Skip to main content

Classification of two-term tilting complexes over Brauer graph algebras

Abstract

Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.

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

Abbreviations

\(z\subset w\) :

Non-empty continuous subsequence

\(S_i\) :

Simple module corresponding to i

\(P_i\) :

Indecomposable projective module corresponding to i

|X|:

Number of isoclasses of the indecomposable summands of X

\({\mathbb {G}}\) :

Ribbon graph or Brauer graph

V :

Set of vertices

H :

Set of half-edges

s :

Emanating vertex specifier

\(\overline{e}\) :

Involution acting on e

\(\sigma \) :

Cyclic ordering

\((e,\sigma (e),\ldots )_v\) :

Cyclic ordering around v

\(\mathrm {val}\) :

Valency

\({{\mathfrak {m}}}\) :

Multiplicity function

\(w=(e_1,\ldots ,e_l)\) :

Half-walk

W :

Walk

\(\epsilon _W\) :

Signature

\((w;\epsilon )\) or \((e_1^+,e_2^-,\ldots )\) :

Signed half-walk

\({\mathsf {SW}}({\mathbb {G}})\) :

Set of signed walks

\(w\cap w'\) :

Set of maximal continuous subsequences common in w and \(w'\)

\(W\cap W'\) :

Set of maximal common subwalks of W and \(W'\)

\(\mathrm {vr}_\pm (e)\) :

Virtual edges associated to e

\({\mathsf {AW}}({\mathbb {G}})\) :

Set of admissible walks

\({\mathsf {tilt}\,}\Lambda \) :

Set of tilting complexes

\(T\ge U\) :

A partial order on tilting complexes

\({\mathsf {n-tilt}}\Lambda \) :

Set of n-term tilting complexes

\({\mathsf {2ipt}\,}\Lambda \) :

Set of indecomposable two-term pretilting complexes

\(\Lambda _{\mathbb {G}}\) :

Brauer graph algebra associated to a Bruaer graph \({\mathbb {G}}\)

\(Q_{\mathbb {G}}\) :

Quiver associated to a Brauer graph \({\mathbb {G}}\)

\(1_E, {[e]}^{0}\) :

Idempotent at \(E=\{e,\overline{e}\}\)

\((e|\sigma (e))\) :

An arrow in \(Q_{\mathbb {G}}\), or irreducible map between projectives

(e|f):

A short path in \(Q_{\mathbb {G}}\), or a short map

[e]:

A Brauer cycle, i.e. shorthand for (e|e)

\(P_M\) :

Minimal projective presentation of M

\(d_M\) :

Differential map in \(P_M\)

\({\mathsf {2scx}}\,\Lambda \) :

Set of certain type of short string complexes and stalk projectives

\(T_w, T_W\) :

Two-term complex associated to a (half-)walk (w or) W

\({\mathsf {CW}}({\mathbb {G}})\) :

Set of complete admissible sets of signed walks on \({\mathbb {G}}\)

\(M_T\) :

Zeroth cohomology of the complex T

\(N_T\) :

\((-1)\)-st cohomology of the complex T

A :

\(\Lambda _{\mathbb {G}}/{\mathsf {soc}\,}\Lambda _{\mathbb {G}}\)

\({\mathfrak {h}}(\alpha )\) :

Head of an arrow or a word

\({\mathfrak {t}}(\alpha )\) :

Tail of an arrow or a word

\({\mathcal {A}}\) :

Set of alphabets associated to A

\(\alpha ^{-1}\) :

Formal inverse of an arrow or a word

\({\mathsf {w}}, {\mathsf {u}}\) :

Word or string of A

\(Q_{\mathsf {w}}\) :

Quiver associated to a string \({\mathsf {w}}\) of A

\(\eta _e\) :

Longest path in the hook module

\(\mu _X^-(T),\mu _X^+(T)\) :

Mutation of the tilting complex T at direct summand X

\({\mathcal {T}}_\Lambda \) :

Hasse quiver of \(({\mathsf {tilt}\,}\Lambda , \le )\), aka tilting quiver of \(\Lambda \)

\(\mu _E^-({\mathbb {G}}),\mu _E^+({\mathbb {G}})\) :

Mutation of Brauer tree

\({\mathbb {G}}^\mathsf {op}\) :

The opposite of \({\mathbb {G}}\)

References

  1. Abe, H., Hoshino, M.: On derived equivalences for selfinjective algebras. Commun. Algebra 34(12), 4441–4452 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  2. Adachi, T.: The classification of \(\tau \)-tilting modules over Nakayama algebras. J. Algebra 452, 227–262 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  3. Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-Tilting theory. Compos. Math. 150(3), 415–452 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  4. Aihara, T.: Tilting-connected symmetric algebras. Algebra Represent. Theory 16(3), 873–894 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. Aihara, T.: Derived equivalences between symmetric special biserial algebras. J. Pure Appl. Algebra 219(5), 1800–1825 (2015)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  7. Aihara, T., Grant, J., Iyama, O.: Private communication

  8. Aihara, T., Mizuno, Y.: Classifying tilting complexes over preprojective algebras of Dynkin type. Algebra Number Theory 11(6), 1287–1315 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  9. Antipov, M.: The Grothendieck group of the stable category of symmetric special biserial algebras. J. Math. Sci. (3) 136, 3833–3836 (2006)

