This volume presents articles from several lectures presented at the school on "Quantum Symmetries in Theoretical Physics and Mathematics" held in Bariloche, Argentina. The various lecturers provided significantly different points of view on several aspects of Hopf algebras, quantum group theory, and noncommutative differential geometry, ranging from analysis, geometry, and algebra to physical models, especially in connection with integrable systems and conformal field theories.

Primary topics discussed in the text include subgroups of quantum \(SU(N)\), quantum ADE classifications and generalized Coxeter systems, modular invariance, defects and boundaries in conformal field theory, finite dimensional Hopf algebras, Lie bialgebras and Belavin-Drinfeld triples, real forms of quantum spaces, perturbative and non-perturbative Yang-Baxter operators, braided subfactors in operator algebras and conformal field theory, and generalized (\(d^N\)) cohomologies.

Readership

Graduate students, research mathematicians, and physicists interested in Hopf algebras, quantum groups, Von Neumann algebras, subfactors and noncommutative differential geometry or on the connections between the previous concepts and conformal field theories or integrable systems.

Table of Contents

• N. Andruskiewitsch -- About finite dimensional Hopf algebras
• M. Dubois-Violette -- Lectures on differentials, generalized differentials and on some examples related to theoretical physics
• J. Böckenhauer and D. E. Evans -- Modular invariants from subfactors
• A. Ocneanu -- The classification of subgroups of quantum \(SU(N)\)
• O. Ogievetsky -- Uses of quantum spaces
• J.-B. Zuber -- CFT, BCFT, ADE and all that