| preface part one quantum sl(2) i preliminaries 1 algebras and modules 2 free algebras 3 the affine line and plane 4 matrix multiplication 5 determinants and invertible matrices 6 graded and filtered algebras 7 ore extensions 8 noetherian rings 9 exercises 10 notes ii tensor products 1 tensor products of vector spaces 2 tensor products of linear maps 3 duality and traces 4 tensor products of algebras 5 tensor and symmetric algebras .6 exercises 7 notes iii the language of hopf algebras 1 coalgebras 2 bialgebras 3 hopf algebras 4 relationship with chapter i. the hopf algebras gl(2) and sl(2) 5 modules over a hopf algebra 6 comodules 7 comodule-algebras. coaction of sl(2) on the affine plane 8 exercises 9 notes iv the quantum plane and its symmetries i the quantum plane 2 gauss polynomials and the q-binomial formula 3 the algebra mq(2) 4 ring-theoretical properties of mq(2) 5 bialgebra structure on mq(2) 6 the hopf algebras glq(2) and slq(2) 7 coaction on the quantum plane 8 hopf *-algebras 9 exercises 10 notes v the lie algebra of sl(2) 1 lie algebras 2 enveloping algebras 3 the lie algebra sl(2) 4 representations of sl(2) 5 the clebsch-gordan formula 6 module-algebra over a bialgebra. action of sl(2) on the affine plane 7 duality between the hopf algebras u(sl(2)) and sl(2) 8 exercises 9 notes vi the quantum enveloping algebra of 5[(2) 1 the algebra uq(sl(2)) 2 relationship with the enveloping algebra of 5[(2) 3 representations of uq 4 the harish-chandra homomorphism and the centre of uq 5 case when q is a root of unity 6 exercises 7 notes vii a hopf algebra structure on uq(sl(2)) 1 comultiplication 2 semisimplicity 3 action of uq(sl(2)) on the quantum plane 4 duality between the hopf algebras uq(sl(2)) and slq(2) 5 duality between uq(sl(2))-modules and slq(2)-comodules 6 scalar products on uq(sl(2))-modules 7 quantum clebsch-gordan 8 exercises 9 notes part two universal r-matrices viii the yang-baxter equation and (co)braided bialgebras 1 the yang-baxter equation 2 braided bialgebras 3 how a braided bialgebra generates r-matrices 4 the square of the antipode in a braided hopf algebra 5 a dual concept: cobraided bialgebras 6 the frt construction 7 application to glq(2) and slq(2) 8 exercises 9 notes ix drinfeld's quantum double 1 bicrossed products of groups 2 bicrossed products of bialgebras 3 variations on the adjoint representation 4 drinfeld's quantum double 5 representation-theoretic interpretation of the quantum double 6 application to uq(sl(2)) 7 r-matrices for uq 8 exercises 9 notes part three low-dimensional topology and tensor categories x knots, links, tangles, and braids 1 knots and links 2 classification of links up to isotopy 3 link diagrams 4 the jones-conway polynomial 5 tangles 6 braids 7 exercises 8 notes 9 appendix. the fundamental group xi tensor categories 1 the language of categories and functors 2 tensor categories 3 examples of tensor categories 4 tensor functors 5 turning tensor categories into strict ones 6 exercises 7 not es xii the tangle category 1 presentation of a strict tensor category 2 the category of tangles 3 the category of tangle diagrams 4 representations of the category of tangles 5 existence proof for jones-conway polynomial 6 exercises 7 notes xiii braidings 1 braided tensor categories 2 the braid category 3 universality of the braid category 4 the centre construction 5 a categorical interpretation of the quantum double 6 exercises 7 notes xiv duality in tensor categories 1 representing morphisms in a tensor category 2 duality 3 ribbon categories 4 quantum trace and dimension 5 examples of ribbon categories 6 ribbon algebras 7 exercises 8 notes xv quasi-bialgebras 1 quasi-bialgebras 2 braided quasi-bialgebras 3 gauge transformations 4 braid group representations 5 quasi-hopf algebras 6 exercises 7 notes part four quantum groups and monodromy xvi generalities on quantum enveloping algebras i the ring of formal series and h-adic topology 2 topologically free modules 3 topological tensor product 4 topological algebras 5 quantum enveloping algebras 6 symmetrizing the universal r-matrix 7 exercises 8 notes 9 appendix. inverse limits xvii drinfeld and jimbo's quantum enveloping algebras 1 semisimple lie algebras 2 drinfeld-jimbo algebras 3 quantum group invariants of links 4 the case of si(2) 5 exercises 6 notes xviii cohomology and rigidity theorems 1 cohomology of lie algebras 2 rigidity for lie algebras 3 vanishing results for semisimple lie algebras 4 application to drinfeld-jimbo quantum enveloping algebras 5 cohomology of coalgebras 6 action of a semisimple lie algebra on the cobar complex 7 computations for symmetric coalgebras 8 uniqueness theorem for quantum enveloping algebras 9 exercises 10 notes 11 appendix. complexes and resolutions xix monodromy of the knizhnik-zamolodchikov equations 1 connections 2 braid group representations from monodromy 3 the knizhnik-zamolodchikov equations 4 the drinfeld-kohno theorem 5 equivalence of uh(g) and ag.t 6 drinfeld's associator 7 construction of the topological braided quasi-bialgebra ag.t 8 verification of the axioms 9 exercises 10 notes 11 appendix. iterated integrals xx postlude. a universal knot invariant 1 knot invariants of finite type 2 chord diagrams and kontsevich's theorem 3 algebra structures on chord diagrams 4 infinitesimal symmetric categories 5 a universal category for infinitesimal braidings 6 formal integration of infinitesimal symmetric categories 7 construction of kontsevich's universal invariant 8 recovering quantum group invariants 9 exercises 10 notes references index |
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