Double Poisson structures on finite dimensional semi-simple algebras

Double Poisson Cohomology of Path Algebras of Quivers

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associ…

移位双泊松结构与非交换泊松扩展

Shifted double Poisson structures and noncommutative Poisson extensions

本文研究了非交换泊松扩展的理论，展示了移位双泊松括号如何诱导渐进李代数结构，并证明非交换泊松扩展与非交换哈密顿约简相容，同时移位双泊松结构与 cofibrant 解决方案无关，并在表示导出模堆上诱导移位泊松结构。

与双重泊松顶点代数相关的函子构造

Functorial constructions related to double Poisson vertex algebras

本文通过jet代数构造为任意双重泊松代数生成双重泊松顶点代数，证明其与表示函子相容，并比较泊松约化与哈密顿约化范畴，给出双重箭图的实例。

非交换导出预栈上的移位双辛结构与双泊松结构

Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks

本文在非交换导出模函子上引入移位双辛与双泊松结构，证明其与移位辛结构的关联及典范等价，并利用形式积分性质揭示Calabi--Yau结构与商预栈上对应结构的联系。

Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems

We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $\mathcal{O}({\rm G}^\ast)$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space $\mathscr{P}_{{\rm G},\mathbb{S}}$ of ${\rm G}$-local systems introduced in \cite{GS3}, and prove th…

泊松K稳定性与常数量曲率的广义凯勒结构

Poisson K-stability and the semiclassical Yau--Tian--Donaldson correspondence

研究紧致凯勒全纯泊松流形的泊松K稳定性，利用无限维动量映射技术，证明了凯勒 - 爱因斯坦法诺流形上泊松结构的“半经典YTD猜想”存在性部分，得到新的常数量曲率辛广义凯勒结构例子并在射影平面证猜想。