Quantum Groups and Quantum Cluster Algebras

PI将研究量子群和量子簇代数的一系列相关问题。首先，他将致力于证明一个非常一般的公理定义类的量子幂零代数的每个成员承认一个典型的量子簇代数结构。在此基础上，他将尝试为所有代数类构建一个统一的分类。许多重要的家族都是这个大类中代数的特例，最著名的是量子双Bruhat胞代数和量子舒伯特胞代数。前一种情况将导致Berenstein-Zelevinsky量子簇代数猜想的证明。在这两种情况下，量子簇代数结构将被用来研究量子群和量子Schubert胞代数的谱的拓扑。在前一种情况下，他将试图证明双连续性，他最近建造的Dixoblets地图从辛叶上一个简单的李群配备了所谓的标准泊松结构的原始频谱的量子群。PI和他的研究生将应用这些想法来证明量子折叠的存在，并从它们构建量子簇代数。他还将他最近的成果刚性的量子环面分类的自同构群的有趣的（量子）集群代数。通过Gekhtman，Shapiro和Vainshtein的工作，可以用Poisson几何来逼近一类经典的簇代数。PI将使用Poisson唯一因子分解域的概念对上述项目的Poisson类似物进行研究。非交换和Poisson代数出现在数学（几何对象上的函数）和物理（经典和量子力学系统中的可观测量）的许多不同方面。这些对象的研究使用许多不同的技术的基础上，代数，几何，分析和组合的方法。PI将通过两种不同的方法研究这些对象。第一个是一个经典的，基于研究这些代数作为运算符（表示）的集合。这种方法使用了代数和几何的技巧。第二种方法是基于最近的组合概念集群代数发明的福明和Zelevinsky。它导致了对象上非常具体的组合结构。然后，突变的思想被用来研究对象的各个部分，这些部分是以前的方法看不到的（他们专注于这些代数的特定“初始侧”）。利用他最近的刚性结果，PI还将研究和分类上述类别中对象的对称性。这样做的动机是对称性降低了客体的复杂性，而对对称性集合的完整描述提供了对客体复杂性的理解。

The PI will work on a series of interrelated problems for quantum groups and quantum cluster algebras. Firstly, he will work on proving that each member of a very general axiomatically defined class of quantum nilpotent algebras admits a canonical quantum cluster algebra structure. Based on this, he will attempt to construct a unified categorification for all algebras in the class. Many important families arise as special cases of algebras in this large class, most notably the quantum double Bruhat cell algebras and the quantum Schubert cell algebras. The former case will lead to a proof of the Berenstein-Zelevinsky quantum cluster algebra conjecture. In both cases the quantum cluster algebra structure will be used to study the topology of the spectra of quantum groups and quantum Schubert cell algebras. In the former case he will attempt to prove bicontinuity of his recently constructed Dixmier map from the symplectic foliation on a simple Lie group equipped with the so called standard Poisson structure to the primitive spectrum of a quantum group. The PI and his graduate students will apply these ideas for proving the existence of quantum foldings and for building quantum cluster algebras from them. He will also apply his recent results on rigidity of quantum tori to the classification of automorphism groups of interesting (quantum) cluster algebras. Via the work of Gekhtman, Shapiro and Vainshtein a certain large class of classical cluster algebras can be approached using Poisson geometry. The PI will work on Poisson analogs of the above projects using a notion of Poisson unique factorization domains.Noncommutative and Poisson algebras arise in many different aspects of mathematics (functions on geometric objects) and physics (observables in classical and quantum mechanical systems). These objects are studied using many different techniques on the basis of algebraic, geometric, analytic and combinatorial methods. The PI will study these objects via two different methods. The first one is a classical one, based on studying the presentations of these algebras as collections of operators (representations). This method uses techniques from algebra and geometry. The second method is based on the recent combinatorial notion of cluster algebras invented by Fomin and Zelevinsky. It leads to a very concrete combinatorial structure on the objects. The idea of mutation is then used to study various parts of the objects which are not seen by the previous methods (they focused on a particular "initial side" of these algebras). Using his recent rigidity results, the PI will also study and classify the symmetries of the objects in the above classes. The motivation for this is that symmetries reduce the complexity of an abject and the full description of the collection of symmetries provides an understanding of the complexity of the object.