广义Landau-Lifshitz系统和与高亏格曲线相关的李代数
Generalized Landau-Lifshitz systems and Lie algebras associated with higher genus curves
摘要：
Wahlquist-Estabrook延拓方法允许对某些偏微分方程得到一个负责拉克斯对和某些类型的Backlund变换的李代数。研究了Landau-Lifshitz方程的n维推广的Wahlquist-Estabrook代数，构造了该代数到亏格为1+（n-3）2^{n-2}的代数曲线上的矩阵值函数的无限维拟广义李代数L（n）的满射.对于n= 3，4，5，我们证明了Wahlquist-Estabrook代数同构于L（n）与一个2维交换李代数的直和.利用这些结果，对任意n，构造了一类新的以上述曲线上的点为参数的Miura型变换（微分替换）.作为副产品，我们得到了一个表示的L（n）在有限数量的发电机和关系，这可能是独立的利益。
英文摘要：
The Wahlquist-Estabrook prolongation method allows to obtain for some PDEs a Lie algebra that is responsible for Lax pairs and Backlund transformations of certain type. We study the Wahlquist-Estabrook algebra of the n-dimensional generalization of the Landau-Lifshitz equation and construct an epimorphism from this algebra onto an infinite-dimensional quasigraded Lie algebra L(n) of certain matrix-valued functions on an algebraic curve of genus 1+(n-3)2^{n-2}. For n=3,4,5 we prove that the Wahlquist-Estabrook algebra is isomorphic to the direct sum of L(n) and a 2-dimensional abelian Lie algebra. Using these results, for any n a new family of Miura type transformations (differential substitutions) parametrized by points of the above mentioned curve is constructed. As a by-product, we obtain a representation of L(n) in terms of a finite number of generators and relations, which may be of independent interest.
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