Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras. (English) Zbl 1104.70011

Summary: The trace variational identity is generalized to zero curvature equations associated with non-semi-simple Lie algebras or, equivalently, Lie algebras possessing degenerate Killing forms. An application of the resulting generalized variational identity to a class of semi-direct sums of Lie algebras in the AKNS case furnishes Hamiltonian and quasi-Hamiltonian structures of the associated integrable couplings. Three examples of integrable couplings for the AKNS hierarchy are presented: one Hamiltonian and two quasi-Hamiltonian.


70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
70H05 Hamilton’s equations
22E70 Applications of Lie groups to the sciences; explicit representations
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