Weak Hopf Algebras and Dynamical Twisting of Quantum Groups

University Of New Hampshire, Durham NH

Investigators

Abstract

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 is equaivalent to the representation category of a semisimple weak Hopf algebra). One goal is to extend the results known for the representation categories of ordinary semisimple Hopf algebras, such as e.g., Larson-Radford theorems, Stefan's finiteness result (Ocneanu's rigidity), and Class Equation to general semisimple tensor categories. Another goal is to use weak Hopf algebra techniques to reveal the structure of modular categories which are important because of their applications to physics and geometry. The investigator also 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 of groups and quantum physics. The theory of groups is the classical mathematical subject that describes in mathematical terms the phenomenon of symmetry (e.g., groups describe the symmetry of molecular structure). Groups are fundamental objects 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 many technological advances of the 20th century such as, e.g., development of the nuclear 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.