
Journal of Lie Theory 30 (2020), No. 4, 10611089 Copyright Heldermann Verlag 2020 On Lie Algebras from Polynomial Poisson Structures Princy Randriambololondrantomalala Dép. de Mathématiques et Informatique, Faculté des Sciences, Université d'Antananarivo, Madagascar princypcpc@yahoo.fr [Abstractpdf] We consider a polynomial Poisson algebra $\mathcal{P}$ on $\mathbb{R}^{2n}$ ($n\geq1$) that is to say $\mathcal{P}$ consists only of polynomials in $\mathbb{R}^{2n}$. We manage the conditions on $\mathcal{P}$ in order to have: every derivation of $\mathcal{P}$ is a differential operator of order one which takes its coefficients in $\mathcal{P}$. Otherwise, this result may not be true. More, we have an analogous result for the derived ideal $[\mathcal{P},\mathcal{P}]$ of $\mathcal{P}$. If $[\mathcal{P},\mathcal{P}] = \mathcal{P}$, derivations of the normalizer $\mathfrak{N}$ of $\mathcal{P}$ are sum of derivations of $\mathcal{P}$ and nonlocal derivations of $\mathfrak{N}$. Without this last hypothesis on $[\mathcal{P},\mathcal{P}]$, we can state a similar theorem about the normalizer of $[\mathcal{P},\mathcal{P}]$. The first ChevalleyEilenberg cohomology of these subalgebras are computed. Moreover, some results from polynomial Hamiltonian vector fields Lie algebras on $\mathbb{R}^{2n}$ has been found out. A special intention to Lie subalgebras of the polynomial Poisson algebra $\mathbb{R}(x,y)$ on $\mathbb{R}^2$ in which the Jacobian conjecture holds is given. We give a definition on a subLie algebra of $\mathbb{R}(x,y)$ verifying the Jacobian conjecture and find that if it is different to $\mathbb{R}(x,y)$, it verifies the Jacobian conjecture. Keywords: Lie algebras, polynomial Poisson structure, Jacobian conjecture, cohomology of ChevalleyEilenberg, differential operators, nonlocal derivations. MSC: 17B66; 53B15, 17B56. [ Fulltextpdf (246 KB)] for subscribers only. 