The 13th International Conference Graduate School of Mathematics, Nagoya University 第13回 名古屋国際数学コンファレンス Perspectives of Representation Theory of Algebras celebrating Kunio Yamagata's 65th birthday November 11-15, 2013 in Nagoya University The 13th International Conference, Graduate School of Mathematics, Nagoya University Perspectives of Representation Theory of Algebras — celebrating Kunio Yamagata’s 65th birthday — Period: November 11–15, 2013 Venue: Sakata–Hirata Hall, Science South Bldg. (Nov. 11–14), ES Hall, E&S Bldg. (Nov. 15), Nagoya University, Nagoya, Japan Organizers: Hideto Asashiba (Shizuoka University), Osamu Iyama (Nagoya University), Jun-ichi Miyachi (Tokyo Gakugei University), Izuru Mori (Shizuoka University), Masahisa Sato (Yamanashi University), Andrzej Skowronski (Nicolaus Copernicus University), Morio Uematsu (Jobu University), Yuji Yoshino (Okayama University) Prof. Kunio Yamagata TIME TABLE Time Nov. 11 (Mon) Nov. 12 (Tue) Nov. 13 (Wed) Nov. 14 (Thu) Nov. 15 (Fri) 09:30–10:20 I. Reiten A. Skowronski K. Erdmann Y. Yoshino H. Lenzing Coffee Break 11:00–11:30 G. Jasso M. Błaszkiewicz T. Itagaki R. Kanda K. Ueyama 11:40–12:30 T. Aihara J. Białkowski D. Zacharia R. Takahashi A. Takahashi Lunch Break Lunch Break 13:00– 14:00–14:50 R. Kase* P. Malicki M. Yoshiwaki* Y. Kimura Y. Mizuno* H. Koga* Coffee Break Excursion Coffee Break 15:40–16:30 L. Demonet A. Skowyrski H. Minamoto M. Wemyss 16:40–17:30 O. Kerner C. Xi J. Miyachi C. M. Ringel 18:00– Banquet • Monday: 14:00–14:30 Kase, 14:35–15:05 Mizuno. • Thursday: 14:00–14:30 Yoshiwaki, 14:35–15:05 Koga. • There will be a Conference Excursion on November 13th (Wed) afternoon. • We will take a group photo just after morning session on November 14th (Thu). • A banquet is planned on November 14th (Thu) from 18:00 at Mei-dining. (See Figure C) 1 PROGRAM November 11th (Mon) (at Sakata–Hirata Hall) 9:00– 9:30 — Registration — 9:30–10:20 Idun Reiten (Norwegian University of Science and Technology) Lattice structure of torsion classes — Coffee Break — 11:00–11:30 Gustavo Jasso (Nagoya University) On the simplicial complex associated with support τ-tilting modules 11:40–12:30 Takuma Aihara (Nagoya University) Tilting mutation theory — Lunch Break — 14:00–14:30 Ryoichi Kase (Osaka University) On the poset of pre-projective tilting modules over path algebras 14:35–15:05 Yuya Mizuno (Nagoya University) τ-tilting modules over preprojective algebras of Dynkin type 15:40–16:30 Laurent Demonet (Nagoya University) Ice quivers with potential associated with triangulations and Cohen-Macaulay modules over orders (case An and Dn) 16:40–17:30 Otto Kerner (University of D¨usseldorf ) From wild hereditary to tilted algebras, and back November 12th (Tue) (at Sakata–Hirata Hall) 9:30–10:20 Andrzej Skowronski (Nicolaus Copernicus University) Selfinjective algebras with deforming ideals — Coffee Break — 11:00–11:30 Marta Błaszkiewicz (Nicolaus Copernicus University) Selfinjective algebras of finite representation type with maximal almost split sequences 11:40–12:30 Jerzy Białkowski (Nicolaus Copernicus University) Periodicity of selfinjective algebras of polynomial growth — Lunch Break — 14:00–14:50 Piotr Malicki (Nicolaus Copernicus University) Finite cycles of indecomposable modules — Coffee Break — 15:40–16:30 Adam Skowyrski (Nicolaus Copernicus University) Homological problems for cycle-finite algebras 2 16:40–17:30 Changchang Xi (Capital Normal University) Higher dimensional tilting modules and recollements November 13th (Wed) (at Sakata–Hirata Hall) 9:30–10:20 Karin Erdmann (University of Oxford) Support varieties via Hochschild cohomology: a necessary condition — Coffee Break — 11:00–11:30 Tomohiro Itagaki (Tokyo University of Science) Cyclic homology of truncated quiver algebras and notes on the no loops conjecture for Hochschild homology 11:40–12:30 Dan Zacharia (Syracuse University) A characterization of the graded center of a Koszul algebra 13:00– — Excursion — November 14th (Thu) (at Sakata–Hirata Hall) 9:30–10:20 Yuji Yoshino (Okayama University) Dependence of total reflexivity conditions — Coffee Break — 11:00–11:30 Ryo Kanda (Nagoya University) Specialization orders on atom spectra of Grothendieck categories 11:40–12:30 Ryo Takahashi (Nagoya University) Existence of cohomology annihilators and strong generation of derived categories — Group Photo — — Lunch Break — 14:00–14:30 Michio Yoshiwaki (Osaka City University) Dimensions of triangulated categories with respect to subcategories 14:35–15:05 Hirotaka Koga (University of Tsukuba) Clifford extension — Coffee Break — 15:40–16:30 Hiroyuki Minamoto (Osaka Prefecture University) Derived Gabriel topology, localization and completion of dg-algebras 16:40–17:30 Jun-ichi Miyachi (Tokyo Gakugei University) Researches on the representation theory of algebras at University of Tsukuba 18:00– — Banquet — 3 November 15th (Fri) (at ES Hall) 9:30–10:20 Helmut Lenzing (University of Paderborn) The ubiquity of the equation x2 + y3 + z5 = 0 — Coffee Break — 11:00–11:30 Kenta Ueyama (Shizuoka University) Ample Group Actions on AS-regular Algebras and Noncommutative Graded Isolated Singularities 11:40–12:30 Atsushi Takahashi (Osaka University) Weyl groups and Artin groups associated to weighted projective lines — Lunch Break — 14:00–14:50 Yoshiyuki Kimura (Osaka City University) Quiver varieties and Quantum cluster algebras — Coffee Break — 15:40–16:30 Michael Wemyss (University of Edinburgh) Some new examples of self–injective algebras 16:40–17:30 Claus Michael Ringel (Bielefeld University) The root posets 4 ABSTRACTS Idun Reiten (Norwegian Univ. of Sci. and Tech.)