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Yang Baxter Equation And Quantum Enveloping Algebras
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Book Synopsis Yang-baxter Equation And Quantum Enveloping Algebras by : Zhong-qi Ma
Download or read book Yang-baxter Equation And Quantum Enveloping Algebras written by Zhong-qi Ma and published by World Scientific. This book was released on 1993-12-30 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: The exact solution of C N Yang's one-dimensional many-body problem with repulsive delta-function interactions and R J Baxter's eight-vertex statistical model are brilliant achievements in many-body statistical physics. A nonlinear equation, now known as the Yang-Baxter equation, is the key to the solution of both problems. The Yang-Baxter equation has also come to play an important role in such diverse topics as completely integrable statistical models, conformal and topological field theories, knots and links, braid groups and quantum enveloping algebras.This pioneering textbook attempts to make accessible results in this rapidly-growing area of research. The author presents the mathematical fundamentals at the outset, then develops an intuitive understanding of Hopf algebras, quantisation of Lie bialgebras and quantum enveloping algebras. The historical derivation of the Yang-Baxter equation from statistical models is recounted, and the interpretation and solution of the equation are systematically discussed. Throughout, emphasis is placed on acquiring calculation skills through physical understanding rather than achieving mathematical rigour.Originating from the author's own research experience and lectures, this book will prove both an excellent graduate text and a useful work of reference.
Book Synopsis Yang-Baxter Equation and Quantum Enveloping Algebras by : Zhongqi Ma
Download or read book Yang-Baxter Equation and Quantum Enveloping Algebras written by Zhongqi Ma and published by World Scientific. This book was released on 1993 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first-ever textbook on the Yang-Baxter equation. A key nonlinear equation for solving two important models in many-body statistical theory - the many-body problem in one dimension with repulsive delta-function interaction presented by Professor Baxter in 1972 - it has become one of the main concerns of physicists and mathematicians in the last ten years. A textbook on this subject which also serves as a reference book is vital for an equation which plays important roles in diverse areas of physics and mathematics like the completely integrable statistical models, conformal field theories, topological field theories, the theory of braid groups, the theory of knots and links, etc. This book arose from lectures given by the author in an attempt to reformulate the results of the rapidly developing research and make the material more accessible. It explains the presentation of the Yang-Baxter equation from statistical models, and expound systematically the meaning and methods of solving for this equation. From the viewpoint of theoretical physics it aims to develop an intuitive understanding of the fundamental knowledge of the Hopf algebras, quantization of Lie bialgebras, and the quantum enveloping algebras, and places emphasis on the introduction of the calculation skill in terms of the physical language.
Book Synopsis Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach by : L.A. Lambe
Download or read book Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach written by L.A. Lambe and published by Springer Science & Business Media. This book was released on 2013-11-22 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: Chapter 1 The algebraic prerequisites for the book are covered here and in the appendix. This chapter should be used as reference material and should be consulted as needed. A systematic treatment of algebras, coalgebras, bialgebras, Hopf algebras, and represen tations of these objects to the extent needed for the book is given. The material here not specifically cited can be found for the most part in [Sweedler, 1969] in one form or another, with a few exceptions. A great deal of emphasis is placed on the coalgebra which is the dual of n x n matrices over a field. This is the most basic example of a coalgebra for our purposes and is at the heart of most algebraic constructions described in this book. We have found pointed bialgebras useful in connection with solving the quantum Yang-Baxter equation. For this reason we develop their theory in some detail. The class of examples described in Chapter 6 in connection with the quantum double consists of pointed Hopf algebras. We note the quantized enveloping algebras described Hopf algebras. Thus for many reasons pointed bialgebras are elsewhere are pointed of fundamental interest in the study of the quantum Yang-Baxter equation and objects quantum groups.
Book Synopsis Yang-Baxter Equation and Quantum Enveloping Algebras by : Chung-chi Ma
Download or read book Yang-Baxter Equation and Quantum Enveloping Algebras written by Chung-chi Ma and published by . This book was released on 1993 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Yang-Baxter Equation in Integrable Systems by : Michio Jimbo
Download or read book Yang-Baxter Equation in Integrable Systems written by Michio Jimbo and published by World Scientific. This book was released on 1990 with total page 740 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume will be the first reference book devoted specially to the Yang-Baxter equation. The subject relates to broad areas including solvable models in statistical mechanics, factorized S matrices, quantum inverse scattering method, quantum groups, knot theory and conformal field theory. The articles assembled here cover major works from the pioneering papers to classical Yang-Baxter equation, its quantization, variety of solutions, constructions and recent generalizations to higher genus solutions.
Book Synopsis Quantum Groups and Lie Theory by : Andrew Pressley
Download or read book Quantum Groups and Lie Theory written by Andrew Pressley and published by Cambridge University Press. This book was released on 2002-01-17 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book comprises an overview of the material presented at the 1999 Durham Symposium on Quantum Groups and includes contributions from many of the world's leading figures in this area. It will be of interest to researchers and will also be useful as a reference text for graduate courses.
