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.
Similar content being viewed by others
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
Abe, H., Hoshino, M.: On derived equivalences for selfinjective algebras. Commun. Algebra 34(12), 4441–4452 (2006)
Adachi, T.: The classification of \(\tau \)-tilting modules over Nakayama algebras. J. Algebra 452, 227–262 (2016)
Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-Tilting theory. Compos. Math. 150(3), 415–452 (2014)
Aihara, T.: Tilting-connected symmetric algebras. Algebra Represent. Theory 16(3), 873–894 (2013)
Aihara, T.: Derived equivalences between symmetric special biserial algebras. J. Pure Appl. Algebra 219(5), 1800–1825 (2015)
Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85(3), 633–668 (2012)
Aihara, T., Grant, J., Iyama, O.: Private communication
Aihara, T., Mizuno, Y.: Classifying tilting complexes over preprojective algebras of Dynkin type. Algebra Number Theory 11(6), 1287–1315 (2017)
Antipov, M.: The Grothendieck group of the stable category of symmetric special biserial algebras. J. Math. Sci. (3) 136, 3833–3836 (2006)
Antipov, M.: Derived equivalence of symmetric special biserial algebras. J. Math. Sci. (5) 147, 6981–6994 (2007)
Antipov, M., Zvonareva, A.: Two-term partial tilting complexes over Brauer tree algebras. J. Math. Sci. (3) 202, 333–345 (2014)
Antipov, M., Zvonareva, A.: On stably biserial algebras and the Auslander-Reiten conjecture for special biserial algebras. arXiv:1711.05021 (2017)
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)
Bekkert, V., Merklen, H.A.: Indecomposables in derived categories of gentle algebras. Algebr. Represent. Theory (3) 6, 285–302 (2003)
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)
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)
Crawley-Boevey, W.W.: Maps between representations of zero-relation algebras. J. Algebra 126(2), 259–263 (1989)
Demonet, L.: Algebras of partial triangulations. arXiv:1602.01592 (2016)
Drozd, Y.A., Kirichenko, V.V.: On quasi-Bass orders. Math. USSR Izv. 6(2), 323–365 (1972)
Erdmann, K.: Blocks of tame representation type and related algebras. In: Lecture Notes in Mathematics, vol. 1428. Springer, New York (1990)
Green, J.A.: Walking around the Brauer tree. J. Aust. Math. Soc. 17, 197–213 (1974)
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)
Kauer, M.: Derivierte Äquivalenz von Graphordnungen und Graphalgebren. PhD thesis, Shaker, Aachen (1998)
Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. (1) 15, 203–271 (2002)
Marsh, R.J., Schroll, S.: The geometry of Brauer graph algebras and cluster mutations. J. Algebra 419, 141–166 (2014)
Membrillo-Hernández, F.H.: Brauer tree algebras and derived equivalence. J. Pure Appl. Algebra 114, 231–258 (1997)
Okuyama, T.: Some examples of derived equivalent blocks of finite groups (1998) (unpublished paper)
Pogorzały, Z.: Algebras stably equivalent to self-injective special biserial algebras. Commun. Algebra 22(4), 1127–1160 (1994)
Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39, 436–456 (1989)
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)
Schaps, M., Zakay-Illouz, E.: Pointed Brauer trees. J. Algebra (2) 246, 647–672 (2001)
Schroll, S.: Trivial extensions of gentle algebras and Brauer graph algebras. J. Algebra 444, 183–200 (2015)
Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. (1) 108, 37–108 (2001)
Skowroński, A.: Selfinjective algebras: finite and tame type. Trends in representation theory of algebras and related topics. Contemp. Math. 406, 169–238 (2006)
Wald, B., Waschbüsch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985)
Zvonareva, A.: Two-term tilting complexes over Brauer tree algebras. J. Math. Sci. (N.Y.) 202(3), 333–345 (2014)
Zvonareva, A.: Mutations and the derived Picard group of the Brauer star algebra. J. Algebra 443, 270–299 (2015)
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
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-017-2006-9