一个Lie代数的子代数及其相关的两类Loop代数

本文构造了Lie代数A2的一个子代数A2,通过选取恰当的基元阶数得到相应的一个loop代数A2,由此设计一个等谱问题,利用屠格式得到了一个新的Liouville可积的Hamilton方程族.作为其约化情形,得到了一个非线性有理分式型演化方程.再由一个矩阵变换,得到了换位运算与A2等价的Lie代数A1的一个子代数A1,将A1再扩展成一个新的高维loop代数G,利用G获得了所得方程族的一类扩展可积系统.

A Subalgebra of Lie Algebra and the Associated Two Types of Loop Algebras

In this paper, a subalgpbra A2 of the Lie algebra A2 is constructed, which gives a corresponding loop algebra A2 by properly choosing the gradation of the basis elements. It follows that an isospectral problem is established. Furthermore, a new Liouville integrable Hamiltonian hierarchy of equations is obtaind by the use of Tu scheme, from which the hierarchy reduces to a nonlinear evolution equation of rational fractal form. By making use of a matrix transformation, a subalgebra A2 of the Lie algebra A1 is presented, which possesses the same communicative operations of basis elements as those in A2. Again we expand the Lie algebra A1 into a high-dimensional loop algebra G, and a type of expanding integrable system of the hierarchy obtained above is worked out.