Quantum Symmetry

Temple University, Philadelphia PA

Investigators

Abstract

Investigation of the symmetries of an object is a central question in mathematics and its applications. Mathematically, this is the study of invertible, property-preserving transformations from an object to itself. It is understood that the symmetries of objects we can visualize (for instance, classical objects such as spaces or manifolds or the functions on such objects) form mathematical structures known as groups. On the other hand, only recently has an appropriate notion of symmetry been developed for quantum objects and their noncommutative algebras of functions, which have been ubiquitous in mathematics and physics since the origin of quantum mechanics. It has been discovered that replacing group actions with actions of Hopf algebras is a natural and effective approach. The goal of this research project is to deepen and extend understanding of such quantum symmetries. This project will advance comprehensively the analysis and applications of quantum symmetry, including tackling the basic question: for a given algebra A, when do genuine Hopf algebra actions on A exist? Moreover, ring-theoretic, homological, and representation-theoretic properties of the Hopf algebras (or quantum groups) that "coact universally" on A will be studied. Beyond the setting of Hopf algebra actions, the investigator will employ the framework of tensor categories to study the occurrence of quantum symmetry, as they serve as a natural categorification of Hopf algebras; one benefit of this framework is that it handles actions of generalized (e.g., weak, quasi) Hopf algebras as well.