# Quantum Groups

This book contains four contributions on various aspects of quantum group theory. The first one contains lecture notes from the course given by P. Etingof on the structure and classification of tensor categories and is written by D. Calaque. It introduces main definitions, basic results and important examples. It contains a discussion of Ocneanu rigidity for fusion categories, module categories and weak Hopf algebras, module categories and their Morita equivalence, applications to representation theory of groups, the lifting theory, and the theory of Frobenius-Perron dimension and its application for classification of fusion categories.

There are three more research papers in the book. The one written by J. Lieberum discusses Drinfeld associators. They are needed for the Kontsevich construction of a universal Vassiliev invariant. The paper contains an explicit description of a rational even Drinfeld associator in (a completion of) the universal enveloping algebra of the Lie superalgebra gl(1|1) and a discussion of its relation with the Viro generalisation of the multivariable Alexander polynomial. The contribution by A. Odesskii and Vl. Rubtsov contains a description of the relation between integrable systems and elliptic algebras. They construct (classical and quantum) integrable systems on a large class of elliptic algebras. The last paper by N. Andruskiewitsch and F. Dumas contains a computation of the group of automorphisms of the positive part of the quantum enveloping algebra of certain simple (complex) finite dimensional Lie algebras.

**Submitted by Anonymous |

**30 / May / 2011