| Preface |
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| Part One Quantum SL(2) |
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1 | (164) |
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3 | (20) |
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3 | (4) |
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7 | (1) |
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3 The Affine Line and Plane |
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8 | (2) |
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10 | (1) |
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5 Determinants and Invertible Matrices |
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10 | (2) |
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6 Graded and Filtered Algebras |
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12 | (2) |
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14 | (4) |
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18 | (2) |
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20 | (2) |
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22 | (1) |
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23 | (16) |
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1 Tensor Products of Vector Spaces |
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23 | (3) |
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2 Tensor Products of Linear Maps |
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26 | (3) |
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29 | (3) |
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4 Tensor Products of Algebras |
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32 | (2) |
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5 Tensor and Symmetric Algebras |
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34 | (2) |
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36 | (2) |
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38 | (1) |
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III The Language of Hopf Algebras |
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39 | (33) |
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39 | (6) |
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45 | (4) |
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49 | (8) |
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4 Relationship with
Chapter I. The Hopf Algebras GL(2) and SL(2) |
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57 | (1) |
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5 Modules over a Hopf Algebra |
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57 | (4) |
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61 | (3) |
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7 Comodule-Algebras. Coaction of SL(2) on the Affine Plane |
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64 | (2) |
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66 | (4) |
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70 | (2) |
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IV The Quantum Plane and Its Symmetries |
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72 | (21) |
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72 | (2) |
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2 Gauss Polynomials and the q-Binomial Formula |
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74 | (3) |
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77 | (4) |
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4 Ring-Theoretical Properties of Mq(2) |
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81 | (1) |
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5 Bialgebra Structure on Mq(2) |
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82 | (1) |
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6 The Hopf Algebras GLq(2) and SLq(2) |
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83 | (2) |
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7 Coaction on the Quantum Plane |
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85 | (1) |
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86 | (2) |
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88 | (2) |
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90 | (3) |
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V The Lie Algebra of SL(2) |
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93 | (28) |
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93 | (1) |
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94 | (5) |
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99 | (2) |
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4 Representations of sl(2) |
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101 | (4) |
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5 The Clebsch-Gordan Formula |
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105 | (2) |
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6 Module-Algebra over a Bialgebra. Action of sl(2) on the Affine Plane |
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107 | (2) |
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7 Duality between the Hopf Algebras U(sl(2)) and SL(2) |
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109 | (8) |
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117 | (2) |
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119 | (2) |
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VI The Quantum Enveloping Algebra of sl(2) |
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121 | (19) |
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121 | (4) |
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2 Relationship with the Enveloping Algebra of sl(2) |
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125 | (2) |
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127 | (3) |
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4 The Harish-Chandra Homomorphism and the Centre of Uq |
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130 | (4) |
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5 Case when q is a Root of Unity |
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134 | (4) |
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138 | (1) |
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138 | (2) |
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VII A Hopf Algebra Structure on Uq(sl(2)) |
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140 | (25) |
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140 | (3) |
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143 | (3) |
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3 Action of Uq(sl(2)) on the Quantum Plane |
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146 | (4) |
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4 Duality between the Hopf Algebras Uq(sl(2)) and SLq(2) |
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150 | (4) |
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5 Duality between Uq(sl(2))-Modules and SLq(2)-Comodules |
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154 | (1) |
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6 Scalar Products on Uq(sl(2))-Modules |
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155 | (2) |
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157 | (5) |
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162 | (1) |
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163 | (2) |
| Part Two Universal R-Matrices |
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165 | (74) |
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VIII The Yang-Baxter Equation and (Co)Braided Bialgebras |
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167 | (32) |
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1 The Yang-Baxter Equation |
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167 | (5) |
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172 | (6) |
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3 How a Braided Bialgebra Generates R-Matrices |
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178 | (1) |
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4 The Square of the Antipode in a Braided Hopf Algebra |
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179 | (5) |
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5 A Dual Concept: Cobraided Bialgebras |
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184 | (4) |
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188 | (6) |
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7 Application to GLq(2) and SLq(2) |
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194 | (2) |
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196 | (2) |
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198 | (1) |
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IX Drinfeld's Quantum Double |
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199 | (40) |
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1 Bicrossed Products of Groups |
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199 | (3) |
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2 Bicrossed Products of Bialgebras |
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202 | (5) |
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3 Variations on the Adjoint Representation |
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207 | (6) |
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4 Drinfeld's Quantum Double |
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213 | (7) |
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5 Representation-Theoretic Interpretation of the Quantum Double |
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220 | (3) |
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6 Application to Uq(sl(2)) |
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223 | (7) |
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230 | (6) |
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236 | (2) |
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238 | (1) |
| Part Three Low-Dimensional Topology and Tensor Categories |
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239 | (144) |
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X Knots, Links, Tangles, and Braids |
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241 | (34) |
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242 | (2) |
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2 Classification of Links up to Isotopy |
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244 | (2) |
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246 | (6) |
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4 The Jones-Conway Polynomial |
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252 | (5) |
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257 | (5) |
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262 | (7) |
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269 | (1) |
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270 | (3) |
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9 Appendix. The Fundamental Group |
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273 | (2) |
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275 | (19) |
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1 The Language of Categories and Functors |
