## Lectures on Quantum GroupsThe Graduate Studies in Mathematics series is made up of books useful as graduate-level course texts. This book is an introduction to the theory of quantum groups. The main topic is the quantized enveloping algebras introduced independently by Drinfeld and Jimbo. Jantzen considers the crystal (or canonical) bases discovered independently by Lusztig and Kashiwara and looks at the quantum analogue of the Lie algebar SL and then at the quantum analogue of arbitrary finite dimensional complex Lie algebras. |

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### Contents

5 | |

9 | |

17 | |

Tensor Products or Uqsl2 as a Hopf Algebra | 31 |

The Quantized Enveloping Algebra Uqg | 51 |

Representations of Uqg | 69 |

Examples of Representations | 87 |

The Center and Bilinear Forms | 105 |

Rmatrices and kqG | 129 |

Braid Group Actions and PBW Type Basis | 141 |

Proof of Proposition 828 | 173 |

Crystal Bases I | 187 |

Crystal Bases II | 217 |

Crystal Bases III | 237 |

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### Common terms and phrases

A C A a t/-module A,av adjoint admissible lattice analogue antiautomorphism apply arbitrary assume assumption automorphism B(oo basis bijective bilinear form Chapter claim follows claim holds commutes Consider construction crystal base crystal graph decomposition define definition denote direct sum dominant weight ea(b element endomorphisms equal equation finite dimensional t/-module formulas Furthermore Gaussian binomial hence Hopf algebra implies induction integers isomorphism kq[G left hand side Lemma Let A E A Lie algebra linearly independent Lusztig Lz(A multiplication nonzero notations obvious polynomial PROOF Proposition prove q is transcendental reduced expression relations REMARK representation resp right hand side root of unity satisfies scalar semisimple short root similarly simple modules simple roots span subalgebra submodule subsection subspace summand Suppose tensor product Theorem transcendental over Q trivial U-module unique Uq(Q vector space weight spaces Weyl group yields

### Popular passages

Page 18 - An endomorphism of a finite dimensional vector space is diagonalizable if and only if its minimal polynomial splits into linear factors, each occurring with multiplicity 1.

Page 18 - Suppose that q is not a root of unity and that char(fc) ^ 2. Let M be a finite dimensional U-module. Then M is the direct sum of its weight spaces.