüsch, J.: Representation-finite biserial algebras. J. Reine Angrew. Math. 345, 172–181 (1983)

    MathSciNet  MATH  Google Scholar 

  41. Skowronski, A., Yamagata, K.: Frobenius Algebras I: Basic Representation Theory. EMS textbooks in mathematics (2011)

  42. Stanley, R. Enumerative Combinatorics, Volume I Second edition. (Cambridge Studies in Advanced Mathematics, Volume 49)

  43. Stroppel, C.: Category \(\mathcal {O}\): quivers and endomorphism rings of projectives. Represent. Theory 7, 322–345 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  44. Tachikawa, H.: Quasi-Frobenius rings and generalizations: QF-3 and QF-1 rings (lecture notes in mathematics 351) Springer (1973)

  45. Thrall, R.M.: Some generalizations of quasi-Frobenius algebras. Trans. Amer. Math. Soc. 64, 173–183 (1948)

    MathSciNet  MATH  Google Scholar 

  46. Vajda, S.: Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications (Dover Books on Mathematics)

  47. Wald, B., Waschbüsch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  48. Westbury, B.W.: The representation theory of the Temperley-Lieb algebras. Math. Z. 219(4), 539–565 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  49. Yamagata, K.: Frobenius Algebras. Handbook of Algebra, vol. 1, pp 841–887. North-Holland, Amsterdam (1996)

    Book  MATH  Google Scholar 

  50. Yamagata, K.: Modules with serial Noetherian endomorphism rings. J. Algebra 127(2), 264–469 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  51. Zimmermann, A.: Representation theory. A homological algebra point of view. Algebra and Applications, 19. Springer, Cham (2014). xx+ 707

    MATH  Google Scholar 

Download references


This research was initiated during the “Conference on triangulated categories in algebra, geometry and topology” and “Workshop on Brauer graph algebras” in Stuttgart University, March 2016. We thank Steffen Koenig for comments on an earlier draught. AC is supported by IAR Research Project. Institute for Advanced Research, Nagoya University, and JSPS International Fellowship.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Aaron Chan.

Additional information

Presented by Yuri Drozd.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chan, A., Marczinzik, R. On Representation-Finite Gendo-Symmetric Biserial Algebras. Algebr Represent Theor 22, 141–176 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Representation theory of finite dimensional algebras
  • Gorenstein dimension
  • Gendo-symmetric algebra
  • Nakayama algebras
  • Almost ν-stable derived equivalence
  • Brauer tree algebras
  • Dominant dimension

Mathematics Subject Classification 2010

  • Primary 16G10
  • 16E10