Poisson Lie groups, integrable systems, and representation theory

本项目的主要目标是研究李群及其相关空间上几大类Poisson结构的几何性质。它们包括复单李群及其Poisson齐性空间上的Belavin-Drinfeld Poisson结构，Kac-Moody群和真实的单群上的Poisson结构.该项目是基于采用新的技术从李理论和环理论，特别是relatedthe上述几何问题的研究的原始谱的universalenveloping代数的李代数，一般既不可解，也不是半单的。主要研究者将致力于将这些问题应用于研究可积系统（其相空间是上述泊松结构的辛叶）和Hopf代数的表示，例如由Etingof-Kazhdan和Etingof-Schedler-Schiffmann通过显式量化Belavin-Drinfeld r-矩阵构建的非标准量子群上的正则函数代数。他将进一步解决应用组合：理论的集群代数的福明和Zelevisky，交叉点的dualSchubert细胞（有关辛叶特殊泊松结构的旗品种）。该项目的第二部分涉及形式伪微分算子的无穷维Poisson李群在Duistermaat和Grunbaum的双谱理论问题中的应用，例如，使用Semenov-Tian-Shansky的相应Dressing作用来构造KP族的无穷小“附加对称”的子代数，以保持双谱波函数的流形。该项目将进一步研究双谱与朗道、波拉克和Slepian的长椭球现象之间的关系，长椭球现象首先出现在时间带限制中，但随后在随机矩阵理论中也发挥了重要作用。特别是这一建议的目标integral运营商有关的无限维类的bispectral algebras的秩1和2，所有这些都是可交换的微分算子，provedin以前的作品的调查。在许多领域的数学和数学物理的主要问题都涉及到研究的对称性（变换群）的对象或模型，正在调查。近年来，广义对称（Hopfalgebras）在许多领域中发挥着越来越重要的作用。本项目研究与经典对称（代数群）到广义对称（Hopf代数）的变形相关的几何结构及其在动力系统、代数、组合学和应用数学中的应用。

The primary goal of this project is to investigate the geometry of severallarge classes of Poisson structures on Lie groups and their homogeneousspaces. They include the Belavin-Drinfeld Poisson structures on complex simpleLie groups and their Poisson homogeneous spaces, and Poisson structures onKac-Moody groups and real simple groups. The project is based on employing noveltechniques from Lie theory and ring theory which in particular relatethe above geometric problems to studies of the primitive spectra of universalenveloping algebras of Lie algebras that are in general neither solvable,nor semisimple. The principal investigator will work on applications of theseproblems to the study of integrable systems (whose phase spaces are symplecticleaves of the above Poisson structures) and representations of Hopf algebras,e.g. algebras of regular functions on non-standard quantum groups constructedby Etingof-Kazhdan and Etingof-Schedler-Schiffmann by explicit quantizations ofBelavin-Drinfeld r-matrices. He will further address applications to combinatorics:the theory of cluster algebras of Fomin and Zelevisky, and intersections of dualSchubert cells (related to symplectic leaves of special Poisson structures onflag varieties). The second part of the project concerns applications of theinfinite dimensional Poisson Lie group of formal pseudo-differential operators toproblems in the theory of bispectrality of Duistermaat and Grunbaum, e.g. using thecorresponding dressing action of Semenov-Tian-Shansky to construct subalgebrasof infinitesimal "additional symmetries" of the KP hierarchy that preserve manifoldsof bispectral wave functions. The project will further investigate relationsbetween bispectrality and the prolate spheroidal phenomenon of Landau, Pollak, andSlepian, which first appeared in time-band limiting but consequently played animportant role in random matrix theory as well. In particular this proposal targetsintegral operators related to the infinite dimensional class of bispectral algebrasof ranks 1 and 2, all of which posses commuting differential operators, provedin previous works of the investigator.The major problems in many areas of mathematics and mathematical physics arerelated to the study of the symmetries (transformation groups) of theobjects or models, under investigation. Recently generalized symmetries (Hopfalgebras) have played an increasingly prominent role in many fields. This projectinvestigates geometric structures related to deformations of classical symmetries(algebraic groups) to generalized symmetries (Hopf algebras) and their applicationsto problems in dynamical systems, algebra, combinatorics, and applied mathematics.