Title: New solutions of the Jacobi equations for three-dimensional Poisson structures Show Chinese title

Title: 雅可比方程的新解法对于三维泊松结构

Abstract: A systematic investigation of the skew-symmetric solutions of the three-dimensional Jacobi equations is presented. As a result, three disjoint and complementary new families of solutions are characterized. Such families are very general, thus unifying many different and well-known Poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution. This unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure, the Casimir invariants and the Darboux canonical form, which are known only for a limited sample of Poisson structures. These common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework, instead of using a case-by-case approach. In addition, the methods developed are valid globally in phase space, thus ameliorating the usual scope of Darboux' reduction which is only of local nature. Finally, the families of solutions found present some new nonlinear superposition principles which are characterized. Show Chinese abstract

Abstract: 对三维雅可比方程的斜对称解进行系统的研究。 作为结果，表征了三个不相交且互补的新解族。 这些族非常普遍，从而统一了许多看似无关的不同且著名的泊松结构，现在它们作为更一般解的特例出现。 这种统一不仅是概念性的，而且允许开发算法以显式确定重要的性质，如辛结构、卡西米尔不变量和达布规范形式，这些性质仅知于有限的泊松结构样本。 因此，这些通用程序同时适用于现在可以在统一且更经济的框架中分析的所有特例，而不是使用逐个情况的方法。 此外，所开发的方法在相空间中是全局有效的，从而改善了通常仅具有局部性质的达布约简的常规范围。 最后，找到的解族具有一些新的非线性叠加原理，这些原理被表征出来了。