Contents -- 1 Introduction -- 1.1 The quantum dynamical Yang-Baxter equation -- 1.1.1 The equation -- 1.1.2 Examples of solutions of QDYBE -- 1.1.3 The QDYBE with spectral parameter -- 1.1.4 Tensor category of representations -- 1.1.5 Gauge transformations and classification -- 1.1.6 Dynamical quantum groups -- 1.1.7 The classical dynamical Yang-Baxter equation -- 1.1.8 Examples of solutions of CDYBE -- 1.1.9 Classification of solutions for CDYBE -- 1.2 The fusion and exchange construction -- 1.2.1 Intertwining operators -- 1.2.2 The fusion and exchange operators -- 1.2.3 Fusion and exchange for quantum groups -- 1.2.4 The ABRR equation -- 1.2.5 The universal fusion operator -- 1.2.6 The dynamical twist equation -- 1.3 Traces of intertwiners and Macdonald functions -- 1.3.1 Trace functions -- 1.3.2 Commuting difference operators -- 1.3.3 Difference equations for the trace functions -- 1.3.4 Macdonald functions -- 1.3.5 Dynamical Weyl groups -- 2 Background material -- 2.1 Facts about sl[sub(2)] -- 2.2 Semisimple finite-dimensional Lie algebras and roots -- 2.3 Inner product on a simple Lie algebra -- 2.4 Chevalley generators -- 2.5 Representations of finite-dimensional semisimple Lie algebras -- 2.6 Irreducible highest weight modules -- Shapovalov form -- 3 Intertwiners, fusion and exchange operators for Lie algebras -- 3.1 Intertwining operators -- 3.2 The fusion operator -- 3.3 The dynamical twist equation -- 3.4 The exchange operator -- 3.5 The ABRR equation -- 3.6 The universal fusion and exchange operators -- 4 Quantum groups -- 4.1 Hopf algebras -- 4.2 Representations of Hopf algebras -- 4.3 The quantum group U[sub(q)](sl[sub(2)]) -- 4.4 The quantum group U[sub(q)](g) -- 4.5 PBW for U[sub(q)](g) -- 4.6 The Hopf algebra structure on U[sub(q)](g) -- 4.7 Representation theory of U[sub(q)](g) -- 4.8 Formal version of quantum groups.

4.9 Quasi-triangular Hopf algebras -- 4.10 Quasi-triangular Hopf algebras and representation theory -- 4.11 Quasi-triangularity and U[sub(q)](g) -- 4.12 Twisting -- 4.13 Quasi-classical limit for the QYBE -- 4.14 Quasi-classical limit for the QDYBE -- 5 Intertwiners, fusion and exchange operators for U[sub(q)](g) -- 5.1 Fusion operator for U[sub(q)](g) -- 5.2 Exchange operator for U[sub(q)](g) -- 5.3 The ABRR equation for U[sub(q)](g) -- 5.4 Quasi-classical limit for ABRR equation for U[sub(q)](g) -- 6 Dynamical R-matrices and integrable systems -- 6.1 Classical mechanics vs. quantum mechanics -- 6.2 Transfer matrix construction -- 6.3 Dynamical transfer matrix construction -- 7 Traces of intertwiners for U[sub(q)](g) -- 7.1 Generalized Macdonald-Ruijsenaars operators -- 7.2 Construction of F[sub(V)] (λ, μ) -- 7.3 Quantum spin Calogero-Moser Hamiltonian -- 7.4 F[sub(V)] (λ, μ) for sl[sub(2)] -- 7.5 Center of U[sub(q)](g) and quantum traces -- 7.6 The functions Z[sub(V)] and X[sub(V)] -- 7.7 The function G -- 7.8 Macdonald-Ruijsenaars equations -- 7.9 Dual Macdonald-Ruijsenaars equations -- 7.10 The symmetry identity -- 8 Traces of intertwiners and Macdonald polynomials -- 8.1 Macdonald polynomials -- 8.2 Vector-valued characters -- 9 Dynamical Weyl group -- 9.1 Dynamical Weyl group (for g = sl[sub(2)]) -- 9.2 Dynamical Weyl group (for any finite-dim. simple g) -- References -- Index.