MOMENT MAPS TO LOOP ALGEBRAS CLASSICAL R-MATRIX AND INTEGRABLE SYSTEMS †
MOMENT MAPS TO LOOP ALGEBRAS CLASSICAL R-MATRIX AND INTEGRABLE SYSTEMS †
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用于循环代数经典 R 矩阵和可积系统的矩图 †
DOI：
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发表时间：
1993
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通讯作者：
J. Harnad
中科院分区：
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作者：
J. Harnad
介绍了一类由经典分裂R矩阵R = P+ P确定的李泊松结构的有限维简化辛向量空间在环代数的对偶空间上的泊松嵌入。这些可以看作是在辛空间上诱导“对偶”群e GR = e G + × e G的环群e G的自然哈密顿作用的等变矩映射。利用Adler-Kostant- Symes定理的r-矩阵形式来推导由Lax型等谱方程决定的交换流。相容条件决定了可积PDE系统的有限维解类，该系统可以用标准的Liouville- Arnold方法进行积分。这涉及到一个适当选择的“光谱达布”（规范）坐标系，其中有一个完全分离的变量。作为一个算例，该方法应用于正弦戈登方程有限维拟周期解的确定。
A class of Poisson embeddings of reduced, finite dimensional symplectic vector spaces into the dual space egof a loop algebra, with Lie Poisson structure determined by the classical split R-matrix R = P+ P is introduced. These may be viewed as equivariant moment maps inducing natural Hamiltonian actions of the "dual" group e GR = e G + × e G of a loop group e G on the symplectic space. The R-matrix version of the Adler-Kostant- Symes theorem is used to induce commuting flows determined by isospectral equations of Lax type. The compatibility conditions determine finite dimensional classes of solutions to integrable systems of PDE's, which can be integrated using the standard Liouville- Arnold approach. This involves an appropriately chosen "spectral Darboux" (canonical) coordinate system in which there is a complete separation of variables. As an example, the method is applied to the determination of finite dimensional quasi-periodic solutions of the sine-Gordon equation.