摘要
向量环代数学和它的扩大的环代数学被建议，它被奉献给获得 Tu 层次。由使用扩大踪迹身份， Tu 层次的 Hamiltonianstructure 被构造。而且，我们把二次形式的身份用于联合 Tu 层次的系统的 integrable。
A vector loop algebra and its extended loop algebra are proposed, which are devoted to obtaining the Tu hierarchy. By making use of the extended trace identity, the Harniltonian structure of the Tu hierarchy is constructed. Furthermore, we apply the quadratic-form identity to the integrable coupling system of the Tu hierarchy.
