Quantum Groups, Quantum Symmetries, and Non-Commutative Geometry

非对易几何是一个纯数学领域，它的起源可以追溯到数学物理中的问题，这些问题是由量子现象的发现推动的。几何实体(例如作为广义相对论基础的4维时空)可以在代数上研究的典型数学框架，如果要与量子现象相协调，就需要扩大和推广。在由此产生的设置中，物理系统的对称性(在结构保持变换的意义上)有时需要被丢弃，取而代之的是新的和更奇异的对称性概念。这些奇异对称性的数学体现被称为“量子群”，它们是这个项目的主要关注点。在“普通的”几何和普通空间上的普通变换的作用下，许多可以被视为理所当然的东西在非对易的环境中变得有问题。这个研究项目的目的是在更大的非对易几何领域内的一系列子领域中阐明其中的一些问题。这个项目研究了围绕量子对称性概念的非对易几何的几个方面。这包括研究量子群，它们的表示理论，它们在代数和几何结构上的作用，以及伴随而来的非对易代数几何中的问题。一个目标是进一步理解量子刚性现象，即某些结构不存在真正的量子对称性。这意味着，当一个行为足够好的Hopf代数(它是量子群的代数体现)以结构保持的方式相互作用时，相互作用因子通过普通群上的函数代数一个接一个地进行。这种情况的许多特例是已知的(积分仿射代数簇、某些光滑非交换射影代数簇、紧连通光滑流形、某些度量空间类等)，但人们对一般现象知之甚少。另一个目标是试图将特定于离散或约化代数群(如Borel子群、极大环面、权系统、紧化、剩余有限性、线性)的工具和概念转移到量子群，以便进一步阐明其结构和表示理论。最后，对称性的考虑允许构造光滑的非交换投影方案的新例子，在某种意义上，这些例子在分类此类方案的模空间内表现为一般的行为。然后，相应代数的表示理论将揭示这些模空间的性质，这些模空间目前还没有被很好地理解。

Noncommutative geometry is a field of pure mathematics that traces its origins to problems in mathematical physics motivated by the discovery of quantum phenomena. The typical mathematical framework in which geometric entities (such as the 4-dimensional space-time that underlies the general theory of relativity) can be studied algebraically needs to be enlarged and generalized if it is to be reconciled with quantum phenomena. In the resulting setup, the symmetries of a physical system (in the sense of structure-preserving transformations) sometimes need to be discarded and replaced with new and more exotic notions of symmetry. The mathematical embodiment of these exotic symmetries are known as "quantum groups," and they are the main focus of this project. Much of what can be taken for granted in the context of "plain" geometry and actions of ordinary transformations on ordinary spaces becomes problematic in the noncommutative setting. This research project aims to shed light on a number of these problems, in a range of subfields within the larger realm of noncommutative geometry. This project investigates several aspects of noncommutative geometry that revolve around the notion of quantum symmetry. This involves studying quantum groups, their representation theory, and their actions on algebraic and geometric structures, as well as attendant problems in non-commutative algebraic geometry. One goal is to further understand the phenomenon of quantum rigidity, whereby certain structures admit no truly quantum symmetries. What this means is that whenever a sufficiently well-behaved Hopf algebra (which is the algebraic embodiment of a quantum group) coacts in a structure-preserving manner, the coaction factors through one by the function algebra on an ordinary group. Many special cases of this are known (integral affine algebraic varieties, certain smooth non-commutative projective algebraic varieties, compact connected smooth manifolds, certain classes of metric spaces, etc.), but the general phenomenon is poorly understood. Another goal is to attempt to transport tools and concepts specific to discrete or reductive algebraic groups (such as Borel subgroups, maximal tori, weight systems, compactifications, residual finiteness, linearity) over to quantum groups in order to further elucidate their structure and representation theory. Finally, symmetry considerations allow for the construction of new examples of smooth noncommutative projective schemes that in some sense behave generically within the moduli spaces that classify such schemes. The representation theory of the corresponding algebras would then shed light into the nature of these moduli spaces that are at the moment not well understood.