On Rational Isomorphisms of Lie Algebras

Let $\mathfrak{n}$ be a finite-dimensional noncommutative nilpotent Lie algebra for which the ring of polynomial invariants of the coadjoint representation is generated by linear functions. Let $\mathfrak{g}$ be an arbitrary Lie algebra. We consider semidirect sums $\mathfrak{n}{\kern1pt\dashv_{\rho}\kern1pt}\mathfrak{g}$ with respect to an arbitrary representation $\rho\colon \mathfrak{g}\to\operatorname{der}\mathfrak{n}$ such that the center $z\mathfrak{n}$ of $\mathfrak{n}$ has a $\rho$-invariant complement.
We establish that some localization $\widetilde{P}(\mathfrak{n}{\kern1pt\dashv_{\rho}\kern1pt}\mathfrak{g})$ of the Poisson algebra of polynomials in elements of the Lie algebra $\mathfrak{n}{\kern1pt\dashv_{\rho}\kern1pt}\mathfrak{g}$ is isomorphic to the tensor product of the standard Poisson algebra of a nonzero symplectic space by a localization of the Poisson algebra of the Lie subalgebra $(z\mathfrak{n})\dashv\mathfrak{g}$. If $[\mathfrak{n},\mathfrak{n}]\subseteq z\mathfrak{n}$, then a similar tensor product decomposition is established for the localized universal enveloping algebra of the Lie algebra $\mathfrak{n}{\kern1pt\dashv_{\rho}\kern1pt}\mathfrak{g}$. For the case in which $\mathfrak{n}$ is a Heisenberg algebra, we obtain explicit formulas for the embeddings of $\mathfrak{g}_P$ in $\widetilde{P}(\mathfrak{n}{\kern1pt\dashv_{\rho}\kern1pt}\mathfrak{g})$. These formulas have applications, some related to integrability in mechanics and others to the Gelfand–Kirillov conjecture.