On some properties of Poisson cohomology: Example of calculation on a Poisson structure

Abstract:

This article explores the field of Poisson cohomology, which intersects algebra, geometry, and physics, offering valuable tools for studying the mathematical structures underlying dynamical systems. Poisson structures appear in a large variety of different contexts, ranging from string theory, classical and quantum mechanics, and differential geometry to abstract algebra, algebraic geometry, and representation theory. In each of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are in basically all cases decisive for the solution to the problem. In general, Poisson cohomology is finer, but it is also more difficult to compute. From a general point of view, the explicit computation of the Poisson cohomology is known to be very hard and, in fact, few examples exist in the literature. In this paper, we propose an explicit calculation of the Poisson cohomology of the Poisson algebra $(\mathbb {F}[x,y],\cdot ,\{x,y\}=y^{n})$ where $n\geq 2$ is an integer, $\mathbb {F}$ is a field of characteristic zero, and $\mathbb {F}[x,y]$ is the algebra of polynomials in the variables $x$ and $y$. Using the classical isomorphism of vector spaces, we prove that except for the cases $k=1$ and $k=2$, every $k$-th Poisson cohomology group of the above Poisson structure is a finite-dimensional vector space over $\mathbb {F}$. More explicitly, we show that the second Poisson cohomology group is an infinite-dimensional $\mathbb {F}$-subspace, while the third Poisson cohomology group is a free-dimensional $\mathbb {F}[x]$-submodule of rank $n-1$. The interaction between the infinite-dimensional first cohomology group and the finite-dimensional nature of higher-order groups illustrates both the richness and complexity of the algebraic structures involved. This work also sets the stage for further exploration of its implications in mathematical physics and geometry.