A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure
A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure
复制标题
芒福德系统的李代数推广、其对称性及其多重哈密顿结构
DOI：
--
复制
发表时间：
2007
期刊：
影响因子：
--
通讯作者：
P. Vanhaecke
中科院分区：
文献类型：
--
作者：
M. Pedroni;P. Vanhaecke
它的对称性和它的多Hamilton结构Marco Pedroni 1和Pol Vanhaecke 2;3摘要本文推广了Mumford系统，它描述了对任意固定g，g维超椭圆Jacobian上的所有线性OWS。Mumford系统的相空间由多项式的三元组组成，受到一定程度的约束，并且自然地被视为sl（2）的循环代数的一个子空间。在我们推广到任意单李代数g的相空间由dimg多项式，再次受到一定程度的限制。该相空间及其多重Hamilton结构是通过沿g的loop代数g（（1））的一个子簇N的Poisson约化沿着得到的。由于N是不是一个泊松子品种的整个多哈密顿结构，我们证明了（代数）泊松减少定理减少沿着任意子品种的一个NE泊松品种，这个定理是类似的精神MarsdenRatiu减少定理。我们还通过引入主对称性，给出了Mumford系统（及其推广）的多Hamilton结构的不同视角;该主对称性可以在循环代数g（（1））上描述为方向的导数，并且被证明在Poisson约化后仍然存在。当作用（作为李导数）在系统的一个泊松结构上时，它产生下一个泊松结构，类似地，当作用在一个哈密顿量（对合）或它们的（交换）向量elds上时，它产生下一个。这样，我们就得到了几个由主对称性建立起来的多哈密顿体系。数学学科分类（1991）：58 F07、17 B81。
its symmetries and its multi-Hamiltonian structure Marco Pedroni1 and Pol Vanhaecke2;3 Abstract In this paper we generalize the Mumford system which describes for any xed g all linear ows on all hyperelliptic Jacobians of dimension g. The phase space of the Mumford system consists of triples of polynomials, subject to certain degree constraints, and is naturally seen as an a ne subspace of the loop algebra of sl(2). In our generalizations to an arbitrary simple Lie algebra g the phase space consists of dimg polynomials, again subject to certain degree constraints. This phase space and its multi-Hamiltonian structure is obtained by a Poisson reduction along a subvariety N of the loop algebra g(( 1)) of g. Since N is not a Poisson subvariety for the whole multi-Hamiltonian structure we prove an (algebraic) Poisson reduction theorem for reduction along arbitrary subvarieties of an a ne Poisson variety; this theorem is similar in spirit to the MarsdenRatiu reduction theorem. We also give a di erent perspective on the multi-Hamiltonian structure of the Mumford system (and its generalizations) by introducing a master symmetry; this master symmetry can be described on the loop algebra g(( 1)) as the derivative in the direction of and is shown to survive the Poisson reduction. When acting (as a Lie derivative) on one of the Poisson structures of the system it produces a next one, similarly when acting on one of the Hamiltonians (in involution) or their (commuting) vector elds it produces a next one. In this way we arrive at several multi-Hamiltonian hierarchies, built up by a master symmetry. Mathematics Subject Classi cation (1991): 58F07, 17B81.
DOI：
--
发表时间：
2005
期刊：
影响因子：
--
作者：
T. Kobayashi;T. Oshima
通讯作者：
T. Oshima