Cover -- Series-title -- Title -- Copyright -- Contents -- Introduction -- Lectures on Cyclotomic Hecke Algebras -- 1 Introduction -- 2 Lecture One -- 2.1 Definitions -- 2.2 Representations -- 2.3 First application -- 3 Lecture Two -- 3.1 Geometric theory -- 3.2 Algorithms -- 3.3 Second application -- 4 Lecture Three -- 4.1 Specht modules and v-deformed Fock spaces -- 4.2 Future direction and Broué's dream -- References -- AN INTRODUCTION TO GROUP DOUBLECROSS PRODUCTS AND SOME USES -- 1 INTRODUCTION -- 2 Group doublecross product -- 3 Hopf algebra bicrossproducts -- 4 Integrable models in field theory -- 5 T-duality in sigma models -- References -- Canonical Bases and Piecewise-linear Combinatorics -- 1 Introduction -- 2 PBW-type Bases and the Canonical Basis -- 3 Rectangle Calculus -- 4 Speculations for type A -- References -- INTEGRABLE AND WEYL MODULES FOR QUANTUM AFFINE… -- 0. INTRODUCTION -- 1. PRELIMINARIES AND SOME IDENTITIES -- 2. THE MODULES… -- 3. PROOF OF LEMMAS 2.1 AND 2.2 -- 4. THE GENERAL CASE: A CONJECTURE -- REFERENCES -- Notes on Balanced Categories and Hopf Algebras -- Introduction -- Preliminaries -- 1 Balanced Hopf Algebras -- 2 Balanced and Sovereign Categories -- 3 Ribbon Braids and Ribbon Links -- Ribbon Braid Group -- Ribbon Braids and Ribbon Links -- References -- Lectures on the dynamical Yang-Baxter equations -- 1 Introduction -- 2 Intertwining operators, fusion and exchange matrices. -- 3 The dynamical Yang-Baxter equations. -- 4 Classification of classical dynamical r-matrices. -- 5 Classical dynamical r-matrices and Poisson-Lie groupoids -- 6 Classification of quantum dynamical R-matrices -- 7 Quantum dynamical R-matrices and quantum groupoids -- 8 The universal fusion matrix and the Arnaudon-Buffenoir-Ragoucy-Roche equation -- 9 Transfer matrices and Generalized Macdonald-Ruijsenaars equations.

10 Appendix A: Classical dynamical r-matrices on a simple Lie algebra g with respect to… -- 11 Appendix B: Proof of the ABRR relation for… -- 12 Review of literature -- References -- QUANTIZED PRIMITIVE IDEAL SPACES AS QUOTIENTS OF AFFINE ALGEBRAIC VARIETIES -- INTRODUCTION -- 1. QUANTUM SEMISIMPLE GROUPS -- 2. QUANTUM TORI -- REFERENCES -- REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS IN POSITIVE CHARACTERISTIC AND QUANTUM GROUPS AT ROOTS OF UNITY -- 1. INTRODUCTION -- 2. GENERALITIES -- 3. ENVELOPING ALGEBRAS -- 4. QUANTISED ENVELOPING ALGEBRAS -- 5. QUANTISED FUNCTION ALGEBRAS AND QUANTUM BORELS -- APPENDIX A -- REFERENCES -- THE YANG-BAXTER EQUATION FOR OPERATORS ON FUNCTION FIELDS -- INTRODUCTION -- 1. THE CONSTANT YANG-BAXTER EQUATION ON RATIONAL FUNCTION FIELDS -- REFERENCES -- Noncommutative Differential Geometry and Twisting of Quantum Groups -- 1 Introduction -- 2 Quantum differential forms -- 2.1 M = k [x] -- 2.2 M = C[G] -- 2.3 M = CG -- 2.4 Proofs -- 2.5 Cotriangular quantum groups and twisting of calculi -- 2.6 Factorisable quantum groups… -- 2.7 Discrete manifolds -- 3 Bundles and connections -- 4 Quantum soldering forms and metrics -- Acknowledgements -- References -- Finite Quantum Groups and Pointed Hopf Algebras -- 1. POINTED HOPF ALGEBRAS AS IMAGES OF QUANTIZED ENVELOPING ALGEBRAS. -- 2. THE LIFTING PROBLEM. -- References -- ON SOME TWO PARAMETER QUANTUM AND JORDANIAN DEFORMATIONS, AND THEIR COLOURED EXTENSIONS* -- I Introduction -- II Two parameter q- deformations -- III Two parameter h- deformations -- IV Coloured Extensions -- V Contractions and Homomorphisms -- VI Conclusions -- Acknowledgments -- References -- TENSOR CATEGORIES AND BRAID REPRESENTATIONS -- 1. LIE TYPE A -- 2. CENTRALIZER ALGEBRAS FOR LIE TYPES B, C AND D -- 3. EXCEPTIONAL GROUPS -- REFERENCES.