A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoids

TL;DR

This paper explores the structure and integrability of Hamiltonian systems on co-adjoint Lie groupoids, establishing their properties and relationships with symplectic Lie groupoids.

Contribution

It introduces the concept of co-adjoint Lie groupoids, examines their structural mappings, and defines integrability of Hamiltonian systems within this framework.

Findings

Co-adjoint Lie groupoids are constructed from Lie groupoids.

The relationship between Lie algebroid structures and co-adjoint structures is analyzed.

Hamiltonian systems on co-adjoint Lie groupoids are shown to be integrable.

Abstract

As we said in our previous work [4], the main idea of our research is to introduce a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called coadjoint Lie groupoids. In this paper, we will examine the relationship between structural mappings of the Lie algebroid associated to Lie groupoid and co-adjoint Lie algebroid. Also, we try to construct and define integrabiliy of Hamiltonian system on the co-adjoint Lie groupoids. In addition, we show that co-adjoint Lie groupoid associated to a symplectic Lie groupoid is a symplectic Lie groupoid.