## Lectures on Representation Theory and Knizhnik-Zamolodchikov EquationsThis text is devoted to mathematical structures arising in conformal field theory and the q-deformations. The authors give a self-contained exposition of the theory of Knizhnik-Zamolodchikov equations and related topics. No previous knowledge of physics is required. The text is suitable for a one-semester graduate course and is intended for graduate students and research mathematicians interested in mathematical physics. |

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### Contents

Lecture 1 Introduction | 1 |

Lecture 2 Representations of finitedimensional and affine Lie algebras | 15 |

Lecture 3 KnizhnikZamolodchikov equations | 29 |

Lecture 4 Solutions of the Knizhnik Zamolodchikov equations | 49 |

Lecture 5 Free field realization | 63 |

Lecture 6 Quantum groups | 79 |

Lecture 7 Local systems and configuration spaces | 97 |

Lecture 8 Monodromy of KnizhnikZamolodchikov equations | 113 |

Lecture 9 Quantum affine algebras | 131 |

Lecture 10 Quantum KnizhnikZamolodchikov equations | 147 |

Lecture 11 Solutions of the quantum KnizhnikZamolodchikov equations for slsub2 | 161 |

Lecture 12 Connection matrices for the quantum KnizhnikZamolodchikov equations and elliptic functions | 171 |

Lecture 13 Current developments and future perspectives | 179 |

References | 189 |

197 | |

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### Common terms and phrases

affine Lie algebras algebra g analogue analytic continuation arbitrary asymptotics axioms basis braid group braided tensor category called category C(g classical coefficients cohomology commutation relations complex number comultiplication conformal field theory connection matrices consider construction COROLLARY correlation functions corresponding defined definition denote difference equations differential equations Drinfeld dual element evaluation representations factors finite-dimensional representations free field realization fundamental solution g-modules highest-weight module highest-weight vector homomorphism Hopf algebra hypergeometric function identity integral formulas intertwining operators irreducible isomorphism Knizhnik-Zamolodchikov equations Lecture lemma lig(g linear lowest-weight manifold Mg(g monodromy morphism normal ordered product notation obtained parameters proof PROPOSITION proved q-analogue q-deformation quantum affine algebras quantum Knizhnik-Zamolodchikov equations quantum KZ equations R-matrix rational function Recall representation theory representations of g root Section simple Lie algebra subalgebra tensor category tensor product THEOREM tion trigonometric unique Varchenko variables vector space Verma module Virasoro algebra Yang-Baxter equation zero