Abstract
In this paper, we study the truncated polynomial algebra L in n variables, and discuss the following four problems in detail: 1) Homology complex and homology group of Poisson algebra L; 2) Given a new Poisson bracket by calculation modular derivation of Frobenius Poisson algebra; 3) Calculate the twisted homology group of Poisson algebra L; 4) Verify the theorem of twisted Poincaré duality between twisted Poisson homology and Poisson Cohomology.
Highlights
Poisson structures appear in a large variety of different contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory
Poisson cohomology appears as one considers deformations of Poisson algebras
Given a Poisson algebra, we can get some vital information about the Poisson algebra structure from calculate its Poisson (Co)homology
Summary
Poisson structures appear in a large variety of different contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. They play an important role in Poisson geometry, in algebraic geometry and non-commutative. It is very important to calculate Poisson cohomology from a given Poisson structure. These researches mainly focused on the smooth algebra and the finite dimension algebra. Launois S and Richard L [7] calculate the Poisson (Co)homology of truncated polynomial algebras in 2 variables, and established the twisted Poincaré duality between them. We want to study infinite dimension situation: a truncated polynomial algebra with n variables
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