KK-theory, quantum groups, and quaternions

KK 理论、量子群和四元数

• 批准号: RGPIN-2021-02746
• 负责人: Kucerovsky, Dan
• 金额: $ 1.31万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758883
• 关键词: KK; theory; quantum; groups; quaternions

This is a project in pure mathematics. Mathematics is about connections and developing provably correct deductions from available information we have about the world. Historically, it was synonymous with higher education itself. Progress in pure mathematics slowly influences the applied sciences; it is a fact that once esoteric concepts in pure mathematics now appear in engineering and physics. For the student or the HQP, pure mathematics teaches advanced logical thinking about complex topics, with possible applications to complicated scientific problems. Thus the benefits of this research program exist at the frontiers of today's science and technology, and may affect tomorrow's science and technology. The basic topic of the program is norm-closed self-adjoint algebras of operators on Hilbert space, called C*-algebras. They have remarkable properties, and connect disparate areas such as functional analysis, algebra, topology, geometry, and dynamical systems. Perhaps the most important single endeavour in this field is to classify such algebras by K-theoretical data (the Elliott program). The time is right to investigate branching the program out to new directions. C*-algebras can be viewed as noncommutative generalizations of topological spaces, and if we look for generalizations of topological groups rather than topological spaces, we obtain C*-algebraic quantum groups (QGs), which may have co-actions on C*-algebras. Our main objective is to generalize the Elliott program to suitable classes of QGs and co-actions. An Elliott style classification of QGs works very differently from existing algebraic classifications, and there is strong potential that this innovative approach will be adopted by an international community of researchers. HQP who obtain an early start in the proposed program will be well positioned for a productive research career. One of the benefits of the above program is that it lets classical algebraic topology be applied to QGs. Algebraic topology uses functors to prove that certain things cannot happen. For C*-algebras, there exists a Cuntz semigroup functor (CSF). We found important structural properties of the CSF of many QGs and some of their co-actions. This discovery may make the CSF a useful tool for studying the algebraic topology of quantum groups. For extending classical algebraic topology to a quantum dynamical framework, it seems desirable to further develop algebraic topological tools for QG and their co-actions. For example, we can construct quantum group isomorphisms from isomorphisms at the CSF level. This interesting phenomenon has no precise counterpart in classical algebraic topology, and is an explicitly `quantum' phenomenon. Jointly with G.A. Elliott and C. Ivanescu, we are studying the Cuntz semigroup of Villadsen algebras. We aim to classify Villadsen algebras through their Cuntz semigroup. This is an exciting, timely, and more importantly, feasible proposal.

这是一个纯数学的项目。数学是关于连接并从我们拥有有关世界的可用信息中开发出可证明的正确扣除的。从历史上看，这是高等教育本身的代名词。纯数学的进展慢慢地影响了应用科学。事实是，纯粹的数学中的深奥概念现在出现在工程和物理学中。对于学生或HQP，纯数学对复杂主题进行了先进的逻辑思考，并可能应用于复杂的科学问题。因此，该研究计划的好处存在于当今科学技术的前沿，并可能影响明天的科学技术。该程序的基本话题是在希尔伯特领域的运营商的规范锁定自我伴侣代数，称为C*-Algebras。它们具有显着的特性，并连接不同区域，例如功能分析，代数，拓扑，几何和动态系统。在该领域中最重要的一项努力也许是通过K理论数据（Elliott程序）对此类代数进行分类。是时候调查将程序分支到新方向的时候了。 C* - 代数可以看作是拓扑空间的非交通概括，如果我们寻找拓扑组而不是拓扑空间的概括，我们会获得C* - 代数量子基团（QGS），这可能在C*-Algebras上具有共同作用。我们的主要目标是将Elliott计划概括为合适的QGS和共同行动。 QGS的Elliott风格分类与现有代数分类的作用差异很大，并且具有强大的潜力是，这种创新的方法将被国际研究人员采用。在拟议计划中获得早期开始的HQP将在生产研究职业中得到很好的位置。上述程序的好处之一是，它可以将古典代数拓扑应用于QGS。代数拓扑使用函子证明某些事情不可能发生。对于C* - 代数，存在一个Cuntz Semigroup函子（CSF）。我们发现许多QGS及其某些共同行动的CSF的重要结构特性。这一发现可能使CSF成为研究量子组代数拓扑的有用工具。为了将经典代数拓扑扩展到量子动力学框架，似乎希望进一步为QG及其共同行动开发代数拓扑工具。例如，我们可以从CSF级别的同构构建量子组同构。这种有趣的现象在古典代数拓扑中没有精确的对应物，并且是一种明确的“量子”现象。与G.A.共同Elliott和C. Ivanescu，我们正在研究Villadsen代数的Cuntz Semigroup。我们的目标是通过其Cuntz Semigroup对Villadsen代数分类。这是一个令人兴奋的，及时的，更重要的是可行的建议。