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

亚基莫夫将研究各种拉格朗日子代数的几何，这些子代数配备了泊松结构，这些结构来自于单李代数的拟三角r-矩阵的Belavin-Drinfeld分类。在特殊情况下，这涉及到研究双旗簇上泊松结构的几何以及de Concini和Procesi的奇妙的群紧化。这是基于来自李理论、组合学和几何学的技术的混合。在相反的方向，雅基莫夫将研究在表示和环论、动力系统和组合学中的应用。其中包括通过显式量化Belavin-Drinfr-矩阵，研究复单李代数的抛物子代数的零根的量子化泛包络代数的谱和单位根处的表示，以及由Etingof-Kazhdan和Etingof-Schedler-Schiffmann构造的非标准量子群上的正则函数的代数。第一部分中的Poisson几何结构通过算子代数的条件期望的代数形式提出了环论中的新方法。在组合学中，PI将致力于从旗簇、双旗簇和奇紧化的泊松结构的叶的环面轨道的坐标环构造簇代数。我们将进一步研究与Kazhdan-Lusztig多项式有关的余迷向分层的性质，以及Richardson簇的显式Poisson退化。在完全可积系统领域，PI将研究Kogan-Zlevinsky可积系统和Schuberts胞格上(双)FLAG簇的某些推广，并将它们与经典可积系统，例如Gelfand-Tsetlin系统联系起来。根据Calogero-Moser系统和威尔逊格拉斯曼系统之间的相互作用，我们将研究Khein和Zakharevich的形式伪微分算子的无限维Poisson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的对称性群。这些对称性的研究是研究这些对象的一项关键技术，因为它可以降低对象的复杂性。雅基莫夫将研究在各种量子化情况下出现的非对易对象的这种对称性，以及它们与动力系统、代数、组合学和应用数学中的问题的关系。

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.