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Algebras and Representation Theory

, Volume 22, Issue 1, pp 141–176 | Cite as

On Representation-Finite Gendo-Symmetric Biserial Algebras

  • Aaron ChanEmail author
  • René Marczinzik
Article
  • 27 Downloads

Abstract

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.

Keywords

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 

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Notes

Acknowledgments

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.

References

  1. 1.
    Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85(3), 633–668 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auslander, M.: Representation theory of Artin algebras I. Comm. Algebra 1, 117–268 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auslander, M., Platzeck, M.I., Todorov, G.: Homological theory of idempotent ideals. Trans. Amer. Math. Soc. 332(2), 667–692 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra III: category \(\mathcal {O}\). Represent. Theory 15, 170–243 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, X.W.: Gorenstein Homological Algebra of Artin Algebras. arXiv:1712.04587 (2017)
  9. 9.
    Chen, H.X., Koenig, S.: Ortho-symmetric modules, Gorenstein algebras and derived equivalences. Int. Math. Res. Not. IMRN. (2016).  https://doi.org/10.1093/imrn/rnv368
  10. 10.
    Chen, H.X., Xi, C.C.: Dominant dimensions, derived equivalences and tilting modules. Israel J. Math. (2016).  https://doi.org/10.1007/s11856-016-1327-4
  11. 11.
    Demonet, L.: Algebras of partial triangulations. arXiv:1602.01592 (2016)
  12. 12.
    Dugas, A.: Tilting mutation of weakly symmetric algebras and stable equivalence. Algebr. Represent. Theory 17(3), 863–884 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Enochs, E., Jenda, O.: Relative Homological Algebra. de Gruyter Expositions in Mathematics, 30. Walter de Gruyter & Co., Berlin (2000). xii+ 339CrossRefGoogle Scholar
  14. 14.
    Fang, M., Hu, W., Koenig, S.: Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions. arXiv:1607.03513 (2016)
  15. 15.
    Fang, M., Koenig, S.: Endomorphism algebras of generators over symmetric algebras. J. Algebra 332, 428–433 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fang, M., Koenig, S.: Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension. Trans. Amer. Math. Soc. 7, 5037–5055 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gabriel, P., Riedtmann, C.: Group representations without groups. Comment. Math. Helv. 54(2), 240–287 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Green, E., Schroll, S.: Brauer configuration algebras: A generalization of Brauer graph algebras. arXiv:1508.03617 (2015)
  19. 19.
    Green, E., Schroll, S.: Almost gentle algebras and their trivial extensions. arXiv:1603.03587 (2016)
  20. 20.
    Green, J.A.: Walking around the brauer tree. J. Austral. Math. Soc. 17, 197–213 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)Google Scholar
  22. 22.
    Hu, W., Xi, C.C.: Derived equivalences and stable equivalences of Morita type. I. Nagoya Math. J. 200, 107–152 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hu, W., Xi, C.C.: Derived equivalences and stable equivalences of Morita type II. arXiv:1412.7301 (2014)
  24. 24.
    Iyama, O.: Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210, 22–50 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Iyama, O.: Cluster tilting for higher Auslander algebras. Adv. Math. 226(1), 1–61 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kerner, O., Yamagata, K.: Morita algebras. J. Algebra 382, 185–202 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Koenig, S., Slungård, I.H., Xi, C.C.: Double centralizer properties, dominant dimension, and tilting modules. J. Algebra 240(1), 393–412 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Külshammer, J.: In the bocs seat: Quasi-hereditary algebras and representation type. To appear in SPP 1388 Conference Proceedings (2016)Google Scholar
  29. 29.
    Marczinzik, R.: Upper bounds for the dominant dimension of Nakayama and related algebras. J. Algebra 496, 216–241 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Marczinzik, R.: Dominant dimension, standardly stratified algebras and Gorenstein homological algebra preprint (2016)Google Scholar
  31. 31.
    Marczinzik, R.: On a conjecture about dominant dimensions of algebras. arXiv:1606.00340 (2016)
  32. 32.
    Marczinzik, R.: Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra preprint (2016)Google Scholar
  33. 33.
    Martin, P.P.: The structure of the partition algebras. J. Algebra 183, 319–358 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Morita, K.: On algebras for which every faithful representation is its own second commutator. Math. Z. 69, 429–434 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mueller, B.: The classification of algebras by dominant dimension. Can. J. Math. 20, 398–409 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Perrina, D., Restivo, A.: Words. Chapter 8 in Handbook of Enumerative Combinatorics (Discrete Mathematics and Its Applications) edited by Miklos BonaGoogle Scholar
  37. 37.
    Rickard, J.: Derived categories and stable equivalence. Pure Appl. Algebra 61 (3), 303–317 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rickard, J.: Equivalences of derived categories for symmetric algebras. J. Algebra 257, 460–481 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Seidel, P.: Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics, p viii+ 326. European Mathematical Society (EMS), Zürich (2008)CrossRefzbMATHGoogle Scholar
  40. 40.
    Skowronski, A., Waschbüsch, J.: Representation-finite biserial algebras. J. Reine Angrew. Math. 345, 172–181 (1983)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Skowronski, A., Yamagata, K.: Frobenius Algebras I: Basic Representation Theory. EMS textbooks in mathematics (2011)Google Scholar
  42. 42.
    Stanley, R. Enumerative Combinatorics, Volume I Second edition. (Cambridge Studies in Advanced Mathematics, Volume 49)Google Scholar
  43. 43.
    Stroppel, C.: Category \(\mathcal {O}\): quivers and endomorphism rings of projectives. Represent. Theory 7, 322–345 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tachikawa, H.: Quasi-Frobenius rings and generalizations: QF-3 and QF-1 rings (lecture notes in mathematics 351) Springer (1973)Google Scholar
  45. 45.
    Thrall, R.M.: Some generalizations of quasi-Frobenius algebras. Trans. Amer. Math. Soc. 64, 173–183 (1948)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Vajda, S.: Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications (Dover Books on Mathematics)Google Scholar
  47. 47.
    Wald, B., Waschbüsch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Westbury, B.W.: The representation theory of the Temperley-Lieb algebras. Math. Z. 219(4), 539–565 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Yamagata, K.: Frobenius Algebras. Handbook of Algebra, vol. 1, pp 841–887. North-Holland, Amsterdam (1996)CrossRefzbMATHGoogle Scholar
  50. 50.
    Yamagata, K.: Modules with serial Noetherian endomorphism rings. J. Algebra 127(2), 264–469 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Zimmermann, A.: Representation theory. A homological algebra point of view. Algebra and Applications, 19. Springer, Cham (2014). xx+ 707zbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.Institute of Algebra and Number TheoryUniversity of StuttgartStuttgartGermany

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