组织者Organizer(YMSC/BIMSA):邓邦明，邱宇，肖杰，徐帆，张贺春，周宇，朱彬

日程安排 Schedule:

摘要Abstracts：

毕映锦 (北京师范大学) On the cohomology of quiver Grassmannians for acyclic quivers

Quiver Grassmannians stem from the categorification of cluster algebras. This notion is related to the cluster monomials in (quantum) cluster algebras and rigid objects in various cluster categories. In this talk, I will show some connections between the cohomology of quiver Grassmannians and the dual canonical bases of quantum groups. This work is closely related to the Positivity Conjecture and Quantization Conjecture in (quantum) cluster algebras. We give an interpretation of the cohomology of quiver Grassmannians in terms of dual canonical bases. As a corollary, we give another proof of Positivity Conjecture for acyclic cluster algebras.

陈慧 (南京医科大学) The realization of a Lie algebra of type BC via Hall   algebras

We define a Lie algebra of type BC by giving its generators and relations. For the gentle algebra Λ(n− 1, 1, 1), we define an Euler form and obtain a similar result as Gabriel’s Theorem, which gives a form-preserving correspondence between the indecomposable representations of Λ(n − 1, 1, 1) and the positive part of the root system of type BC. Further, we establish an isomorphism between the Ringel–Hall Lie algebra of Λ(n − 1, 1, 1) and the positive part of the type BC Lie algebra. This is a joint work with Dong Yang.

费主睿 (上海交通大学) Tropical F-polynomials and General  Presentations

We developed the theory of general presentations for any finite-dimensional ba- sic algebra in parallel with that of general representations of quivers (due to Kac and Schofield). For example we proved an analogue of Kac’s canonical decomposition, and the special case for rigid presentations implies most results in the tau-tilting theory.

We introduce the tropical F -polynomial fM of a quiver representation M , and explain its interplay with the general presentation. We provide a tropical formula to compute the dimensions of generic Homs and Exts. As a consequence, we give a presentation of the Newton polytope N (M ) of M .  We propose an algorithm to determine the generic Newton polytopes, and show it works for path algebras. In the end, we mention some applications in cluster algebras.

李利平 (湖南师范大学) Representations over diagrams of abelian categories: a preliminary introduction

Diagrams of categories, which are functor categories from a fixed index category to 2-categories of categories equipped with extra structure, are widely applied in sheaf theory of manifolds and algebraic varieties, topos theory, and representation theory of categories. In this talk I will describe some preliminary knowledge on diagrams of abelian categories and representations (modules) over diagrams, and introduce several functorial constructions on these module categories over diagrams. These results will lay the foundation for our further investigation on cotorsion theory in functor categories.

This is a joint work with Zhenxing Di, and Nina Yu.

卢明 (四川大学) Semi-derived Ringel–Hall algebras and quantum loop algebras

Semi-derived Ringel–Hall algebras for arbitrary hereditary abelian categories are introduced by L. Peng and the speaker. In this talk, we use the semi-derived Ringel–Hall algebra of the category of coherent sheaves on a weighted projective line to realize the quantum loop algebra in its Drinfeld’s new presentation. The semi-derived Ringel–Hall algebra of the quiver algebra of affine type A was known earlier to realize the same algebra in its Serre presentation. The Geigle–Lenzing’s derived equivalence induces an isomorphism of these two semi-derived Ringel–Hall algebras, explaining Drinfeld–Beck’s isomorphism of the quantum group of affine type A under the two presentations. This is joint work with Shiquan Ruan.

万金奎 (北京理工大学) Zelevinsky involution and degenerate affine Hecke–Clifford algebras

We define an analog of Zelevinsky involution on degenerate affine Hecke– Clif- ford algebras and give an algorithm to compute its action on irreducible completely splittable representations over an algebraically closed field of characteristic not equal to 2 via placed skew shifted Young diagrams. This is a joint work with Zhekun He.

吴笑林 (Universit ́e de Paris, 华东师范大学) Relative cluster categories and Higgs categories

Cluster categories were introduced in 2006 by Buan–Marsh–Reineke–Reiten– Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot and Plamondon to arbitrary cluster algebras associated with quivers (2009 and 2011). A higher dimensional generalization is due to Guo (2011). Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells,.... The work of Geiss–Leclerc–Schro¨er often yields Frobenius exact categories which allow to categorify such cluster algebras. In this talk, we will present the construction of the Higgs category (generalizing GLS’ Frobenius categories E) and of the relative cluster category (generalizing the derived category of E). The Higgs category is no longer exact but still extriangulated in the sense of Nakaoka–Palu (2019).

叶郁 (中国科技大学) A generalized Knorrer’s periodicity theorem

In this talk, I will discuss the graded isolated singularity obtained from ten- sor products of quadric hypersurfaces, and show a generalized version of Knorrer’s periodicity theorem. This is based on a joint work with Jiwei He and Xinchao Ma.

张笑婷 (首都师范大学) Geometric classification of spaces of totally stable stability conditions

We classify non-empty space ToStD of totally stable stability conditions on a triangulated category D, where D must be D(Q) for some Dynkin quiver Q. To do so, we construct geometric model for D(Q) and realize ToStD(Q) as moduli space of certain stable hQ-gon for Coxeter number hQ. In the talk, I will focus on the type D case. This is a joint work with Yu Qiu.

周国 (华东师范大学) The homotopy theory of differential algebras of arbitrary weights

We study the homotopy theory of differential algebras of arbitrary weights. We determine the minimal model and the Koszul dual homotopy cooperad of theoperad governing differential algebras of arbitrary weights. We propose a definition of homotopy differential algebras and we construct the deformation complex for differential algebras of arbitrary weights as well as the L∞-algebra structure on it. Our result recovers previous results of Loday in weight zero case and thus completely solves a question imposed by Loday. This talk is based on a joint work with Jun Chen, Li Guo and Kai Wang.

周远扬 (华中师范大学) Inertial blocks and coefficient extensions

Inertial blocks over big enough fields are due to Puig. I generalize these blocks to arbitrary fields. Such a generalization has its own interest. In this talk, I will give a characterization of the algebraic structure of inertial blocks over arbitrary fields and investigate the relationship between inertial blocks over arbitrary fields and coefficient extensions.