Poisson Lie groups, representation theory, combinatorics, and integrable systems

Yakimov将研究各种拉格朗日子代数的几何，这些子代数配备了由简单李代数的准三角形r-矩阵的Belavin-Drinfeld分类导出的泊松结构。在特殊情况下，这涉及到双旗变异上泊松结构的几何研究和De Concini和Procesi的奇妙群紧化。这是基于李论、组合学和几何技术的混合。在相反的方向上，Yakimov将研究在表示和环理论、动力系统和组合学中的应用。这些研究包括复单李代数的抛物子代数的零根的量子化普遍包络代数的谱和单位根上的表示，以及由Etingof-Kazhdan和etingof - schler - schiffmann通过显式量化的Belavin-Drinfeld r-矩阵构造的非标准量子群上的正则函数代数。第一部分的泊松几何构造通过算子代数的条件期望的代数版本提出了环理论的新方法。在组合学方面，PI将研究从旗型、双旗型和奇妙紧化上泊松结构叶的环面轨道的坐标环构造聚类代数。将进一步研究与Kazhdan-Lusztig多项式有关的共同性层理的性质，以及理查德森变异的显泊松退化。在完全可积系统领域，本项目将研究Kogan-Zelevinsky可积系统及其在（双）标志变体中Schuberts细胞上的某些推广，并将其与经典可积系统，如Gelfand-Tsetlin系统联系起来。本文将根据Calogero-Moser系统与Wilson’s adelic Grassmannian系统之间的相互作用，研究Khesin和Zakharevich的形式伪微分算子的无限维泊松李群的类似问题。几何、代数、数学物理和组合学中的许多对象都具有大量的对称性。研究这些对称性是研究这些物体的一项关键技术，因为它可以降低物体的复杂性。亚基莫夫将研究出现在各种量子化情况下的非交换对象的对称性及其与动力系统、代数、组合学和应用数学问题的关系。

Yakimov will investigate the geometry of varieties of Lagrangian subalgebras, equipped with Poisson structures derived from the Belavin-Drinfeld classification of quasitriangular r-matrices for simple Lie algebras. In particular cases this involves the study of the geometry of Poisson structures on double flag varieties and the wonderful group compactifications of De Concini and Procesi. This is based on a blend of of techniques from Lie theory, combinatorics, and geometry. In the opposite direction, Yakimov will investigate applications to representation and ring theory, dynamical systems, and combinatorics. These include the study of the spectra and representations at roots of unity of the quantized universal enveloping algebras of nilradicals of parabolic subalgebras of complex simple Lie algebras and the algebras of regular functions on non-standard quantum groups constructed by Etingof-Kazhdan and Etingof-Schedler-Schiffmann by explicit quantizations of Belavin-Drinfeld r-matrices. The Poisson geometric constructions from the first part suggest novel approaches in ring theory via an algebraic version of conditional expectation for operator algebras. In combinatorics, the PI will work on the construction of cluster algebras from coordinate rings of torus orbits of leaves of Poisson structures on flag varieties, double flag varieties, and wonderful compactifications. Further properties of coisotropic stratifications, related to Kazhdan-Lusztig polynomials, and explicit Poisson degenerations of Richardson varieties will be studied. In the field of completely integrable systems, the PI will study Kogan-Zelevinsky integrable systems and certain generalizations of those on Schuberts cells in (double) flag varieties and relate them to classical integrable systems, e.g. Gelfand-Tsetlin systems. Similar questions for the infinite dimensional Poisson Lie group of formal pseudo-differential operators of Khesin and Zakharevich will be studied, in the light of the interplay between Calogero-Moser systems and Wilson's adelic Grassmannian.Many objects in geometry, algebra, mathematical physics, and combinatorics posses large groups of symmetries. The investigation of theses symmetries is a key technique in the study of these objects, since it leads to a reduction of the complexity of the objects. Yakimov will study such symmetries of noncommutative objects which appear in various quantized situation and their relations to problems in dynamical systems, algebra, combinatorics, and applied mathematics.