# On rational isomorphisms of Lie algebras

Research paper by S. T. Sadetov

Indexed on: 01 Jan '07Published on: 01 Jan '07Published in: Functional Analysis and Its Applications

#### 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.