Poisson Lie groups, integrable systems, and representation theory

本课题的主要目的是研究李群及其齐次空间上几类Poisson结构的几何性质。它们包括复单Lie群及其Poisson齐次空间上的Belavin-Drinfeld Poisson结构，以及Kac-Moody群和实单群上的Poisson结构。这个项目是基于利用李论和环论中的新技术来研究一般既不可解也不半单的李代数的泛包络代数的本原谱。主要的研究人员将致力于这些问题在可积系统(相空间是上述Poisson结构的辛叶)和Hopf代数表示的研究中的应用，例如由Belavin-Drinfeld r-矩阵的显式量化构造的非标准量子群上的正则函数的代数。他将进一步讨论在组合学中的应用：Fomin和Zlevisky的簇代数理论，以及对偶Schubert胞格的交集(与FLAG簇上特殊泊松结构的辛叶有关)。项目的第二部分涉及形式拟微分算子的无穷维泊松李群在Duistermaat和Grunbaum双谱理论中的应用，例如，利用Semenov-Tian-Shansky的相应的修饰作用来构造Kp族的保持双谱波函数流形的无穷小“附加对称”的子代数。该项目将进一步研究双谱与朗道、波拉克和斯莱宾的长椭球现象之间的关系，这些现象最初出现在时间带限制中，但随后也在随机矩阵理论中发挥了重要作用。具体地说，这一建议针对的是与第一和第二阶双谱代数的无穷维类有关的积分算子，这些算子都具有可交换的微分算子。许多数学和数学物理领域的主要问题与研究对象或模型的对称性(变换群)有关。近年来，广义对称(Hopfagebras)在许多领域中发挥着越来越重要的作用。这个项目研究与经典对称(代数群)到广义对称(Hopf代数)的变形有关的几何结构及其在动力系统、代数、组合学和应用数学问题中的应用。

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.