....... November 11th (Mon), 9:30–10:20 Lattice structure of torsion classes This lecture is based on work with Iyama, Thomas and Todorov. A general problem for a finite dimensional algebra A over an algebraically closed field k is to investigate when the partially ordered set of functorially finite torsion classes are a lattice. Some work by Mizuno gives a motivation for considering this problem. We give the answer when A is the path algebra kQ for a finite acyclic quiver Q. Gustavo Jasso (Nagoya University).................. November 11th (Mon), 11:00–11:30 On the simplicial complex associated with support τ-tilting modules The class of support τ-tilting modules was introduced recently by Adachi–Iyama–Reiten so as to provide a completion of the class of tilting modules from the point of view of mutation. Let A be a finite dimensional algebra with n simple modules. In this talk, I will explain how to construct an abstract simplical complex ∆(A) whose maximal faces are in bijection with the isomorphism classes of basic support τ-tilting modules. Using established results, we will describe the combinatorial properties of ∆(A). In particular, if ∆(A) is finite, then the geometric realization of ∆(A) is homeomorphic to a (n − 1)- dimensional sphere. This work is part of a joint project with Osamu Iyama. Takuma Aihara (Nagoya University)................ November 11th (Mon), 11:40–12:30 Tilting mutation theory In representation theory of algebras, the notion of mutation plays crucial roles. Three kinds of mu- tation are well-known: silting mutation [AI], quiver-mutation [FZ] and cluster tilting mutation [BMRRT, IY]. Moreover these are closely related with each other, that is, in hereditary case there are one-to-one correspondences compatible with mutation, among silting objects, clusters and cluster tilting objects [AIR]. In this talk, we focus on silting mutation, which is a generalization of tilting mutation studied first by Bernstein–Gelfand–Ponomarev and Auslander–Platzeck–Reiten, it was investigated also by Riedtmann– Schofield and Happel–Unger. We note that silting mutation is always possible, but tilting mutation is not. We mainly discuss silting quivers to observe the behavior of silting mutation. A finite dimensional algebra over a field is said to be silting-connected if its silting quiver is connected. Now we raise the following question: Question. When is a finite dimensional algebra over a field silting-connected? Our goal of this talk is to give a partial answer of this question. Now we state the main theorem. Main Theorem. Any of the following algebras is silting-connected: (1) local algebras [AI]; 5 (2) hereditary algebras [AI]; (3) representation-finite symmetric algebras [A]; (4) Brauer graph algebras of type odd [AAC]. In particular, I will explain the strategy of the proof of (3) and (4) in this theorem. References [AAC] T. Adachi; T. Aihara; A. Chan, On tilting complexes of Brauer graph algebras I: combina- torics arising from two-term tilting complexes and mutation quivers of tilting complexes. in preparation. [AIR] T. Adachi; O. Iyama; I. Reiten, τ-tilting theory. to appear in Compos. Math. [A] T. Aihara, Tilting-connected symmetric algebras. Algebr. Represent. Theory 16 (2013), no. 3, 873–894. [AI] T. Aihara; O. Iyama, Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668. [BMRRT] A. B. Buan; R. Marsh; M. Reineke; I. Reiten; G. Todorov, Tilting theory and cluster combi- natorics. Adv. Math. 204 (2006), no. 2, 572–618. [FZ] S. Fomin; A. Zelevinsky, Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. [IY] O. Iyama; Y. Yoshino, Mutation in triangulated categories and rigid Cohen–Macaulay mod- ules. Invent. Math. 172 (2008), no. 1, 117–168. Ryoichi Kase (Osaka University).................... November 11th (Mon), 14:00–14:30 On the poset of pre-projective tilting modules over path algebras To classify tilting modules is an important problem of the representation theory of finite dimensional algebras. Theory of tilting-mutation introduced by Riedtmann and Schofield is one of the approach to this problem. Riedtmann and Schofield defined the tilting quiver related with tilting-mutation. Happel and Unger defined the partial order on the set of (isomorphic classes of) basic tilting modules and showed that tilting quiver is coincided with Hasse quiver of this poset. In this talk we consider the poset of pre-projective tilting modules over path algebras.