Quantum Groups, Quantum Categories and Quantum Field Theory [electronic resource] / by Jürg Fröhlich, Thomas Kerler.

and survey of results -- Local quantum theory with braid group statistics -- Superselection sectors and the structure of fusion rule algebras -- Hopf algebras and quantum groups at roots of unity -- Representation theory of U q red (s? 2) -- Path representations of the braid groups for quantum groups at roots of unity -- Duality theory for local quantum theories, dimensions and balancing in quantum categories -- The quantum categories with a generator of dimension less than two.

This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.