Factorizable Sheaves and Quantum Groups [Paperback]

The book is devoted to the geometrical construction of the representations of Lusztig's small quantum groups at roots of unity. These representations are realized as some spaces of vanishing cycles of perverse sheaves over configuration spaces. As an application, the bundles of conformal blocks over the moduli spaces of curves are studied. The book is intended for specialists in group representations and algebraic geometry.TABLE OF CONTENTS Introduction 1 Acknowledgement 5 Part 0. OVERVIEW 1. Introduction Chapter 1. Local ????? 2. The category $?CC$ 11 3. Braiding local systems 15 4. Factorizable sheaves 19 5. Tensor product 20 6. Vanishing cycles 23 Chapter 2. Global (genus 0) 7. Cohesive local systems 27 8. Gluing 30 9. Semi-infinite cohomology 31 Bindestrich einfuegen, auch im Ms 10. Conformal blocks (genus 0) 33 11. Integration 35 12. Regular representation 36 13. Regular sheaf 37 Chapter 3. Modular 14. Heisenberg local system 40 15. Fusion structures on $?FS$ 46 16. Conformal blocks (higher genus) 48 Part I. INTERSECTION COHOMOLOGY OF REAL ARRANGEMENTS 1. Introduction 50 2. Topological preliminaries 51 3. Vanishing cycles functors 55 4. Computations for standard sheaves 63 Part II. CONFIGURATION SPACES AND QUANTUM GROUPS 1. Introduction 71 Chapter 1. Algebraic discussion 2. Free algebras and bilinear forms 73 3. Hochschild complexes 82 4. Symmetrization 84 5. Quotient algebras 87 Chapter 2. Geometric discussion 6. Diagonal stratification and related algebras 89 7. Principal stratification 94 8. Standard sheaves 100 Chapter 3. Fusion 9. Additivity theorem 110 10. Fusion and tensor products 112 Chapter 4. Category $?CC$ 11. Simply laced case 116 12. Non-simply laced case 119 Part III. TENSOR CATEGORIES ARISING FROM CONFIGURATION SPACES 1. Introduction 122 Chapter 1. Category $?FS$ 2. Space $?CA$ 124 3. Braiding local system $?CI$ 126 4. Factorizable sheaves 128 5. Finite sheaves 130 6. Standard sheaves 132 Chapter 2. Tensor structure 7. Marked disk operad 134 8. l“8