An extended trace identity and applications

For the loop algebras in the form of non-square matrices, their commuting operations can be used to set up linear isospectral problems. In order to look for the Hamiltonian structures of the corresponding integrable evolution hierarchies of equations, an extended trace identity is obtained by means of commutators, which undoes the constraint on the known trace identity proposed by Tu [Guizhang Tu. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30(2):330–8], and has an obvious simplicity comparing with the quadratic-form identity given by Guo and Zhang [Fukui Guo, Yufeng Zhang. The quadratic-form identity for constructing the Hamiltonian structure of integrable systems. J Phys A 2005;38:8537–48] with the aspect of applications.