#### 第二十届全国代数表示论研讨会

**June 21 - June 26, 2018**

**Nanjing, Jiangsu, China**

## Welcome to our Conference!

**会议专题**

**Topic 1: **Frobenius-Schur indicators and modular tensor categories

**Plenary speakers **: 黄华林、刘公祥、杨毓萍和叶郁

**Abstract **: Frobenius-Schur indicators appeared in the group representation theory at first. Then it was generalized to semisimple Hopf algebras and more generally to pivotal tensor categories. Modular tensor categories came from many mathematical branches such as conformal field theory, VOA etc. Both of them are active research objectives recently. We want to provide a short introduction to them and in particular we want to use the indicator theory as a basic tool to study modular tensor categories.

**Content **: **共五讲 **

1. Groups, tensor categories, indicator ( 一讲 )

2. Indicators for pivotal categories ( 两讲 )

3. Equivariant indicator and some applications ( 两讲 )

**References ****： **

1) Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor Tensor

categories. Mathematical Surveys and Monographs, 205.

2) Sommerhäuser, Yorck; Zhu, Yongchang Hopf algebras and congruence

subgroups. Mem. Amer. Math. Soc. 219 (2012), no. 1028, vi+134 pp.

3) Kashina, Yevgenia; Sommerhäuser, Yorck; Zhu, Yongchang On higher

Frobenius-Schur indicators. Mem. Amer. Math. Soc. 181 (2006), no. 855,

viii+65 pp.

4) Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric C.; Wang, Zhenghan

Rank-finiteness for modular categories. J. Amer. Math. Soc. 29 (2016), no. 3,

857–881.

5) Ng, Siu-Hung; Schauenburg, Peter Congruence subgroups and generalized

Frobenius-Schur indicators. Comm. Math. Phys. 300 (2010), no. 1, 1–46.

** **

**Topic 2: ****Gentle algebras **

**Plenary speakers **: 1 . 陈学庆，威斯康辛大学

2. 张 杰，北京理工大学

3. 章 超， 贵州大学

**Abstract **: We can find gentle algebras in many places in mathematics. The study of gentle algebras was initiated by Assem and Skowroński in order to study iterated tilted algebras of type A and became an active area of research in representation theory in the recent years. Gentle algebras form an interesting class of examples because of the simple combinatorial definition and because of many connections to other classes of algebras such as string algebras, biserial algebras, special biserial algebras, tame algebras, Gorenstein algebras, Koszul algebras, Brauer graph algebras, and surface algebras .

We will give a brief introduction on gentle algebras by definitions and examples. The first goal of our talks is to understand the classification of the indecomposable modules (up to isomorphism) and the Auslander-Reiten theory over a gentle algebra.

The second goal is to learn derived categories of gentle algebras. The indecomposable objects in derived categories of gentle algebras and the morphisms between them have been explicitly described by Bekkert-Merklen [BM] and Arnesen - Laking - Pauksztello [ALP] respectively. Avella-Alaminos and Gei ss [ AG ] provided combinatorial derived invariants for gentle algebras . As a result, many explicit computations can be carried out , and gentle algebras can be seen as a useful test-subject for many more general conjectures related to derived categories. In this part, we will give a brief introduction on the indecomposable objects, the morphisms between them, and the Avella-Alaminos - Gei ss invariant in derived categories of gentle algebras with some concrete examples. The third goal of our talks is to understand how gentle algebras arise from surfaces. We give a geometric model for module category of a gentle algebra and its bounded derived category.

**Content: ****共三讲 **

(1) Basics of gentle algebras

(2) Derived categories of gentle algebras

(3) Gentle algebras arising from surfaces

**References **

[ABCP] I. Assem, T. Brustle, G. Charbonneau-Jodoin, P.-G. Plamondon: Gentle algebras arising from surface triangulations, Algebra. Number Theory 4 (2010), no. 2, 201-229.

[BR] M. C. R. Butler, C. M. Ringel: Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145-179.

[ALP] K. K. Arnesen, R. Laking, D. Pauksztello: Morphisms between indecomposable objects in the derived category of a gentle algebra , J. Algebra 467 (2016), 1 - 46.

[AG] D. Avella-Alaminos, C. Gei ss: Combinatorial derived invariants for gentle algebras , J. Pure Appl. Algebra 212 (2008), no.1, 228-243.

[BM] V. Bekkert, H. A. Merklen: Indecomposables in derived categories of gentle algebras , Algebr. Represent. Theory 6 (2003), no. 3, 285-302.

