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

Yakimov将研究拉格朗日子代数的各种几何，配备来自简单李代数的准三角r-矩阵的Belavin-Drinfeld分类的泊松结构。在特殊情况下，这涉及到研究几何泊松结构的双旗品种和精彩的团体紧德康西尼和Procesi。这是基于混合的技术，从李理论，组合数学和几何。在相反的方向，Yakimov将研究应用表示和环理论，动力系统和组合。其中包括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单位根的谱和表示，以及Etingof-Kazhdan和Etingof-Schedler-Schiffmann通过显式量化Belavin-Drinfeld r-矩阵构造的非标准量子群上的正则函数代数。第一部分中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论中的新方法。在组合学中，PI将致力于从旗簇、双旗簇和奇妙的紧化上的Poisson结构叶的环面轨道的坐标环来构造簇代数。进一步的性质的共各向同性分层，有关Kazhdan-Lusztig多项式，明确泊松退化的理查森品种将被研究。在完全可积系统领域，PI将研究Kogan-Zelevinsky可积系统和（双）旗簇中Schuberts细胞上的可积系统的某些推广，并将其与经典可积系统联系起来，例如Gelfand-Tsetlin系统。类似的问题，无穷维泊松李群的形式伪微分算子的Khesin和Zakharevich将研究，在光之间的相互作用Calogero-Moser系统和威尔逊的adelic格拉斯曼。许多对象在几何，代数，数学物理和组合数学中的大群的对称性。这些对称性的研究是研究这些对象的关键技术，因为它导致减少对象的复杂性。Yakimov将研究这种对称性的非交换对象出现在各种量化的情况和他们的关系问题的动力系统，代数，组合学和应用数学。

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.