Quadratic Deformations of Lie-Poisson Structures

Abstract

In this letter, first we give a decomposition for any Lie-Poisson structure $\ddot{I}{€}_g$ associated to the modular vector. In particular, $\ddot{I}{€}_g$ splits into two compatible Lie-Poisson structures if $dimg\le 3$. As an application, we classified quadratic deformations of Lie-Poisson structures on ${\mathbb{R}}^3$ up to linear diffeomorphisms.

14 pages, no figure