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
gorbanali@azaruniv.ac.ir
Rezvaneh Ayoubi
Dept. of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
rezvaneh.ayoubi@azaruniv.ac.ir
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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