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

Ma, Wen-Xiu; Chen, Min

J. Phys. A, Math. Gen. 39, No. 34, 10787-10801 (2006).

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.

MSC:

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

Keywords:

trace variational identity; degenerate Killing forms; AKNS hierarchy