Weak Hopf Algebras and Dynamical Twisting of Quantum Groups

本课题的主要研究对象是弱Hopf代数(在文献中也称为量子群胚)。它们是类群体，推广了Hopf代数和通常的群胚，自然地出现在半单张量刚性(融合)范畴理论、共形场理论、代数扩张的对称性和量子动力学的Yang-Baxter方程中。研究人员发展了弱Hopf代数及其表示的一般理论，并将其应用于融合范畴的研究和分类(每个弱Hopf代数等价于半单弱Hopf代数的表示范畴)，一个目的是将已知的关于二元半单Hopf代数的表示范畴的结果推广到一般的半单张量范畴，如Larson-Radford定理、Stefan的有限性结果(Ocneanu刚性)和类方程.另一个目标是使用弱Hopf代数技术来揭示模范畴的结构，这是因为它们在物理和几何中的应用是重要的。研究人员还研究了由动力学扭曲引起的弱Hopf代数(这些扭曲导致了量子动力学Yang-Baxter方程的解)。本文的目的是从概念上解释与动力学扭度相关的新的有趣现象，例如，群代数中的非极小动力学扭度和动力学量子群的自对偶性。该理论创立于20世纪80年代，起源于经典的群论和量子物理。群论是经典的数学学科，用数学术语描述对称性现象(例如，群描述分子结构的对称性)。群是基本的对象，因为它们提供了在所有数学领域使用的通用语言。量子理论揭示了决定非常小粒子的行为和它们之间的相互作用的物理定律。这一理论导致了20世纪的许多技术进步，例如核物理的发展。为了对量子物理系统的结构给予充分的数学描述，经典群理论是不够的，这就是为什么新的量子群理论被发明的原因。在这些新对象中，弱Hopf代数和量子群胚的概念特别令人感兴趣，因为它们最近在数学-物理分裂的两边都有许多应用。本建议致力于弱Hopf代数的研究和分类，并探索它们在数学和物理的各个领域中的应用。

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.