    MathSciNet  Article  Google Scholar 

  10. Antipov, M.: Derived equivalence of symmetric special biserial algebras. J. Math. Sci. (5) 147, 6981–6994 (2007)

    MathSciNet  Article  Google Scholar 

  11. Antipov, M., Zvonareva, A.: Two-term partial tilting complexes over Brauer tree algebras. J. Math. Sci. (3) 202, 333–345 (2014)

    Article  MATH  Google Scholar 

  12. Antipov, M., Zvonareva, A.: On stably biserial algebras and the Auslander-Reiten conjecture for special biserial algebras. arXiv:1711.05021 (2017)

  13. Ariki, S., Iijima, K., Park, E.: Representation type of finite quiver Hecke algebras of type \(A^{(1)}_{\ell }\) for arbitrary parameters. Int. Math. Res. Not. IMRN (15), 6070–6135 (2015)

  14. Bekkert, V., Merklen, H.A.: Indecomposables in derived categories of gentle algebras. Algebr. Represent. Theory (3) 6, 285–302 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  15. Broomhead, N., Pauksztello, D., Ploog, D.: Discrete derived categories II: the silting pairs CW complex and the stability manifold. J. Lond. Math. Soc. (2) 93(2), 273–300 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  16. Butler, M.C.R., Ringel, C.M.: Auslander–Reiten sequences with few middle terms and applications to string algebras. Commun. Algebra 15(1–2), 145–179 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  17. Crawley-Boevey, W.W.: Maps between representations of zero-relation algebras. J. Algebra 126(2), 259–263 (1989)

    MathSciNet  Article  MATH  Google Scholar 

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

  19. Drozd, Y.A., Kirichenko, V.V.: On quasi-Bass orders. Math. USSR Izv. 6(2), 323–365 (1972)

    Article  MATH  Google Scholar 

  20. Erdmann, K.: Blocks of tame representation type and related algebras. In: Lecture Notes in Mathematics, vol. 1428. Springer, New York (1990)

  21. Green, J.A.: Walking around the Brauer tree. J. Aust. Math. Soc. 17, 197–213 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  22. Kauer, M.: Derived equivalence of graph algebras. Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997). Contemp. Math. 229, 201–213 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  23. Kauer, M.: Derivierte Äquivalenz von Graphordnungen und Graphalgebren. PhD thesis, Shaker, Aachen (1998)

  24. Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. (1) 15, 203–271 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  25. Marsh, R.J., Schroll, S.: The geometry of Brauer graph algebras and cluster mutations. J. Algebra 419, 141–166 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  26. Membrillo-Hernández, F.H.: Brauer tree algebras and derived equivalence. J. Pure Appl. Algebra 114, 231–258 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  27. Okuyama, T.: Some examples of derived equivalent blocks of finite groups (1998) (unpublished paper)

  28. Pogorzały, Z.: Algebras stably equivalent to self-injective special biserial algebras. Commun. Algebra 22(4), 1127–1160 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  29. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39, 436–456 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  30. Roggenkamp, K.W.: Biserial algebras and graphs. Algebras and modules, II (1996), pp. 481–496. In: CMS Conference Proceedings, vol. 24. American Mathematical Society, Providence (1998)

  31. Schaps, M., Zakay-Illouz, E.: Pointed Brauer trees. J. Algebra (2) 246, 647–672 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  32. Schroll, S.: Trivial extensions of gentle algebras and Brauer graph algebras. J. Algebra 444, 183–200 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  33. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. (1) 108, 37–108 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  34. Skowroński, A.: Selfinjective algebras: finite and tame type. Trends in representation theory of algebras and related topics. Contemp. Math. 406, 169–238 (2006)

    Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  36. Zvonareva, A.: Two-term tilting complexes over Brauer tree algebras. J. Math. Sci. (N.Y.) 202(3), 333–345 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  37. Zvonareva, A.: Mutations and the derived Picard group of the Brauer star algebra. J. Algebra 443, 270–299 (2015)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We owe our deepest gratitude to Osamu Iyama for many fruitful discussions, as well as the financial support for the third author’s visit to Nagoya University, which nurtured this research. This article is typed up during AC’s subsequent visits at Nagoya University, and finished during the first author’s visit at Uppsala University. We are thankful for the hospitality of these institutions. We thank the referee for pointing out missing references and known results in the literature. We thank also Ryoichi Kase and Alexandra Zvonareva for various discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Aaron Chan.

Additional information

T. Aihara and A. Chan was partly supported by IAR Research Project, Institute for Advanced Research, Nagoya University. T. Aihara was supported by Grant-in-Aid for Young Scientists 15K17516. T. Adachi is supported by Grant-in-Aid for JSPS Research Fellow 17J05537. A. Chan is supported by JSPS International Research Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Adachi, T., Aihara, T. & Chan, A. Classification of two-term tilting complexes over Brauer graph algebras. Math. Z. 290, 1–36 (2018). https://doi.org/10.1007/s00209-017-2006-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-2006-9

Keywords

  • Tilting complex
  • Brauer graph algebra
  • Ribbon graph

Mathematics Subject Classification

  • 16G10
  • 16G20
  • 18E30