丛代数及其相关问题研讨会

讲座时间 Datetime: 2022年6月20日-25日，星期一—星期六，   10:00-11:30

地点 Venue: 腾讯线上会议：946-628-9201

 主持人 Host：黄敏

报告人 Speaker:徐帆、段冰、于世卓、周雁、刘思阳

单位 Affiliation:清华大学、兰州大学、南开大学、北京大学、浙江大学、杭州师范大学

报告摘要 Abstract:

丛代数及其相关问题研讨会

 

 

2022年6月20日

下午：学术交流（地点：线上）
时间	报告题目	报告人
14:30-15:30	Derived Hall algebras and acyclic quantum cluster algebras	徐帆
15:30-16:30	Progress on monoidal categorification of (quantum) cluster algebras	段冰

 

2022年6月22日

下午：学术交流（地点：线上）
时间	报告题目	报告人
14:30-15:30	Kazhdan-Lusztig isomorphism and  Bott-Samelson atlas  on  flag varieties	于世卓

 

2022年6月25日

上午：学术交流（地点：线上）
时间	报告题目	报告人
9:30-10:30	Stokes data, the WKB approximation and periods on spectral curves.	周雁
10:30-11:30	On the cluster automorphisms of cluster algebras	刘思阳

会议报告摘要

 

Derived Hall algebras and acyclic quantum cluster algebras

（导出Hall代数和无圈量子丛代数）

徐帆

清华大学

 

In this talk, we introduce the categorification of the derived Hall algebra of an acyclic quiver and then construct a bar-invariant basis. As corollaries, the categorifications of quantum groups and quantum cluster algebras can be achieved. This talk is based on joint work with Chen Xueqing, Ding Ming and Zhang Haicheng..

（本报告会首先介绍无圈箭图的到处Hall代数的范畴化，并给出一组bar不变的基。作为推论，我们会得到量子群以及量子丛代数的范畴化。）

 

 

                                                        Progress on monoidal categorification of (quantum) cluster algebras

                               （量子丛代数的Monoidal范畴化进展）

段冰 兰州大学

D. Hernandez and B. Leclerc introduced the notion of monoidal categorification of cluster algebras, and conjectured that the monoidal category of finite-dimensional $U_q(\hat{\mathfrak{g}})$-modules provides a monoidal categorification of a cluster algebra with infinite rank. L. Bittmann developed a quantum cluster algebra approach to representations of simply-laced quantum affine algebras. We first recall the work of Hernandez-Leclerc and Bittmann, and then introduce our recent work on monoidal categorification of cluster algebras.

（D. Hernandez and B. Leclerc引入了丛代数的monoidal范畴化并猜测有限维$U_q(\hat{\mathfrak{g}})$-模范畴能够范畴化一类无线秩的丛代数。Bittmann发展了一套利用量子丛代数去逼近单边量子仿射代数的表示范畴。我们首先会解散他们的工作，接下来介绍我们在这方面的进展。）

 

           Kazhdan-Lusztig isomorphism and  Bott-Samelson atlas  on  flag varieties（Kazhdan-Lusztig同构和旗簇上的Bott-Samelson坐标卡）

                      于世卓 南开大学

 

On a flag variety, Kazhdan-Lusztig isomorphisms can be defined on shifted big cells and applied to construct the Bott-Samelson atlas. In this talk, we introduce the compatibility between the standard Poisson structures, the induced cluster structures and the Lusztig's total positivity on flag varieties based on Bott-Samelson atlas.  This is the joint work with Jiang-Hua Lu.

（在一个旗簇上，Kazhdan-Lusztig可以定义在一个大胞腔上并且可以用来构造Bott-Samelson坐标卡。本报告中，我们会引入Poisson结构的相容性，诱导出的丛代数结构以及Lusztig的全正性。）

 

 

 

 

                                                                    Stokes data, the WKB approximation and periods on spectral curves.

                               （Stokes数据，谱曲线上的WKB逼近和周期）

周雁 北京大学

 

We study, using the extended isomonodromy deformation, the WBK approximation of Stokes matrices of meromorphic linear ODE systems of Poincare rank 1 on the projective line that appear in the theory of semisimple Frobenius manifolds. We show that, via the degenerate Riemann Hilbert map, the WKB approximation of Stokes matrices recovers the Gelfand-Tsetlin integrable systems whose action variables match with period integrals on spectral curves. We also propose how the study of global aspects of the ODE systems might give us new insight into the Exact WKB analysis and the Stokes geometry of higher order ODE’s and their relationship to cluster theory and spectral networks as developed by Gaiotto-Moore-Neitzke. This is joint work with Anton Alekseev and Xiaomeng Xu.

（利用等单峰形变，我们研究一类亚纯线性ODE系统的Stokes矩阵。我们证明Stokes矩阵的WKB逼近能够还原Gelfand-Tsetlin可积系统。）

 

 

 

                                                                                     On the cluster automorphisms of cluster algebras

                                    （关于丛代数的丛自同构）

刘思阳 浙江大学/杭州师范大学

 

Cluster automorphisms were introduced by Assem, Schiffler and Shamchenko, as $Z$-automorphisms of cluster algebras mapping a cluster to another cluster and preserving the mutations. I will report some equivalent conditions for cluster autormorphisms and directed cluster automorphism groups. In particular, we show that the second condition of the definition of cluster automorphisms (i.e., preserving the mutations) could be implied by the first condition (i.e., mapping a cluster to another cluster). This is a joint work with Cao Peigen, Li Fang and Pan Jie.

（丛自同构是由Assem, Schiffler and Shamchenko引入的一类具有较好性质的代数同构。我将介绍一些丛自同构的等价条件。）