# The Dynamical Yang-Baxter Equation, Representation Theory, and Quantum Integrable Systems

This nice small book, based on a course given at MIT in 2001, is devoted to connections of quantum dynamical Yang-Baxter equations with certain integrable systems and representations of semisimple Lie algebras and of quantum groups. A condense summary of these interrelations can be found in the introductory chapter of the book. To make it more accessible, the authors review basic facts on finite-dimensional representations of semisimple Lie algebras and the generalization to the case of quantum groups. The main topics treated in the book are intertwining operators, fusion operators and exchange operators. It leads quickly to quantum dynamical Yang-Baxter equation and the ABRR equation. The transfer matrix construction shows how to create a quantum integrable system corresponding to an R-matrix. In particular, it leads to integrable systems of Macdonald-Ruijsenaars type. It is then possible to give an interpretation of certain Macdonald polynomials in terms of representation theory. The book ends with a description of the theory of the dynamical Weyl group for a semisimple Lie algebra. To read the book, it is helpful to understand the theory of finite-dimensional representations of semisimple Lie algebras (a short summary is included). The contents of the book is very nicely presented and the book offers a readable introduction to a very interesting, interdisciplinary field.

**Submitted by Anonymous |

**21 / Oct / 2011