On rational isomorphisms of Lie algebras

Abstract

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 $\mathfrak{n} \dashv _\rho \mathfrak{g}$ with respect to an arbitrary representation ρ: $\mathfrak{g}$ → der $\mathfrak{n}$ such that the center z $\mathfrak{n}$ of $\mathfrak{n}$ has a ρ-invariant complement.

We establish that some localization $\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})$ of the Poisson algebra of polynomials in elements of the Lie algebra $\mathfrak{n} \dashv _\rho \mathfrak{g}$ 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 $\mathfrak{n} \dashv _\rho \mathfrak{g}$. For the case in which $\mathfrak{n}$ is a Heisenberg algebra, we obtain explicit formulas for the embeddings of $\mathfrak{g}_P$ in $\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})$. These formulas have applications, some related to integrability in mechanics and others to the Gelfand-Kirillov conjecture.