摘要
新谎言代数学，它是远不同的形式已知的 A <SUB > n-1 </SUB>, 被建立，为哪个相应的环代数学被给。从这，二个 isospectral 问题被揭示，状况其相容性读一种零个弯曲方程，它允许 soliton 方程的宽松的 integrable 层次。到在产生如此的 soliton 方程层次的 Hamiltonian 结构的目的，一种美丽的杀死 Cartan 形式，矩阵功能的概括踪迹，被给一个概括 Tu 公式(GTF ) 它被获得，当踪迹身份由 Tu Guizhang 求婚了时[J。数学。Phys。30 (1989 ) 330 ] 是 GTF 的一种特殊情况。坚定出现在 GTF 的经常的 γ
上的计算公式被得出，它在它上保证准确、简单的计算。最后，我们举二个例子揭示在文章介绍的理论的应用。在细节，第一个例子与二潜在的功能和 Hamiltonian 结构一起揭示 soliton 方程的一个新 Liouville-integrable 层次。获得 soliton 方程的第二个 integrable 层次，高度维的环代数学首先被构造。因此，第二个例子与 5 潜力部件功能和 bi-Hamiltonian 结构显示出另一个新 Liouville integrable 层次。在纸介绍的途径可以广泛地被用来与 multi-Hamiltonian 结构产生另外的新 integrable soliton 方程层次。
A new Lie algebra, which is far different form the known An-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.
