Quantum GroupsThis book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on Drinfeld's recent fundamental contributions. The first part presents in detail the quantum groups attached to SL[subscript 2] as well as the basic concepts of the theory of Hopf algebras. Part Two focuses on Hopf algebras that produce solutions of the Yang-Baxter equation, and on Drinfeld's quantum double construction. In the following part we construct isotopy invariants of knots and links in the three-dimensional Euclidean space, using the language of tensor categories. The last part is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations, culminating in the construction of Kontsevich's universal knot invariant. |
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A-linear A-Mod A-module a₁ adjoint algebra morphism antipode automorphism basis bialgebra braid group braided quasi-bialgebra braided tensor category Chapter chord diagram coalgebra cocommutative commutes comodule comultiplication consequence Corollary counit crossing point defined definition denote Drinfeld dual endomorphism equivalent exists a unique Figure finite finite-dimensional formal series gauge transformation given highest weight vector Hopf algebra Hopf algebra structure implies induces integer inverse limit isotopy invariant knot Lemma linear map link diagram M₁ Math module monodromy morphism morphism of algebras natural isomorphism objects polynomial proof of Theorem Proposition quantum double quantum groups quasi-bialgebra R-matrix Reidemeister Transformation Relation representation resp ribbon category satisfying scalar Section semisimple Lie algebra strict tensor category submodule subspace surjective symmetric tangle diagrams tensor category tensor product topological braided trivial universal R-matrix V₁ V₂ vector space Yang-Baxter equation