Quadratic Deformations of Lie-Poisson Structures

TL;DR

This paper explores quadratic deformations of Lie-Poisson structures, providing a decomposition related to the modular vector and classifying such deformations in three-dimensional space.

Contribution

It introduces a decomposition for Lie-Poisson structures based on the modular vector and classifies quadratic deformations on  up to linear diffeomorphisms.

Findings

Lie-Poisson structures decompose into compatible structures in low dimensions

Quadratic deformations on  are classified up to linear transformations

Decomposition linked to the modular vector enhances understanding of Lie-Poisson structures

Abstract

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