Toda lattices and Toric varieties for real semisimple Lie algebras

DMS-0071523 Luis Casian这个项目关注的是某些真实的环面簇的拓扑（积分同调、上同调和胞腔分解），这些环面簇是在（有符号的）户田格的等谱流形被紧致化时产生的。户田格可以作为一个可积的哈密顿系统显式求解，但其解的几何特征还没有得到明确的解释。在李理论中，这些环面簇由作用在半单李代数的真实的旗流形上的分裂Cartan子群的一般轨道的闭包组成。一个有趣的问题，然后详细描述，他们的结构，这与真实的旗流形的结构有一些相似之处。这些品种的拓扑结构是众所周知的复杂的情况下，然而，真实的情况下提出了新的困难，以前没有得到解决。这个主要问题的扩展也被认为是其中包括一些卡茨-穆迪版本的原始问题，完整的Kostant-Toda格，并在一般情况下，结构的真实的旗流形。Wess-Zumino-Novikow-Witten（WZNW）模型是共形场论中最重要的模型方程之一，在对称约化的背景下，不确定（带符号）户田晶格的出现激发了对这些环面簇的研究。然后可以看到，研究中的复曲面变种给出了这些不确定的户田晶格的积分流形的（预期的）正则化的具体描述，其中这些户田系统的解的无限性（即爆破点）将所有内容粘合到光滑的紧凑流形中。研究户田格的等谱流形有助于理解基于QR或LU分解的矩阵特征值算法的几何结构。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