Toda lattices and Toric varieties for real semisimple Lie algebras

luis Casian这个项目研究了（有符号的）Toda格的等谱流形紧化时产生的某些实环变体的拓扑结构（积分同调、上同调和单元分解）。Toda格可以显式地解为可积哈密顿系统，但解的几何特征尚未明确。在李论中，这些环变异体由作用于半单李代数的实标志流形上的分裂Cartan子群的一般轨道的闭包组成。然后，一个有趣的问题是详细描述它们的结构，它与实际旗流形的结构有一些相似之处。在复杂情况下，这些品种的拓扑结构是众所周知的；然而，实际情况提出了以前没有解决的新困难。本文还考虑了这一主要问题的扩展，包括原问题的一些Kac-Moody版本，完整的Kostant-Toda格，以及实旗流形的一般结构。在共形场论中最重要的模型方程之一wess - zumino - novikowo - witten （WZNW）模型的对称约简背景下，不确定（有符号）Toda格的出现是研究这些环变分的物理动机。所研究的环面变异体可以被看作是对这些不定Toda格的积分流形的（预期的）正则化的具体描述，其中这些Toda系统的无穷解（即爆炸点）将所有的东西粘合到光滑紧化流形中。此外，对Toda格的等谱流形的研究有助于理解基于QR或LU分解的矩阵特征值算法的几何性质。本项目将阐明可积系统的一个全局方面

DMS-0071523Luis Casian This project concerns the topology (integral homology, cohomology and cell decompositions) of certain real toric varieties that arise when isospectral manifolds of a (signed) Toda lattice are compactified. The Toda lattice can be solved explicitly as an integrable hamiltonian system, but the geometrical feature of the solutions has not been clarified. In Lie-theoretic terms, these toric varieties consist of closures of generic orbits of a split Cartan Subgroup acting on a real flag manifold of a semisimple Lie algebra. An interesting problem is then to describe, in detail, their structure, which has some similarities with the structure of real flag manifolds. The topology of these varieties is well-known in the complex case; however the real case poses new difficulties which have not been tackled before. Extensions of this main problem are also considered which include some Kac-Moody versions of the original problem, the full Kostant-Toda lattice and, in general, the structure of real flag manifolds. The study of these toric varieties is physically motivated by the appearance of the indefinite (signed) Toda lattices in the context of symmetry reduction of the Wess-Zumino-Novikow-Witten (WZNW) model which is one of the most important model equation for conformal field theory. The toric varieties under Study can then be seen to give a concrete description of (an expected) regularization of the integral manifolds of these indefinite Toda lattices, where infinities (i.e. blow up points) of the solutions of these Toda systems glue everything into a smooth compact manifold. Also the study of isospectral manifolds of the Toda lattices is useful to understand the geometry of matrix eigenvalue algorithm based on QR or LU factorization. The present project will clarify a global aspect of the integrable systems