Beijing Institute of Mathematical Sciences and Applications

D. Happel introduced the root category as a two-periodic orbit triangulated category of the derived category of Dynkin quiver. The Gabriel Theorem can be stated with the Auslander-Reiten quiver of , not only for the positive roots but also the whole root system . We introduce here a process to build up semi-simple Lie algebras and Chevalley groups via Hall algebra approach. The construction can be applied to a realization of compact real form and maximal compact subgroups from the root category , and obtain the Peter-Weyl Theorem and the Plancherel Theorem for compact groups. This is a joint work with Buyan Li.

In this talk, I will explain how Auslander--Reiten theory plays a central role in the categorical understanding of cluster algebras and in the construction of their bases. I will review some classical results as well as recent developments.

We will present new insights in alterfold theory to study modular tensor categories. We provide streamlined quick proofs and broad generalizations of a wide range of results on modular invariance of modular tensor categories. It is based on joint work with M. Shuang, Y. Wang and J. Wu, arXiv:2412.12702.

In this talk, we study the relations between groups related to cluster automorphism groups which are defined by Assem, Schiffler and Shamchenko in \cite{ASS}. We establish the relationship among (strict) direct cluster automorphism groups and those groups consisting of periodicities of respectively labeled seeds and exchange matrices in the language of short exact sequences. As an application, we characterize automorphism-finite cluster algebras in the cases with bipartite seeds or finite mutation type. Finally, we study the relation between the groups Aut(A) and AutM_n(S) and give the negative answer via counter-examples to King and Pressland's a problem. This is a joint work with Siyang Liu.

The representation theory of finite-dimensional algebras is characterized by numerous interconnected concepts, such as tilting modules, torsion pairs, semibricks, and wide subcategories. In this talk, we discuss the introduction of these notions into a higher setting through the framework of extended module categories. We will demonstrate how the classical relationships between these structures are maintained in the higher setting while also revealing the novel changes and new phenomena that arise.

We study the irreducible representations of the quantum groups at roots of unity, in the setting of affine Hopf algebras admitting a large central Hopf subalgebra. Such kind of Hopf algebras H can be endowed with a Cayley-Hamilton Hopf algebra structure in the sense of De Concini-Procesi-Reshetikhin-Rosso. By a standard fact, the set of irreducible representations of H is the disjoint union of the irreducible representations over its fiber algebras. The irreducible representations of H have strong relation with the (lowest) discriminant ideal of H. The talk is based on a joint work with Mi and Yakimov, and some recent work with Huang, Mi and Qi.

We investigate a central question in the representation theory of algebras: Under what conditions does a self-orthogonal generator-cogenerator over an algebra become projective? This is closely related to the Nakayama conjecture. In this talk, we focus on modules whose endomorphism algebras are virtually Gorenstein. In particular, we show that if an algebra has infinite dominant dimension and satisfies certain Ext-vanishing conditions, then being virtually Gorenstein forces the algebra to be self-injective. This talk reports a joint work with Changchang Xi.

The fact that every finite-dimensional algebra over a field is isomorphic to the centralizer algebra of two matrices, suggests to study topics on finite-dimensional algebras by the ones on centralizer algebras of matrices. In this talk, we investigate the centralizer algebra of one matrix, called a centralizer matrix algebra. We present a characterization of the equivalence of singularity categories of centralizer matrix algebras in terms of matrix equivalence. Furthermore, we show that all homological conjectures hold true for centralizer matrix algebras. This talk reports a joint work with Zhenxian Chen (see arXiv:2603.20643).

We provide a classification of irreducible completely splittable representations of the affine Hecke-Clifford superalgebras $\mathcal{H}_n^{\mathrm{aff}}(q)$ when $q^2$ is a primitive $h$-th root of unity. As an application, we derive a necessary and sufficient condition for the finite Hecke-Clifford $\mathcal{H}_n(q)$ to be semisimple. In particular, we show that $\mathcal{H}_n(q)$ is semisimple if and only $h > n$ in the case $h$ is odd and $h > 2n$ in the case $h$ is even.

I would like to explain a program which aims to build relationships between affine Springer theory and representations of simple affine vertex algebras. We will discuss some explicit correspondence for admissible levels, and formulate some new conjectures about representations of simple affine vertex algebras at integer nonadmissible levels and their associated varieties. This is based on joint work with Dan Xie, Wenbin Yan and Qixian Zhao.

本报告主要关注平面三次曲线的牛顿标准型和魏尔斯特拉斯标准型。传统的处理方式主要基于牛顿的几何方法，需要借助奇点、拐点和裴祖定理等相对复杂的工具。我们利用多项式的中心理论，直接处理多项式的分离变量化简，以初等的纯代数方法导出三次曲线的牛顿标准型、魏尔斯特拉斯标准型。