Quantum Groups, Poisson Lie Groups, and Combinatorics

本文讨论了量子群、泊松李群、泊松齐次空间及其与组合学的关系等一系列相关问题。这些包括:确定了De Concini-Kac-Procesi和Lusztig以及Berenstein-Zwicknagl（平面）编织对称代数的幂零代数的量子化普适包络代数的谱，实现了函数的量子化代数的轨道法，对量子幂零代数的自同构群进行了分类，研究了与Belavin-Drinfeld分类相关的函数的量子化代数以及相应的拉格朗日子代数的变体。证明了Kogan-Zelevinsky可积系统的完备性，研究了旗簇和双旗簇上的簇代数及其与这些对象上的全正性的相互作用。李群和代数群理论是数学的一个主要分支，用于以统一的方式研究物理理论的对称性以及几何和代数对象。近年来，这些技术的非交换版本开始扮演越来越重要的角色。Hopf代数出现在数学物理、代数、组合学、非交换几何和拓扑学的许多研究中。

This proposal addresses series of interrelated problems on quantum groups, Poisson Lie groups, Poisson homogeneous spaces, and their relations to combinatorics. These include: determining the spectra of quantized universal enveloping algebras of nilpotent algebras of De Concini-Kac-Procesi and Lusztig and the Berenstein-Zwicknagl (flat) braided symmetric algebras, carring out the orbit method for quantized algebras of functions, classifying automorphism groups of quantum nilpotent algebras, investigating quantized algebras of functions related to the Belavin-Drinfeld classification and the corresponding varieties of Lagrangian subalgebras, proving completeness of Kogan-Zelevinsky integrable systems, investigating cluster algebras on flag varieties and double flag varieties and their interaction with total positivity on these objects.The theory of Lie and algebraic groups is a major branch of mathematics, used to study in a unified way symmetries of physical theories, and geometric and algebraic objects. In recent years noncommutative versions of these techniques started playing an increasingly prominent role. Hopf algebras appeared in numerous studies in mathematical physics, algebra, combinatorics, noncommutative geometry, and topology.