扭仿射李代数的Drinfeld-Sokolov方程簇

发布时间：2018-08-19 14:05

【摘要】：Drinfeld-Sokolov方程簇在可积系统领域中有着重要的意义.这一构造把无穷维李代数的理论和可积系统领域联系在了一起,并进一步地和其它许多数学研究分支产生了联系.简单地说,对任意一个给定的仿射型李代数和其Dynkin图上的一个固定的顶点,都可以给出一个相应的可积方程簇.这些方程簇都是哈密顿系统,其中大部分还可以表示成标量Lax方程的形式.对于非扭的仿射李代数来说,其对应的Drinfeld-S okolov系统具有很多较好的性质,例如这些系统实际上都具有双哈密顿结构.但是这样的性质对于扭仿射李代数来说则不成立.因此非扭的仿射李代数所对应的Drinfeld-Sokolov方程簇已经得到了广泛的研究,但是对于扭仿射李代数所对应的Drinfeld-Sokolov方程簇则远非如此.在本文中我们研究了对应于扭仿射李代数的Drinfeld-Sokolov方程簇.这里个关键的步骤是,对于一个扭的仿射李代数,可以找到一个适当的非扭的仿射李代数和其上的一个图自同构,使得其不动点子代数即对应于前述的扭仿射李代数.我们考察了这样的一对系统之间的关系.对于A2n(2),A2n-1(2)和Dn+1(2)型的仿射李代数,可以具体地证明其Drinfeld-Sokolov方程簇可以看成是从相应的非扭仿射李代数对应的某个推广的Drinfeld-Sokolov方程簇约化下来得到的.这种约化在相应的标量Lax算子上的体现也被仔细地考察了.这样我们就可已通过研究非扭的Drinfeld-Sokolov方程簇的某些性质来得到扭的Drinf eld-Sokolov方程簇的相应性质.我们还观察到,这样的约化过程还可推广到更广义的Drinfeld-Sokolov方程簇上去.这一类约化的一个典型的例子就是从约束KP方程簇到A2n-1(2)这一扭仿射李代数所对应的Drinfeld-Sokolov方程簇上的约化.我们详细地计算了约束KP方程簇所对应的双哈密顿系统的中心不变量,其结果表明这一系统是其无色散极限的一个拓扑形变.
[Abstract]:Drinfeld-Sokolov equations are of great significance in the field of integrable systems. This structure links the theory of infinite dimensional lie algebras with the field of integrable systems and further links with many other branches of mathematical research. Simply speaking, for any given affine lie algebra and a fixed vertex on its Dynkin graph, we can give a family of integrable equations. These equations are all Hamiltonian systems, and most of them can be expressed in the form of scalar Lax equations. For non-torsional affine lie algebras, the corresponding Drinfeld-S okolov systems have many good properties, for example, these systems actually have double Hamiltonian structures. But this property is not true for torsional affine lie algebras. Therefore, the Drinfeld-Sokolov equation family corresponding to the non-torsional affine lie algebra has been extensively studied, but the Drinfeld-Sokolov equation family corresponding to the torsional affine lie algebra is far from the same. In this paper, we study the families of Drinfeld-Sokolov equations corresponding to torsional affine lie algebras. A key step here is that for a twisted affine lie algebra, we can find an appropriate non-torsional affine lie algebra and an automorphism of a graph on it, so that its fixed point algebra corresponds to the torsional affine lie algebra mentioned above. We examined the relationship between such a pair of systems. For the affine lie algebras of A2n (2) A2n-1 (2) and DN1 (2), it is proved that the Drinfeld-Sokolov equation family can be regarded as the reduction of some generalized Drinfeld-Sokolov equations corresponding to the corresponding non-torsional affine lie algebras. The representation of this reduction on the corresponding scalar Lax operator is also carefully examined. In this way, we have obtained the corresponding properties of the torsional Drinfeld-Sokolov equation family by studying some properties of the non-torsional Drinf eld-Sokolov equation family. It is also observed that the reduction process can be extended to the more generalized Drinfeld-Sokolov equations. A typical example of this kind of reduction is from the constrained KP equation family to the Drinfeld-Sokolov equation family corresponding to the torsional affine lie algebra A2n-1 (2). The central invariants of double Hamiltonian systems corresponding to constrained KP equations are calculated in detail. The results show that the system is a topological deformation of its dispersion free limit.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O152.5

【相似文献】