威尼斯人

LieWorkshop（Soochow◎online）

1.报告题目: Quantized GIM algebras      

报告人:  郜云教授(York University)      

报告时间: 2020年8月1日, 8: 00--8: 50      

报告地点：腾讯会议(ID: 401669273  密码: 202008)      

报告摘要: A GIM Lie algebra of $n\times n$ generalized intersection matrix is a fixed point subalgebra of Kac-Moody algebra of  $2n\times2n$ GCM. We will show a quantum version of this.      

2.报告题目: Lie algebras over quantum torus and twisted multiloop Lie algebras      

报告人:  常智华教授(华南理工大学)      

报告时间: 2020年8月1日, 9: 00--9: 50      

报告地点：腾讯会议(ID: 401669273  密码: 202008)      

报告摘要: According to S. Berman, Y. Gao and Y. Krylyuk's remarkable work, the centerless core of an extended affine Lie algebra of type $A_l$ is isomorphic to the Lie algebra $\mathfrak{sl}_{l+1}(\mathbb{C}_Q)$ coordinated by a quantum torus $\mathbb{C}_Q$. We consider the two-variable case $\mathbb{C}_q=\mathbb{C}_q[x^{\pm1},y^{\pm1}]$. If $q$ is a root of unity, the Lie algebra $\mathfrak{sl}_{l+1}(\mathbb{C}_q)$ is isomorphic to a twisted loop Lie algebra. I will explain the reasoning to this fact in my talk.      

3.报告题目: Degenerate quantum general linear groups      

报告人:  成锦博士(山东师范大学)      

报告时间: 2020年8月1日, 10: 00--10: 50      

报告地点：腾讯会议(ID: 401669273  密码: 202008)      

报告摘要: We construct a degenerate version of the Drinfeld-Jimbo quantum group of $\mathfrak{gl}_{m+n}$, which is a new Hopf algebra over $\mathbb{C}(q)$ with a triangular decomposition and has a rich representation theory.  We study its structures and apply results to develop highest weight representations. A classification of the finite dimensional simple modules is obtained in terms of highest weights, which are essentially characterised by $m+n-2$ nonnegative integers and an arbitrary nonzero scalar.      

The degenerate quantum group has an m+n dimensional irreducible representation analogous to the natural representation of $\mathfrak{gl}_{m+n}$. It admits an R-matrix that satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite. We construct a topological invariant of knots from this R-matrix, which reproduces the celebrated HOMFLY polynomial.      

4.报告题目: Dualities in category O over infinite dimensional Lie algebras      

报告人:  刘兴鹏博士(中国科学技术大学)      

报告时间: 2020年8月1日, 11: 00--11: 50      

报告地点：腾讯会议(ID: 401669273  密码: 202008)      

报告摘要: It is different from the case of finite dimensional reductive Lie algebra, the       

category O of infinite dimensional Kac-Moody algebra has blocks with two different       

types, positive level and negative level. They are related to two different types of geometry.

 However we can use the Ringel duality to relate those two different types. In this talk, we 

firstly review some basic notions on 

category O over Kac-Moody Lie algebras, then talk       

about a result given by Soergel, i.e., a version of Ringel duality on category O. Finally, we      

extend this Ringel duality to a duality between finite truncations of parabolic category over Kac-Moody Lie algebras.      

5.报告题目: Representations of the mirror-twisted Heisenberg-Virasoro algebra      

报告人:  赵开明教授(Wilfrid Laurier University)      

报告时间: 2020年8月2日, 8: 00--8: 50      

报告地点：腾讯会议(ID: 561141212  密码: 202008)      

报告摘要: I will give all irreducible Harish-Chandra modules over the mirror-twisted Heisenberg-Virasoro algebra ${\mathfrak{D}}$, all unitary irreducible highest weight modules, and   all irreducible restricted modules over ${\mathfrak{D}}$ of level zero. This talk is based on a joint paper with D. Liu, Y. Pei, L. Xia.      

6.报告题目: Tensor products of $\C[K^{\pm1}]$-free modules for $U_q(\sl_2)$      

报告人:  郭向前教授(郑州大学)      

报告时间: 2020年8月2日, 9: 00--9: 50      

报告地点：腾讯会议(ID: 561141212  密码: 202008)      

报告摘要: We study the $\C[K^{\pm1}]$-free modules for $U_q(\sl_2)$ and their tensor products with finite-dimensional modules. We obtain some direct sum decomposition formulas for the tensor products, which are similar to the Clebsch-Gordan formula. This is joint work with Yan-an Cai, Hongjia Chen, Yao Ma and Mianmian Zhu.      

7.报告题目: Using Weyl algebras in the representations of Lie algebras      

报告人:  刘根强教授(河南大学)      

报告时间: 2020年8月2日, 10: 00--10: 50      

报告地点：腾讯会议(ID: 561141212  密码: 202008)      

报告摘要: In this talk, I will recall the representation theory of the Weyl algebras, and talk about using Weyl modules in the representation theory of several Lie algebras: finite dimensional simple Lie algebras, non-semisimple Lie algebras and so on.      

