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普遍的公理定义的量子幂零代数类的每个成员都承认一个规范的量子聚类代数结构。在此基础上，他将尝试为班上所有代数建立一个统一的分类。在这一大类中，许多重要的代数家族作为代数的特殊情况出现，最著名的是量子双布鲁哈特细胞代数和量子舒伯特细胞代数。前一种情况将导致Berenstein-Zelevinsky量子聚类代数猜想的证明。在这两种情况下，量子簇代数结构将用于研究量子群和量子舒伯特细胞代数的光谱拓扑。在前一种情况下，他将试图证明他最近构造的Dixmier映射的双连续性，从具有所谓标准泊松结构的简单李群上的辛叶理到量子群的原始谱。PI和他的研究生将应用这些想法来证明量子折叠的存在，并以此构建量子簇代数。他还将把他关于量子环面的刚性的最新成果应用于有趣的（量子）簇代数的自同构群的分类。通过Gekhtman， Shapiro和Vainshtein的工作，可以用泊松几何逼近某一大类经典聚类代数。PI将使用泊松唯一分解域的概念来处理上述项目的泊松类比。非交换代数和泊松代数出现在数学（几何对象上的函数）和物理学（经典和量子力学系统中的可观测值）的许多不同方面。在代数、几何、解析和组合方法的基础上，使用许多不同的技术来研究这些对象。PI将通过两种不同的方法研究这些物体。第一个是经典的，基于研究这些代数作为算子集合（表示）的表示。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由Fomin和Zelevinsky发明的簇代数的组合概念。它导致了对象上非常具体的组合结构。突变的思想随后被用于研究物体的不同部分，这些部分是以前的方法所看不到的（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降低了物体的复杂性，而对称集合的完整描述提供了对物体复杂性的理解。

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.