Component-trace identities for Hamiltonian structures

Abstract

We show that on a particular class of semi-direct sums of matrix Lie algebras, component traces of the matrix product can produce bilinear forms which are non-degenerate, symmetric and invariant under the Lie product. The corresponding variational identities are called component-trace identities and provide tools in generating Hamiltonian structures of integrable couplings including the perturbation equations. An illustrative example of applying component-trace identities is given for the KdV hierarchy.

Keywords:

• Hamiltonian structures
• integrable couplings
• zero-curvature equations
• matrix Lie algebras

AMS Subject Classifications::

• 37K05
• 35Q53
• 37K10
• 35Q58

Acknowledgements

The work was supported in part by the Established Researcher Grant and the CAS faculty development grant of the University of South Florida and Chunhui Plan of the Ministry of Education of China.