The Poisson cohomology of \(\mathfrak{sl}^{\ast}_2 (\mathbb{R})\) (Q6183444)

Computing the Poisson cohomology poses challenges due to the absence of general methods, making it an intriguing and difficult task. This paper presents a comprehensive calculation of the smooth Poisson cohomology of the linear Poisson structure associated with the dual Lie algebra \({sl}^{\ast}_2 (\mathbb{R})\), which features a standard linear Poisson structure. The paper provides concrete and explicit descriptions of the Poisson cohomology in different categories and offers geometric explanations of the cohomology elements. Of particular value is the extraction of a general strategy for calculating the Poisson cohomology locally around a singularity of the Poisson structure, which can be applied in computing other Poisson cohomologies as well.