On Stably Biserial Algebras and the Auslander–Reiten Conjecture for Special Biserial Algebras

  • M. A. AntipovEmail author
  • A. O. Zvonareva

According to a result claimed by Pogorza_ly, selfinjective special biserial algebras can be stably equivalent to stably biserial algebras only, and these two classes coincide. By an example of Ariki, Iijima, and Park, the classes of stably biserial and selfinjective special biserial algebras do not coincide. In these notes based on some ideas from the Pogorzały paper, a detailed proof is given for the fact that a selfinjective special biserial algebra can be stably equivalent to a stably biserial algebra only. The structure of symmetric stably biserial algebras is analyzed. It is shown that in characteristic other than 2, the classes of symmetric special biserial (Brauer graph) algebras and symmetric stably biserial algebras coincide. Also a proof of the Auslander–Reiten conjecture for special biserial algebras is given.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. Adachi, T. Aihara, and A. Chan, “Classification of two-term tilting complexes over Brauer graph algebras,” arXiv:1504.04827 (2015).Google Scholar
  2. 2.
    T. Aihara, “Derived equivalences between symmetric special biserial algebras,” J. Pure App. Algebra, 219, No. 5, 1800–1825 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Antipov, “Derived equivalence of symmetric special biserial algebras,” Zap. Nauchn. Semin. POMI, 343, 5–32 (2007).MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Antipov and A. Zvonareva, “Two-term partial tilting complexes over Brauer tree algebras,” Zap. Nauchn. Semin. POMI, 413, 5–25 (2013).zbMATHGoogle Scholar
  5. 5.
    S. Ariki, K. Iijima, and E. Park, “Representation type of finite quiver Hecke algebras of type \( {A}_l^{(1)} \) for arbitrary parameters,” Int. Math. Research Notices, 15, 6070–6135 (2015).Google Scholar
  6. 6.
    M. Auslander and I. Reiten, “Representation theory of Artin algebras V: Invariants given by almost split sequences,” Commun. Algebra, 5, No. 5, 519–554 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    M. Auslander and I. Reiten, “Representation theory of Artin algebras VI: A functorial approach to almost split sequences,” Commun. Algebra, 6, No. 3, 257–300 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. C. R. Butler and C. M. Ringel, “Auslander–Reiten sequences with few middle terms and applications to string algebras,” Commun. Algebra, 15, No. 1–2, 145–179 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Kauer, “Derived equivalence of graph algebras,” Contemp. Math., 229, 201–214 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    R. J. Marsh and S. Schroll, “The geometry of Brauer graph algebras and cluster mutations,” J. Algebra, 419, 141–166 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R. Martínez-Villa, “Algebras stably equivalent to l-hereditary,” Lect. Notes Math., 832, 396–431 (1980).Google Scholar
  12. 12.
    R. Martínez-Villa, “Properties that are left invariant under stable equivalence,” Commun. Algebra, 18, No. 12, 4141–4169 (1990).MathSciNetzbMATHGoogle Scholar
  13. 13.
    I. Muchtadi-Alamsyah, “Braid action on derived category Nakayama algebras,” Commun. Algebra, 36, No. 7, 2544–2569 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Z. Pogorzały, “Algebras stably equivalent to selfinjective special biserial algebras,” Commun. Algebra, 22, No. 4, 1127–1160 (1994).CrossRefzbMATHGoogle Scholar
  15. 15.
    Z. Pogorzały, “On the stable Grothendieck groups,” in: CMS Conf. Proc., 14 (1993), pp. 393–406.Google Scholar
  16. 16.
    J. Rickard and M. Schaps, “Folded tilting complexes for Brauer tree algebras,” Adv. Math., 171, No. 2, 169–182 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    R. Rouquier and A. Zimmermann, “Picard groups for derived module categories,” Proc. London Math. Soc., 87, No. 1, 197–225 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Schaps and E. Zakay-Illouz, “Pointed Brauer trees,” J. Algebra, 246, No. 2, 647–672 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S. Schroll, “Trivial extensions of gentle algebras and Brauer graph algebras,” J. Algebra, 444, 183–200 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. Skowroński and J. Waschb¨usch, “Representation-finite biserial algebras,” J. Reine Angew. Math., 345, 172–181 (1983).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Y. Volkov and A. Zvonareva, “Derived Picard groups of selfinjective Nakayama algebras,” Manuscripta Math., 152, No. 1-2, 199–222 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    B. Wald and J. Waschb¨usch, “Tame biserial algebras,” J. Algebra, 95, No. 2, 480–500 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Zimmermann, “Self-equivalences of the derived category of Brauer tree algebras with exceptional vertex,” Anal. S¸tiint¸. Univ. Ovidius, 9, No. 1, 139–148 (2001).MathSciNetzbMATHGoogle Scholar
  24. 24.
    A. Zvonareva, “Mutations and the derived Picard group of the Brauer star algebra,” J. Algebra, 443, 270–299 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. Zvonareva, “Two-term tilting complexes over Brauer tree algebras,” Zap. Nauchn. Semin. POMI, 423, 132–165 (2014).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations