标题： 泊松上同调在可诱导性问题和变形映射研究中的应用 显示英文标题

标题： Applications of Poisson cohomology to the inducibility problems and study of deformation maps

摘要： 本文提供了Flato、Gerstenhaber和Voronov引入的Poisson上同调群的一些应用。 给定一个由表示引起的Poisson代数的可交换扩张，我们首先研究一对Poisson代数自同构的可诱导性，并表明相应的障碍位于第二Poisson上同调群中。 因此，我们得到了连接各种自同构群和第二Poisson上同调群的Wells精确序列。 随后，我们还考虑了一对Poisson代数导子的可诱导性，获得了相应的障碍并构造了对应的Wells型精确序列。 为了得到另一个应用，我们在一个原始双倍Poisson代数中引入了“形变映射”的概念。 一个形变映射统一了各种著名的算子，如Poisson同态、Poisson导子、交叉同态、任何权值的Rota-Baxter算子、扭曲的Rota-Baxter算子、Reynolds算子和修改的Rota-Baxter算子在Poisson代数上的表现。 我们证明了一个形变映射$r$会诱导一个新的Poisson代数结构及其适当的表现形式。 相应的Poisson上同调被定义为形变映射$r$的上同调。 最后，我们通过上同调研究了算子$r$的形式形变。 显示英文摘要

摘要： This paper provides some applications of the Poisson cohomology groups introduced by Flato, Gerstenhaber and Voronov. Given an abelian extension of a Poisson algebra by a representation, we first investigate the inducibility of a pair of Poisson algebra automorphisms and show that the corresponding obstruction lies in the second Poisson cohomology group. Consequently, we obtain the Wells exact sequence connecting various automorphism groups and the second Poisson cohomology group. Subsequently, we also consider the inducibility for a pair of Poisson algebra derivations, obtain the obstruction and construct the corresponding Wells-type exact sequence. To get another application, we introduce the notion of a `deformation map' in a proto-twilled Poisson algebra. A deformation map unifies various well-known operators such as Poisson homomorphisms, Poisson derivations, crossed homomorphisms, Rota-Baxter operators of any weight, twisted Rota-Baxter operators, Reynolds operators and modified Rota-Baxter operators on Poisson algebras. We show that a deformation map $r$ induces a new Poisson algebra structure and a suitable representation of it. The corresponding Poisson cohomology is defined to be the cohomology of the deformation map $r$. Finally, we study the formal deformations of the operator $r$ in terms of the cohomology.