Toda lattices and Toric varieties for real semisimple Lie algebras

DMS-0071523 Luis Casian这个项目涉及当(带符号的)Toda格的等谱流形被紧致时产生的某些实环面簇的拓扑(积分同调、上同调和胞格分解)。Toda格子可以显式地作为一个可积的哈密顿系统来求解，但是解的几何特征还没有被阐明。在李论术语中，这些环簇由作用在半单李代数的实旗流形上的分裂Cartan子群的一般轨道的闭包组成。一个有趣的问题是详细描述它们的结构，这与实际旗形的结构有一些相似之处。在复杂的情况下，这些变种的拓扑结构是众所周知的；然而，真实的情况带来了以前从未解决过的新困难。我们还考虑了这一主要问题的推广，包括原问题的一些Kac-Moody形式，全Kostant-Toda格，以及一般的实旗流形的结构。在Wess-Zumino-Novikow-Witten(WZNW)模型对称性约化的背景下，不定(带符号)Toda晶格的出现推动了对这些环簇的研究，WZNW模型是共形场论中最重要的模型方程之一。然后，所研究的环变簇可以被看作是这些不定Toda格的积分流形的具体描述(预期的)正则化，其中这些Toda系统的解的无穷大(即爆破点)将一切粘合到一个光滑的紧致流形中。此外，对Toda格等谱流形的研究也有助于理解基于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全球方面。

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