Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations | Zendy

Boris Kolev | Zendy

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Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

Author(s) -

Boris Kolev

Publication year - 2007

Publication title -

philosophical transactions of the royal society a mathematical physical and engineering sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.074

H-Index - 169

eISSN - 1471-2962

pISSN - 1364-503X

DOI - 10.1098/rsta.2007.2012

Subject(s) - mathematics , lie algebra , vector field , poisson manifold , lie conformal algebra , lie bracket of vector fields , adjoint representation , affine lie algebra , graded lie algebra , poisson bracket , pure mathematics , lie coalgebra , poisson distribution , hamiltonian (control theory) , lie theory , algebra over a field , poisson algebra , affine transformation , adjoint representation of a lie algebra , current algebra , geometry , symplectic geometry , mathematical optimization , statistics

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, we investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures, called affine or modified Lie-Poisson structures, are involved in the integrability of certain Euler equations that arise as models for shallow water waves.

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