8.报告题目: A class of non-weight modules of affine Kac-Moody algebras      

报告人:  谭海军博士(东北师范大学)      

报告时间: 2020年8月2日, 11: 00--11: 50      

报告地点：腾讯会议(ID: 561141212  密码: 202008)      

报告摘要: Let  $\tilde{\mathfrak{h}}$ be  a Cartan subalgebra of an affine Kac-Moody algebra without containing  the center.  We classify the module structures on $U(\tilde{\mathfrak{h}})$ for all affine Kac-Moody algebras which are free $U(\tilde{\mathfrak{h}})$-modules of rank 1. Actaully these modules exist only for   type $A_l^{(1)}(l\ge 1)$ or  $C_l^{(1)}(l\ge 2)$ affine Kac-Moody algebras. Most of these modules are irreducible non-weight modules. This is a joint work with Yan-an Cai and Kaiming Zhao.      

9.报告题目: Presentations for toroidal Lie algebras and twisted loop algebras      

报告人:  谭绍滨教授(厦门大学)      

报告时间: 2020年8月3日, 8: 00--8: 50      

报告地点：腾讯会议(ID: 868752068   密码: 202008)      

报告摘要: In this talk, we start with the presentations for Kac-Moody algebras, and then recall certain presentations for the toroidal Lie algebras. Finally we provide the Moody-Rao-Yokohama presentation and Drinfeld type presentation for the twisted loop algebras, which are related to the quantization of the twisted toroidal Lie algebras.      

10.报告题目: Parafermion Vertex Operator Algebra      

报告人:  王清教授(厦门大学)      

报告时间: 2020年8月3日, 9: 00--9: 50      

报告地点：腾讯会议(ID: 868752068   密码: 202008)      

报告摘要: Parafermion vertex operator vertex algebra (VOA) is a coset VOA related to an affine VOA. In this talk, I will present our resent results on the rational parafermion VOA and the parafermion VOA beyond rational cases. These results are based on the joint work with Cuipo Jiang, Drazen Adamovic and Antun Milas.      

11.报告题目: Classification of finite-dimensional Hopf algebras over dual Radford algebras      

报告人:  胡乃红教授(华东师范大学)      

报告时间: 2020年8月3日, 10: 00--10: 50      

报告地点：腾讯会议(ID: 868752068   密码: 202008)      

报告摘要:  We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property. This a joint work with Dr. Xiongrong Chuan      

12.报告题目: Simple weight modules on the differential operator superalgebra      

报告人:  刘东教授(湖州师范学院)      

报告时间: 2020年8月3日, 11: 00--11: 50      

报告地点：腾讯会议(ID: 868752068   密码: 202008)      

报告摘要: In this talk, we introduce some progresses on simple weight modules with finite dimensional weight spaces over the differential operator superalgebra.      

13.报告题目: R-matrix presentations of quantum affine algebras of classical type      

报告人:  景乃桓教授(North Carolina State University)      

报告时间: 2020年8月4日, 8: 00--8: 50      

报告地点：腾讯会议(ID: 757242442  密码: 202008)      

报告摘要: Drinfeld's new realization of quantum affine algebras and Yangians are fundamental in representation theory of the quantum algebras. Quantum groups have its origin from the quantum inverse scattering method lead by the Leningrad school, who also has the R-matrix formulation known as the RTT method.      

In 1994 Ding and Frenkel showed the identification of the two formulations for type A quantum affine algebras and Brundon-Kleshchev in 2004 proved the isomorphism for the Yangians in type A. Two years ago Liu, Molev and the speaker have solved the problem for Yangians in other classical types. In this talk I will report our work on the identification for quantum affine algebras of classical types.      

14.报告题目: Product and complex structures on hom-Lie superalgebras and their applications      

报告人:  陈良云教授(东北师范大学)      

报告时间: 2020年8月4日, 9: 00--9: 50      

报告地点：腾讯会议(ID: 757242442  密码: 202008)      

报告摘要:  In this talk, we mainly introduce  product and complex structures  on hom-Lie superalgebras. We  give a necessary and sufficient condition for the existence of product structures on hom-Lie superalgebras and obtain some properties of para-K$\ddot{\rm {a}}$hler hom-Lie superalgebras, which is equipped with a product structure on pseudo-Riemann hom-Lie superalgebra. Further, we introduce the phase space of a hom-Lie superalgebra and show that para-K\{a}hler hom-Lie superalgebra and phase space can be obtained from each other through the hom-left symmetric product. Similarly,   we obtain some properties  of complex structures on hom-Lie superalgebras.  This talk is a report on joint work with  Ying Hou.      

