Quadratic Deformations of Lie–Poisson Structures

Abstract

In this letter, first we give a decomposition for any Lie–Poisson structure $$\pi_{\mathfrak g}$$ associated to the modular vector. In particular, $$\pi_{\mathfrak g}$$ splits into two compatible Lie–Poisson structures if $${\rm dim}{\mathfrak g} \le 3$$ . As an application, we classified quadratic deformations of Lie– Poisson structures on $$\mathbb R^3$$ up to linear diffeomorphisms.