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ویرایش:
نویسندگان: Kassel. Christian
سری:
ISBN (شابک) : 9781461269007, 1461207835
ناشر: Springer
سال نشر: 2012
تعداد صفحات: 540
زبان: English
فرمت فایل : PDF (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود)
حجم فایل: 36 مگابایت
در صورت تبدیل فایل کتاب Quantum groups به فرمت های PDF، EPUB، AZW3، MOBI و یا DJVU می توانید به پشتیبان اطلاع دهید تا فایل مورد نظر را تبدیل نمایند.
توجه داشته باشید کتاب گروه های کوانتومی نسخه زبان اصلی می باشد و کتاب ترجمه شده به فارسی نمی باشد. وبسایت اینترنشنال لایبرری ارائه دهنده کتاب های زبان اصلی می باشد و هیچ گونه کتاب ترجمه شده یا نوشته شده به فارسی را ارائه نمی دهد.
در اینجا مقدمهای بر نظریه گروههای کوانتومی با تأکید بر ارتباطات دیدنی با نظریه گره و مشارکتهای بنیادی اخیر درینفلد آورده شده است. این گروه های کوانتومی متصل به SL2 و همچنین مفاهیم اساسی تئوری جبرهای Hopf را ارائه می دهد. پوشش همچنین بر جبرهای Hopf متمرکز است که راهحلهای معادله یانگ-باکستر را تولید میکنند و شرحی از برخورد ظریف درینفلد با تکدرومی معادلات Knizhnik-Zamolodchikov ارائه میدهد.
Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.
Title page 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 1. The Hopf Algebras GL(2) and SL(2) 5 Modules over a Hopf Algebra 6 Comodules 7 Comodule-Algebras. Coaction of 5L(2) on the Affine Plane 8 Exercises 9 Notes IV The Quantum Plane and Its Symmetries 1 The Quantum Plane 2 Gauss Polynomials and the q-Binomial Formula 3 The Algebra M_q(2) 4 Ring-Theoretical Properties of M_q(2) 5 Bialgebra Structure on M_q(2) 6 The Hopf Algebras GL_q(2) and SL_q(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 sl(2) 1 The Algebra U_q(sl(2)) 2 Relationship with the Enveloping Algebra of sl(2) 3 Representations of U_q 4 The Harish-Chandra Homomorphism and the Centre of U_q 5 Case when q is a Root of Unity 6 Exercises 7 Notes VII A Hopf Algebra Structure on U_q(sl(2)) 1 Comultiplication 2 Semisimplicity 3 Action of U_q(sl(2)) on the Quantum Plane 4 Duality between the Hopf Algebras U_q(sl(2)) and SL_q(2) 5 Duality between U_q(sl(2))-Modules and SL_q(2)-Comodules 6 Scalar Products on U_q(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 GL_q(2) and SL_q(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 U_q(sl(2)) 7 R-Matrices for U_q 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 Croup 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 Notes 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 1 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 sl(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 Croup Representations from Monodromy 3 The Knizhnik-Zamolodchikov Equations 4 The Drinfeld-Kohno Theorem 5 Equivalence of U_h(g) and A_{g,t} 6 Drinfeld's Associator 7 Construction of the Topological Braided Quasi-Bialgebra A_{g,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 Croup Invariants 9 Exercises 10 Notes References Index