[OPS] S. Opper , P. Plamondon and S.Schroll : A geometric model for the derived category of gentle algebras, https://arxiv.org/abs/1801.09659

[BS] K. Baur , R. Simoes : A geometric model for the module category of a gentle algebra, https://arxiv.org/abs/1803.05802

** **

**Topic 3: ****Cluster and Calabi-Yau categories from surfaces **

**Plenary speakers **: 1 . 邱 宇，清华大学

2. 周 宇，清华大学

**Abstract **: In the series of lectures, we will introduce two classes of Calabi-Yau categories from Riemann surfaces with marked points: cluster categories and 3-Calabi-Yau categories. We interpret the stuff in categories via geometric/topological terms: e.g. objects as curves, Ext^1 group as intersection numbers of curves, Auslander-Reiten translation as (tagged) rotations, cluster-tilting objects as (tagged) triangulations, spherical twist groups as braid twists groups. As applications, we show the connectedness of the cluster exchange graphs and the simply connectedness of the space of stability conditions.

**Content: ****共五讲 **

1. Quivers with potential from triangulated surfaces

2. Cluster categories of marked surfaces I

3. Cluster categories of marked surfaces II

4. Decorated marked surfaces: Spherical objects via string model

5. Decorated marked surfaces: Applications

**References **

[BZ] Brüstle, Thomas; Zhang, Jie On the cluster category of a marked surface without punctures. Algebra Number Theory 5 (2011), no. 4, 529–566.

[ZZZ] Zhang, Jie; Zhou, Yu; Zhu, Bin Cotorsion pairs in the cluster category of a marked surface. J. Algebra 391 (2013), 209–226.

[QZ] Qiu, Yu; Zhou, Yu Cluster categories for marked surfaces: punctured case. Compos. Math. 153 (2017), no. 9, 1779–1819.

[Q] Qiu, Yu Decorated marked surfaces: spherical twists versus braid twists, Math. Ann. 365(2016), pp 595-633.

[QZ] Qiu, Yu; Zhou, Yu Decorated marked surfaces II: Intersection numbers and dimensions of Homs , arXiv:1411.4003.

** **

**Topic 4: ****Quivers with relations for symmetrizable Cartan matrices **

**Plenary speakers **: 1. 盛洁 , 中国农业大学

2. 卢明 , 四川大学

3. 付昌建 , 四川大学

4. 庄晓，清华大学

**Abstract **: This series of talks gives a quick introduction to GLS' work on a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. Their motivation was to realize the enveloping algebra of the positive part of a semisimple complex Lie algebra for the non-simply laced types. The goal was finally achieved in [3] via a convolution algebra of constructible functions on module varieties of aforementioned Iwanaga-Gorenstein algebras. They also obtain generalizations of classical results on finite dimensional hereditary algebras and associated preprojective algebras. Moreover, they generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case.

**Content: ****共四讲 **

Talk 1: We introduce the definitions of the Iwanaga-Gorenstein algebras and the corresponding generalized preprojective algebras associated to quivers with relations for symmetrizable Cartan matrices. We also emphasize an analogy to the representation theory of modulated graphs.

Talk 2: We introduce the generalization of classical results of Gabriel, Dlab-Ringel and Gelfand-Ponomarev.

Talk 3: We introduce how to realize the enveloping algebra of the positive part of a semisimple complex Lie algebra as a convolution algebra of constructible functions on module varieties of some Iwanaga-Gorenstein algebras of dimension 1.

Talk 4: We introduce the Caldero-Chapoton map for cluster algebras of finite type via locally free modules of certain Iwanaga-Gorenstein algebras( {\it cf.} Talk 1&2). And we compute an example in details.

**References **

[1] C. Geiß, B. Leclerc, J. Schrӧer, Quivers with relations for symmetrizable Cartan matrices I: Foundations. Invent. Math. 209 (2017), 61-158.

[2] C. Geiß, B. Leclerc, J. Schrӧer, Quivers with relations for symmetrizable Cartan matrices II: Change of symmetrizer. Int. Math. Res. Not. (2017), doi

10.1093/imrn/rnw299.

[3] C. Geiß, B. Leclerc, J. Schrӧer, Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras. Represent. Theory 20 (2016), 375-413.

[4] C. Geiß, B. Leclerc, J. Schrӧer, Quivers with relations for symmetrizable Cartan matrices IV: Crystal graphs and semicanonical functions. Preprint (2017), 49 pp., arXiv:1702.07570.

[5] C. Geiß, B. Leclerc, J. Schrӧer, Quivers with relations for symmetrizable Cartan matrices V. Caldero-Chapoton formula. Preprint (2017), 24 pp., arXiv:1704.06438.