## Foundations of Quantum Group TheoryThis is a graduate-level text that systematically develops the foundations of the subject. Quantum groups (i.e. Hopf algebras) are treated as mathematical objects in their own right; basic properties and theorems are proven in detail from this standpoint, including the results underlying key applications. After formal definitions and basic theory, the book goes on to cover such topics as quantum enveloping algebras, matrix quantum groups, combinatorics, cross products of various kinds, the quantum double, the semiclassical theory of Poisson-Lie groups, the representation theory, braided groups and applications to q-deformed physics. The explicit proofs and a great many worked examples and exercises will allow readers to quickly pick up the techniques needed for working in this exciting new field. |

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### Contents

vii | |

II | xi |

III | 1 |

IV | 5 |

V | 6 |

VI | 12 |

VII | 15 |

VIII | 17 |

XLIII | 225 |

XLIV | 239 |

XLV | 262 |

XLVI | 286 |

XLVII | 297 |

XLVIII | 302 |

XLIX | 304 |

L | 314 |

IX | 22 |

X | 28 |

XI | 36 |

XII | 38 |

XIII | 39 |

XIV | 50 |

XV | 55 |

XVI | 65 |

XVII | 70 |

XVIII | 72 |

XIX | 73 |

XX | 85 |

XXI | 95 |

XXII | 102 |

XXIII | 105 |

XXIV | 108 |

XXV | 111 |

XXVI | 128 |

XXVII | 143 |

XXVIII | 149 |

XXIX | 153 |

XXXI | 163 |

XXXII | 166 |

XXXIII | 173 |

XXXIV | 178 |

XXXV | 180 |

XXXVI | 191 |

XXXVII | 192 |

XXXVIII | 197 |

XXXIX | 204 |

XL | 209 |

XLI | 220 |

XLII | 223 |

### Common terms and phrases

2-cocycle adjoint action alge algebra homomorphism algebra map algebra of observables algebra structure analogous antipode axioms bialgebra bialgebra or Hopf bicrossproduct bosonisation braid statistics braided category braided covector braided groups braided matrices Chapter classical coaction coadjoint coalgebra cobracket cocommutative cocycle comodule compute condition construction coproduct corresponding counit covariant covector cross product cross relations defined definition deformation denote double cross product dual quasitriangular structure duality element enveloping algebra Example extended factorisable finite finite-dimensional formulae functions functor g-deformed generalisation gives Hence inverse isomorphism Lemma Lie algebra Lie bracket linear map matched pair matrix module algebra monoidal category morphism noncommutative normalisation notation obeys obtain point of view product algebra Proof properties Proposition quantisation quantum double quantum group quantum matrices quasitriangular Hopf algebra quotient QYBE R-matrix random walk representation right action Section self-dual sense subalgebra tensor product Theorem theory twisting Uq(g usual vector space verify