We study the $\mathbb{C}[K^{\pm 1}]$-free modules for the quantum group $U_q(\mathfrak{sl}_2)$ and their tensor products with finite-dimensional simple modules. We obtain some direct sum decomposition formulas for the tensor products, which are similar to the classical Clebsch-Gordan formulas. This talk is based on joint works with Yan-an Cai, Xiangqian Guo, Yao Ma and Mianmian Zhu.

Keller's conjecture establishes a link between the singular Hochschild cohomology and the Hochschild cohomology of the dg singularity category on the $B_\infty$-level. Using Leavitt path algebras, we confirm Keller's conjecture for any radical-square-zero algebra: there is an isomorphism in the homotopy category of $B_\infty$-algebras between the Hochschild cochain complex of the dg singularity category and the singular Hochschild cochain complex of the algebra. We prove that Keller's conjecture is invariant under one-point (co)extensions and singular equivalences with levels. This is joint with Huanhuan Li and Zhengfang Wang.

We make a comparison between Lusztig's symmetries and the isomorphisms defined by Sevenhant and Van den Bergh via combining BGP-reflection isomorphisms and Fourier transforms on the double Ringel-Hall algebra of a quiver. We also give an example of a quiver for which Sevenhant-Van den Bergh's isomorphisms do not satisfy the braid relations on the whole double Ringel-Hall algebra. This talk is based on joint work with Chenyang Ma.

介绍雅可比猜想、多项式导子的像、Mathiue子空间的联系. 证明一类三元多项式导子的像是Mathieu子空间. (与田海峰, Wenhua Zhao合作).

In this talk, I will introduce quantum Borcherds-Bozec algebras and their recent developments, including classical limit , representations and geometric realization etc.

In this talk, we want to present a combinatorial identity which is closely related to the multi-dimensional integral $\gamma_{m}$ in the study of divisor functions. As an application, we determine the finite dual of the group algebra of infinite dihedral group.

We establish a structure theorem for connected graded Hopf algebras over a field of characteristic 0 by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some excellent conditions. The approach to the structure theorem is constructive based on the combinatorial properties of Lyndon words and the standard bracketing on words. As a consequence of the structure theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic 0 are all iterated Hopf Ore extensions of the base field.

1932年，Prüfer引入了一类整环，即每个非零的有限生成理想是可逆理想的整环，等价地，$\operatorname{w.gl.dim}(R)\leq 1$。 1936年，Krull 将其命名为 Prüfer 整环。由于 Prüfer 整环有许多好的环结构理论和同调性质，1970年，Griffin 将这一概念建立一般交换环上，引入了 Prüfer 环的概念。交换环 $R$ 被称为 Prüfer 环，是指每个有限生成正则理想是可逆理想。 2006年 Bazzoni 和 Glaz 对 Prüfer 环的研究成果进行了收集和整理，但这些结果都是从乘法理想理论研究理论得到的。 在 2014年 Cahen-Fontana-Frisch-Glaz 提出了如下公开问题

Problem 1a: Let R be a Prüfer ring. Is $\operatorname{fPD}(R)\leq 1$ ?

Problem 1b: Let R be a total ring of quotients. Is $\operatorname{fPD}(R)=0$ ?

我们利用有限生成半正则理想的乘法系，建立了一套与 $\omega$-模类似的 Lucas 模系统，再构造反例，对以上两个公开问题给出了否定的回答。

We present natural connections among trigonometric Lie algebras, affine Lie algebras, and vertex algebras. More specifically, we prove that restricted modules for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebra. Furthermore, we prove that every quasi-finite unitary highest weight irreducible module of type A trigonometric Lie algebra gives rise to an irreducible equivariant quasi module for the simple affine vertex algebra. This is a joint work with Haisheng Li and Shaobin Tan.