A note on symplectic and Poisson linearization of semisimple Lie algebra actions

TL;DR

This paper proves local linearization of analytic symplectic actions of semisimple Lie algebras and extends results to Poisson manifolds, highlighting differences between smooth and analytic cases.

Contribution

It establishes analytic linearization results for semisimple Lie algebra actions on symplectic and Poisson manifolds, including new equivariant theorems.

Findings

Analytic symplectic actions of semisimple Lie algebras can be locally linearized in Darboux coordinates.

Smooth linearization only holds for compact semisimple Lie algebras.

Provides new equivariant Darboux and Weinstein splitting theorems in the Poisson setting.

Abstract

In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson setting.