New Directions in Hopf Algebras

This book contains survey articles for graduate students and researchers on various topics surrounding Hopf algebras.

These articles form a broad survey of this exciting field. Topics include pointed Hopf algebras, triangular Hopf algebras, Hopf algebra extensions and cohomology, quantum groups and groupoids, quantum symmetric pairs, monoidal categories, and the Brauer group of a Hopf algebra.Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.