Linear Poisson structures and Hom-Lie algebroids

Esmaeil Peyghan,Amir Baghban

doi:10.33044/revuma.v60n2a01

Abstract

Considering Hom-Lie algebroids in some special cases, we obtain some results of Lie algebroids for Hom-Lie algebroids. In particular, we introduce the local splitting theorem for Hom-Lie algebroids. Moreover, linear Hom-Poisson structure on the dual Hom-bundle will be introduced and a one-to-one correspondence between Hom-Poisson structures and Hom-Lie algebroids will be presented. Also, we introduce Hamiltonian vector fields by using linear Poisson structures and show that there exists a relation between these vector fields and the anchor map of a Hom-Lie algebroid.

Highlights

Hom-Lie algebroids were introduced by Laurent-Gengoux and Teles in [4] using the notion of Hom-Gerstenhaber algebras
Many results in Hamiltonian dynamics are indebted to Poisson geometry, where they serve as phase spaces
Though Poisson geometry was an outcome of symplectic geometry, it is a powerful theory in mathematics

Summary

Introduction

Hom-Lie algebroids were introduced by Laurent-Gengoux and Teles in [4] using the notion of Hom-Gerstenhaber algebras. In [1], the authors could fix the definition of Hom-Lie algebroid in a more suitable way by introducing the notion of Hom-bundle. In this sense, there is a fundamental example. The pull back bundle φ!TM of the tangent Lie algebroid TM is a Hom-Lie algebroid This example is based on the concept of the Hom bundle, and using it the authors introduced the Hom-Poisson tensor structures. The structure of this paper is organized as follows: In Section 2, we recall the notions of Hom-algebra and Hom-Lie algebroid and present some examples of these concepts.

Preliminaries

Local splitting theorem

Linear Hom-Poisson structures