Weak Hopf Algebras and Dynamical Twisting of Quantum Groups

弱Hopf代数与量子群的动态扭曲

• 批准号: 0200202
• 负责人: Dmitri Nikshych
• 金额: $ 9万
• 依托单位: University of New Hampshire
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200202&HistoricalAwards=false
• 关键词: Weak; Hopf; Algebras; Dynamical; Twisting

The main objects of investigation of this project are weak Hopf algebras(also called quantum groupoids in literature). These are group-like objects that generalize both Hopf algebras and usual groupoids and that appear naturally in the theory of semisimple tensor rigid (fusion) categories, conformal field theory, symmetries of algebra extensions, and the quantum dynamical Yang-Baxter equation. The investigator develops the general theory of weak Hopf algebras and their representations and applies it to the study and classification of fusion categories (every such a category isequaivalent to the representation category of a semisimple weak Hopf algebra).One goal is to extend the results known for the representation categories ofordinary semisimple Hopf algebras, such as e.g., Larson-Radford theorems, Stefan's finiteness result (Ocneanu's rigidity), and Class Equation to generalsemisimple tensor categories. Another goal is to use weak Hopf algebratechniques to reveal the structure of modular categories which are importantbecause of their applications to physics and geometry. The investigatoralso studies weak Hopf algebras arising from dynamical twists (these twists give rise to solutions of the quantum dynamical Yang-Baxter equation of Felder). The goal here is to get a conceptual explanation of new intriguing phenomena related to dynamical twists, such as, e.g., non-minimizable dynamical twists in group algebras and self-duality of dynamical quantum groups.In this proposal the investigator studies problems of the theory of quantum groups. The origins of this theory which was created in 1980s are the classical theory ofgroups and quantum physics. The theory of groups is the classical mathematicalsubject that describes in mathematical terms the phenomenon of symmetry(e.g., groups describe the symmetry of molecular structure). Groups are fundamentalobjects as they provide a universal language used in all areas of mathematics.The quantum theory reveals the laws of physics that determine the behavior of very small particles and interactions between them. This theory led to manytechnological advances of the 20th century such as, e.g., development of thenuclear physics. In order to give an adequate mathematical description of the structure of quantum physical systems the theory of classical groups is not sufficient, this is why the new theory of quantum groups was invented. Among these new objects the notions of weak Hopf algebras and quantum groupoids are of special interest, as they have recently seen many applications on both sides of the mathematics-physics divide. The present proposal is devoted to the study and classification of weak Hopf algebras and exploring their applications to various areas of mathematics and physics.

本项目的主要研究对象是弱Hopf代数（文献中也称为量子群群）。这些是类群对象，它们概括了 Hopf 代数和通常的群群，并且自然地出现在半单张量刚性（融合）范畴、共形场论、代数扩展的对称性和量子动力学 Yang-Baxter 方程的理论中。研究人员发展了弱Hopf代数及其表示的一般理论，并将其应用于融合类别的研究和分类（每个这样的类别都相当于半简单弱Hopf代数的表示类别）。一个目标是扩展已知的结果对于普通半简单 Hopf 代数的表示类别，例如 Larson-Radford 定理、Stefan 的有限性结果（Ocneanu 的刚性），以及一般半简单张量类别的类方程。另一个目标是使用弱 Hopf 代数技术来揭示模范畴的结构，这很重要，因为它们在物理和几何中的应用。研究人员还研究了由动力学扭曲产生的弱霍普夫代数（这些扭曲产生了费尔德的量子动力学杨-巴克斯特方程的解）。这里的目标是获得与动态扭曲相关的新有趣现象的概念解释，例如群代数中的不可最小化动态扭曲和动态量子群的自对偶性。在这个提案中，研究者研究了理论问题的量子群。这个产生于20世纪80年代的理论起源于经典群论和量子物理。群论是一门经典的数学学科，它用数学术语描述对称现象（例如群描述分子结构的对称性）。群是基本对象，因为它们提供了数学所有领域使用的通用语言。量子理论揭示了决定非常小的粒子的行为以及它们之间相互作用的物理定律。 这一理论带来了 20 世纪的许多技术进步，例如核物理学的发展。为了对量子物理系统的结构给出充分的数学描述，经典群理论是不够的，这就是发明新的量子群理论的原因。在这些新物体中，弱霍普夫代数和量子群群的概念特别令人感兴趣，因为它们最近在数学和物理鸿沟的两侧都有许多应用。 目前的提案致力于弱Hopf代数的研究和分类，并探索它们在数学和物理各个领域的应用。