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Journal of Lie Theory 31 (2021), No. 2, 493--516
Copyright Heldermann Verlag 2021

Hamiltonian Systems on Co-Adjoint Lie Groupoids

Ghorbanali Haghighatdoost
Dept. of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

Rezvaneh Ayoubi
Dept. of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

Our purpose is to introduce by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid a class of Lie groupoids. In other words, we show that the orbits of the co-adjoint representation on the isotropy Lie algebroid of a Lie groupoid are Lie groupoid. We will call this type of Lie groupoid, co-adjoint Lie groupoid. Also, we try to construct and define Hamiltonian systems on the co-adjoint Lie groupoids. By considering the trivial Lie groupoid as an example, we show that our construction can be considered as a generalization of the construction of the Lie groups to the Lie groupoids. Finally we present the types I and II of Hamilton-Jacobi theorem of the Hamiltonian system corresponding to the co-adjoint Lie algebroid.

Keywords: Lie groupoids, Lie algebroids, Hamiltonian system, Hamilton-Jacobi equation.

MSC: 18B40, 53D17, 70H08, 70H20.

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