代数学研讨会

2022年9月20日

2022年9月22日

会议报告摘要

Nakajima's graded quiver varieties and the triangular bases of cluster algebras

（Nakajima分次箭图簇以及丛代数的三角基）

李理

奥克兰大学

Nakajima's graded quiver varieties are smooth complex algebraic varieties associated with quivers, and can be used to study cluster algebras. In the talk, I will explain how to precisely locate the supports of the triangular basis of skew-symmetric rank 2 quantum cluster algebras by applying the decomposition theorem to various morphisms related to quiver varieties, thus prove a conjecture proposed by Lee-Li-Rupel-Zelevinsky in 2014.

（Nakajima分次箭图簇是定义在一个箭图上的复代数簇，可以被用来研究丛代数。本报告中，我们将解释如何用其来讨论秩为2的量子丛代数的三角基的支撑集的分布。）

Twists of graded Poisson algebras and applications

（分次泊松代数的扭及应用）

王兴庭

霍华德大学

In noncommutative projective algebraic geometry, twistings of homogenous coordinate rings give equivalences between noncommutative projective schemes. We introduce a Poisson version of such twisting of any graded Poisson algebra. This is used to prove every graded Poisson algebra is the graded twist of a unimodular one, which is a decomposition theorem for graded Poisson structures. We also study various new concepts in Poisson twisting related to the computation of Poisson homology and Poisson cohomology. This is joint work with Hongdi Huang, Xin Tang and James Zhang.

（在分交换投射代数几何，齐次坐标环的扭能够给出非交换投射概型的等价。我们引入任意分次泊松代数的扭的泊松版本。我们证明了任意分次泊松代数是一个unimodular分次泊松代数的分次扭,我们研究了泊松扭中和泊松(上)同调相关概念。）

Twisting of quantum groups and the related properties

（量子群的扭以及相关性质）

黄红娣

莱斯大学

Let $H$ be a Hopf algebra and $\sigma$ be a 2-cocycle on $H$. There exists a monoidal equivalence between the comodule categories of $H$ and of its 2-cocycle twist $H^\sigma$. In this talk, we will describe the connection between a Zhang twist $H^{\phi}$ and a 2-cocyle twist $H^{\sigma}$ of $H$, where $\phi$ is a graded automorphism of $H$. Moreover, we will discuss various twistings of superpotentials, preregular forms, and their associated $H$- comodule algebras $A$.

（令H是一个Hopf代数，$\sigma$是一个2-余圈。H的余模范畴和H^\sigma的余模范畴存在一个monoidal范畴等价。本报告中，我们会描述张扭H^\phi和H^\sigma的关系，其中\phi是H的分次自同构。进一步，我们会讨论它们的H余模代数。