A Type of New Loop Algebra and a Generalized Tu Formula

A new Lie algebra, which is far different form the known A n- 1 , is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing–Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula (GTF) is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 (1989) 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi-Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.