Poisson structures on associated with rigid Lie algebras

TL;DR

This paper explores the Poisson-Lichnerowicz cohomology of polynomial Poisson algebras, focusing on structures linked to rigid Lie algebras, and computes cohomology spaces for specific brackets.

Contribution

It introduces methods to compute Poisson cohomology for polynomial algebras associated with rigid Lie algebra brackets, expanding understanding of their algebraic structures.

Findings

Computed Poisson cohomology spaces for specific brackets

Analyzed non homogeneous Poisson structures on polynomial algebras

Linked Poisson structures to rigid Lie algebra properties

Abstract

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials $\mathbb{C}[X_{1},..., X_{n}]$ using exterior calculus. After presenting some non homogeneous Poisson brackets on this algebra, we compute Poisson cohomological spaces when the Poisson structure corresponds to a bracket of a rigid Lie algebra.