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275 | (6) |
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281 | (3) |
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3 Examples of Tensor Categories |
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284 | (3) |
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287 | (1) |
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5 Turning Tensor Categories into Strict Ones |
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288 | (3) |
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291 | (2) |
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293 | (1) |
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294 | (20) |
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1 Presentation of a Strict Tensor Category |
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294 | (5) |
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2 The Category of Tangles |
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299 | (3) |
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3 The Category of Tangle Diagrams |
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302 | (3) |
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4 Representations of the Category of Tangles |
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305 | (6) |
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5 Existence Proof for Jones-Conway Polynomial |
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311 | (2) |
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313 | (1) |
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313 | (1) |
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314 | (25) |
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1 Braided Tensor Categories |
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314 | (7) |
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321 | (1) |
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3 Universality of the Braid Category |
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322 | (8) |
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4 The Centre Construction |
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330 | (3) |
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5 A Categorical Interpretation of the Quantum Double |
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333 | (4) |
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337 | (1) |
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338 | (1) |
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XIV Duality in Tensor Categories |
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339 | (29) |
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1 Representing Morphisms in a Tensor Category |
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339 | (3) |
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342 | (6) |
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348 | (6) |
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4 Quantum Trace and Dimension |
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354 | (4) |
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5 Examples of Ribbon Categories |
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358 | (3) |
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361 | (4) |
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365 | (1) |
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366 | (2) |
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368 | (15) |
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368 | (3) |
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2 Braided Quasi-Bialgebras |
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371 | (1) |
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372 | (5) |
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4 Braid Group Representations |
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377 | (2) |
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379 | (2) |
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381 | (1) |
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381 | (2) |
| Part Four Quantum Groups and Monodromy |
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383 | (123) |
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XVI Generalities on Quantum Enveloping Algebras |
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385 | (18) |
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1 The Ring of Formal Series and h-Adic Topology |
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385 | (3) |
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2 Topologically Free Modules |
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388 | (2) |
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3 Topological Tensor Product |
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390 | (2) |
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392 | (3) |
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5 Quantum Enveloping Algebras |
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395 | (3) |
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6 Symmetrizing the Universal R-Matrix |
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398 | (2) |
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400 | (1) |
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401 | (1) |
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9 Appendix. Inverse Limits |
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401 | (2) |
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XVII Drinfeld and Jimbo's Quantum Enveloping Algebras |
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403 | (17) |
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1 Semisimple Lie Algebras |
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403 | (3) |
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2 Drinfeld-Jimbo Algebras |
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406 | (4) |
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3 Quantum Group Invariants of Links |
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410 | (2) |
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412 | (6) |
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418 | (1) |
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418 | (2) |
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XVIII Cohomology and Rigidity Theorems |
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420 | (29) |
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1 Cohomology of Lie Algebras |
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420 | (4) |
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2 Rigidity for Lie Algebras |
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424 | (3) |
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3 Vanishing Results for Semisimple Lie Algebras |
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427 | (3) |
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4 Application to Drinfeld-Jimbo Quantum Enveloping Algebras |
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430 | (1) |
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5 Cohomology of Coalgebras |
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431 | (3) |
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6 Action of a Semisimple Lie Algebra on the Cobar Complex |
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434 | (1) |
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7 Computations for Symmetric Coalgebras |
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435 | (7) |
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8 Uniqueness Theorem for Quantum Enveloping Algebras |
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442 | (4) |
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446 | (1) |
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446 | (1) |
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11 Appendix. Complexes and Resolutions |
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447 | (2) |
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XIX Monodromy of the Knizhnik-Zamolodchikov Equations |
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449 | (35) |
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449 | (2) |
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2 Braid Group Representations from Monodromy |
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451 | (4) |
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3 The Knizhnik-Zamolodchikov Equations |
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455 | (3) |
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4 The Drinfeld-Kohno Theorem |
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458 | (3) |
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5 Equivalence of Uh(g) and Ag,t |
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461 | (2) |
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463 | (5) |
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7 Construction of the Topological Braided Quasi-Bialgebra Ag,t |
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468 | (3) |
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8 Verification of the Axioms |
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471 | (8) |
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479 | (1) |
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479 | (1) |
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11 Appendix. Iterated Integrals |
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480 | (4) |
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XX Postlude. A Universal Knot Invariant |
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484 | (22) |
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1 Knot Invariants of Finite Type |
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484 | (2) |
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2 Chord Diagrams and Kontsevich's Theorem |
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486 | (5) |
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3 Algebra Structures on Chord Diagrams |
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491 | (3) |
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4 Infinitesimal Symmetric Categories |
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494 | (2) |
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5 A Universal Category for Infinitesimal Braidings |
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496 | (2) |
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6 Formal Integration of Infinitesimal Symmetric Categories |
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498 | (1) |
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7 Construction of Kontsevich's Universal Invariant |
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499 | (3) |
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8 Recovering Quantum Group Invariants |
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502 | (3) |
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505 | (1) |
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505 | (1) |
| References |
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506 | (17) |
| Index |
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523 | |
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