15. 报告题目: Some advances in the study of irreducible modules of $\mathfrak{sl}(mp)$ in characteristic $p$      

报告人:  舒斌教授(华东师范大学)      

报告时间: 2020年8月4日, 10: 00--10: 50      

报告地点：腾讯会议(ID: 757242442  密码: 202008)      

报告摘要: I will introduce recent advances in the study of irreducible modules of $\mathfrak{sl}(n)$ in characteristic $p$ with $p$ dividing $n$, which are owing to works by Lusztig 2011, Wen Xin 2016, Jantzen 2017, Liu Bin-Shu Bin 2019.      

Previously, Friedlander-Parshall 1988, and Jantzen 1999 described all irreducible modules of $\mathfrak{sl}(n)$ in characteristic $p$ with $p$ not dividing $n$ for regular nilpotent representations and  subregular nilpotent representations, respectively. However, the situation becomes complicated when $p$ divides $n$, and the question of describing irreducible modules has been open for a long time.  In this talk, we will tell you the answer to the question for regular nilpotent representations and subregular nilpotent representations.      

16.报告题目: Whittaker modules for the planar Galilean conformal algebra and its central extension      

报告人:  姚裕丰教授(上海海事大学)      

报告时间: 2020年8月4日, 11: 00--11: 50      

报告地点：腾讯会议(ID: 757242442  密码: 202008)      

报告摘要: Let $\mathcal{G}$ be the planar Galilean conformal algebra and $\widetilde{\mathcal{G}}$ be its universal central extension. In this talk, we study universal and generic Whittaker $\mathcal{G}$-modules (resp. $\widetilde{\mathcal{G}}$-modules) of type $\phi$. We classify the isomorphism classes of universal and generic Whittaker modules. Moreover, we show that a generic Whittaker modules of type $\phi$ is irreducible if and only if $\phi$ is nonsingular. For the nonsingular case, we completely determine the Whittaker vectors in universal and generic Whittaker modules. For the singular case, we concretely construct some Whittaker vectors, which generate proper submodules of generic Whittaker modules. This is a joint work with Qiufan Chen and Hengyun Yang.      

17.报告题目: Deformations and cohomologies of relative Rota-Baxter Lie algebras      

报告人:  生云鹤教授(吉林大学)      

报告时间: 2020年8月5日, 8: 00--8: 50      

报告地点：腾讯会议(ID: 712317791  密码: 202008)      

报告摘要: First we determine the dg Lie algebra that controls deformations of a relative Rota-Baxter operator and develop the cohomology thoery of a relative Rota-Baxter operator. Then we determine the $L_\infty$-algebra that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the  cohomology  of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. In particular, we introduce the cohomologies of Rota-Baxter Lie algebras and triangular Lie bialgebras using this general framework.      

18.报告题目: Serre-Lusztig relations and braid group symmetries of i-quantum groups      

报告人:  卢明博士(四川大学)      

报告时间: 2020年8月5日, 9: 00--9: 50      

报告地点：腾讯会议(ID: 712317791  密码: 202008)      

报告摘要: A quantum symmetric pair consists of a quantum group and its coideal subalgebra (called an i-quantum group). In this talk, we present higher order Serre relations (called Serre-Lusztig relations) of quasi-split i-quantum groups, and give applications to braid group symmetries of i-quantum groups. This is joint work with Xinhong Chen and Weiqiang Wang.      

19.报告题目: Recent Progress on the Geometry of Affine Nash Groups      

报告人:  包益欣博士(哈尔滨工业大学(深圳))      

报告时间: 2020年8月5日, 10: 00--10: 50      

报告地点：腾讯会议(ID: 712317791  密码: 202008)      

报告摘要: Nash manifolds and Nash groups play an important role in representation theory of real algebraic groups for several reasons.      

First, they naturally appear in the study of real algebraic groups and their actions on real algebraic manifolds, for example, identity components and covering groups of real algebraic groups are Nash groups.      

Another reason is that it is possible to define Schwartz functions on Nash manifolds. These reasons stimulated recently the research of Nash groups.      

In particular, almost linear Nash groups have been systemitically studied by B. Sun. In this lecture, we introduce some recently obtained classification results on affine Nash groups.      

Our approach is a direct application of algebraization method introduced by B. Sun and Y. Fang.      

20.报告题目: Jantzen coefficients and radical filtrations for generalized Verma modules      

报告人:  肖维博士(深圳大学)      

报告时间: 2020年8月5日, 11: 00--11: 50      

报告地点：腾讯会议(ID: 712317791  密码: 202008)      

报告摘要: We give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) integral blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the usual BGG category O. Combined with Jantzen coefficients, we determine the radical filtrations for all basic generalized Verma modules. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of generalized Verma modules in any (possibly singular) integral blocks of the parabolic BGG category are also given. This is a joint work with Prof. Hu Jun.

组织者：吕仁才，白占强，蔡延安