Quantum Groups And Knot Invariants PDF Download
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Author | : David M. Jackson |
Publisher | : Springer |
Total Pages | : 422 |
Release | : 2019-05-04 |
Genre | : Mathematics |
ISBN | : 3030052133 |
Download An Introduction to Quantum and Vassiliev Knot Invariants Book in PDF, ePub and Kindle
This book provides an accessible introduction to knot theory, focussing on Vassiliev invariants, quantum knot invariants constructed via representations of quantum groups, and how these two apparently distinct theories come together through the Kontsevich invariant. Consisting of four parts, the book opens with an introduction to the fundamentals of knot theory, and to knot invariants such as the Jones polynomial. The second part introduces quantum invariants of knots, working constructively from first principles towards the construction of Reshetikhin-Turaev invariants and a description of how these arise through Drinfeld and Jimbo's quantum groups. Its third part offers an introduction to Vassiliev invariants, providing a careful account of how chord diagrams and Jacobi diagrams arise in the theory, and the role that Lie algebras play. The final part of the book introduces the Konstevich invariant. This is a universal quantum invariant and a universal Vassiliev invariant, and brings together these two seemingly different families of knot invariants. The book provides a detailed account of the construction of the Jones polynomial via the quantum groups attached to sl(2), the Vassiliev weight system arising from sl(2), and how these invariants come together through the Kontsevich invariant.
Author | : Tomotada Ohtsuki |
Publisher | : World Scientific |
Total Pages | : 516 |
Release | : 2002 |
Genre | : Invariants |
ISBN | : 9789812811172 |
Download Quantum Invariants Book in PDF, ePub and Kindle
This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The ChernOCoSimons field theory and the WessOCoZuminoOCoWitten model are described as the physical background of the invariants. Contents: Knots and Polynomial Invariants; Braids and Representations of the Braid Groups; Operator Invariants of Tangles via Sliced Diagrams; Ribbon Hopf Algebras and Invariants of Links; Monodromy Representations of the Braid Groups Derived from the KnizhnikOCoZamolodchikov Equation; The Kontsevich Invariant; Vassiliev Invariants; Quantum Invariants of 3-Manifolds; Perturbative Invariants of Knots and 3-Manifolds; The LMO Invariant; Finite Type Invariants of Integral Homology 3-Spheres. Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics."
Author | : Vladimir G. Turaev |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 608 |
Release | : 2016-07-11 |
Genre | : Mathematics |
ISBN | : 3110435225 |
Download Quantum Invariants of Knots and 3-Manifolds Book in PDF, ePub and Kindle
Due to the strong appeal and wide use of this monograph, it is now available in its third revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics. Contents:Invariants of graphs in Euclidean 3-space and of closed 3-manifoldsFoundations of topological quantum field theoryThree-dimensional topological quantum field theoryTwo-dimensional modular functors6j-symbolsSimplicial state sums on 3-manifoldsShadows of manifolds and state sums on shadowsConstructions of modular categories
Author | : Christian Kassel |
Publisher | : |
Total Pages | : 0 |
Release | : 1997 |
Genre | : Categories (Mathematics) |
ISBN | : 9782856290552 |
Download Quantum Groups and Knot Invariants Book in PDF, ePub and Kindle
This book provides a concise introduction to quantum groups, braided monoidal categories and quantum invariants of knots and of three-dimensional manifolds. The exposition emphasizes the newly discovered deep relationships between these areas.
Author | : S. Chmutov |
Publisher | : Cambridge University Press |
Total Pages | : 521 |
Release | : 2012-05-24 |
Genre | : Mathematics |
ISBN | : 1107020832 |
Download Introduction to Vassiliev Knot Invariants Book in PDF, ePub and Kindle
A detailed exposition of the theory with an emphasis on its combinatorial aspects.
Author | : Christian Kassel |
Publisher | : Springer Science & Business Media |
Total Pages | : 540 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461207835 |
Download Quantum Groups Book in PDF, ePub and Kindle
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.
Author | : Shahn Majid |
Publisher | : Cambridge University Press |
Total Pages | : 183 |
Release | : 2002-04-04 |
Genre | : Mathematics |
ISBN | : 0521010411 |
Download A Quantum Groups Primer Book in PDF, ePub and Kindle
Self-contained introduction to quantum groups as algebraic objects, suitable as a textbook for graduate courses.
Author | : Cisar Gómez |
Publisher | : Cambridge University Press |
Total Pages | : 477 |
Release | : 1996-04-18 |
Genre | : Mathematics |
ISBN | : 0521460654 |
Download Quantum Groups in Two-Dimensional Physics Book in PDF, ePub and Kindle
A 1996 introduction to integrability and conformal field theory in two dimensions using quantum groups.
Author | : Pavel Etingof |
Publisher | : American Mathematical Soc. |
Total Pages | : 344 |
Release | : 2016-08-05 |
Genre | : Algebraic topology |
ISBN | : 1470434415 |
Download Tensor Categories Book in PDF, ePub and Kindle
Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.
Author | : Boris L. Feigin |
Publisher | : American Mathematical Soc. |
Total Pages | : 214 |
Release | : 1998 |
Genre | : Mathematics |
ISBN | : 9780821810842 |
Download Topics in Quantum Groups and Finite-Type Invariants Book in PDF, ePub and Kindle
Presents the first collection of articles consisting entirely of work by the faculty and students at the Higher Mathematics College at the Independent University of Moscow. The 11 contributions cover symmetry groups of regular polyhedra over finite fields, vector bundles on an elliptical curve and Skylanin algebras, Tutte decomposition for graphs and symmetric matrices, and invarians and homology of spaces of knots in arbitrary manifolds. The focus of the text is on quantum groups and low-dimensional topology. No index. Annotation copyrighted by Book News, Inc., Portland, OR.