Book Synopsis Quantum Groups in Three-Dimensional Integrability by : Atsuo Kuniba
Download or read book Quantum Groups in Three-Dimensional Integrability written by Atsuo Kuniba and published by Springer Nature. This book was released on 2022-09-25 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum groups have been studied intensively in mathematics and have found many valuable applications in theoretical and mathematical physics since their discovery in the mid-1980s. Roughly speaking, there are two prototype examples of quantum groups, denoted by Uq and Aq. The former is a deformation of the universal enveloping algebra of a Kac–Moody Lie algebra, whereas the latter is a deformation of the coordinate ring of a Lie group. Although they are dual to each other in principle, most of the applications so far are based on Uq, and the main targets are solvable lattice models in 2-dimensions or quantum field theories in 1+1 dimensions. This book aims to present a unique approach to 3-dimensional integrability based on Aq. It starts from the tetrahedron equation, a 3-dimensional analogue of the Yang–Baxter equation, and its solution due to work by Kapranov–Voevodsky (1994). Then, it guides readers to its variety of generalizations, relations to quantum groups, and applications. They include a connection to the Poincaré–Birkhoff–Witt basis of a unipotent part of Uq, reductions to the solutions of the Yang–Baxter equation, reflection equation, G2 reflection equation, matrix product constructions of quantum R matrices and reflection K matrices, stationary measures of multi-species simple-exclusion processes, etc. These contents of the book are quite distinct from conventional approaches and will stimulate and enrich the theories of quantum groups and integrable systems.
Book Synopsis Quantum Groups by : Christian Kassel
Download or read book Quantum Groups written by Christian Kassel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 540 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Quantum Groups by : Vladimir K. Dobrev
Download or read book Quantum Groups written by Vladimir K. Dobrev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-07-10 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: With applications in quantum field theory, general relativity and elementary particle physics, this three-volume work studies the invariance of differential operators under Lie algebras, quantum groups and superalgebras. This second volume covers quantum groups in their two main manifestations: quantum algebras and matrix quantum groups. The exposition covers both the general aspects of these and a great variety of concrete explicitly presented examples. The invariant q-difference operators are introduced mainly using representations of quantum algebras on their dual matrix quantum groups as carrier spaces. This is the first book that covers the title matter applied to quantum groups. Contents Quantum Groups and Quantum Algebras Highest-Weight Modules over Quantum Algebras Positive-Energy Representations of Noncompact Quantum Algebras Duality for Quantum Groups Invariant q-Difference Operators Invariant q-Difference Operators Related to GLq(n) q-Maxwell Equations Hierarchies
Book Synopsis Algebraic Analysis of Solvable Lattice Models by : Michio Jimbo
Download or read book Algebraic Analysis of Solvable Lattice Models written by Michio Jimbo and published by American Mathematical Soc.. This book was released on 1995 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the NSF-CBMS Regional Conference lectures presented by Miwa in June 1993, this book surveys recent developments in the interplay between solvable lattice models in statistical mechanics and representation theory of quantum affine algebras. Because results in this subject were scattered in the literature, this book fills the need for a systematic account, focusing attention on fundamentals without assuming prior knowledge about lattice models or representation theory. After a brief account of basic principles in statistical mechanics, the authors discuss the standard subjects concerning solvable lattice models in statistical mechanics, the main examples being the spin 1/2 XXZ chain and the six-vertex model. The book goes on to introduce the main objects of study, the corner transfer matrices and the vertex operators, and discusses some of their aspects from the viewpoint of physics. Once the physical motivations are in place, the authors return to the mathematics, covering the Frenkel-Jing bosonization of a certain module, formulas for the vertex operators using bosons, the role of representation theory, and correlation functions and form factors. The limit of the XXX model is briefly discussed, and the book closes with a discussion of other types of models and related works.
Download or read book Quantum Groups written by Steven Shnider and published by International Press of Boston. This book was released on 1993 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the field of quantum groups, including topology and statistical mechanics, based on lectures given at the Sackler Institute for Advanced Studies at Tel-Aviv University. Detailed proofs of the main results are presented and the bibliography contains more than 1260 references.
Download or read book Quantum Groups written by Petr P. Kulish and published by . This book was released on 1992 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of Quantum Groups is a rapidly developing area with numerous applications in mathematics and theoretical physics, e.g. in link and knot invariants in topology, q-special functions, conformal field theory, quantum integrable models. The aim of the Euler Institute's workshops was to review and compile the progress achieved in the different subfields. Near 100 participants came from 14 countries. More than 20 contributions written up for this book contain new, unpublished material and half of them include a survey of recent results in the field (deformation theory, graded differential algebras, contraction technique, knot invariants, q-special functions). FROM THE CONTENTS: V.G. Drinfeld: On Some Unsolved Problems in Quantum Group Theory.- M. Gerstenhaber, A. Giaquinto, S.D. Schack: Quantum Symmetry.- L.I. Korogodsky, L.L. Vaksman: Quantum G-Spaces and Heisenberg Algebra.-J. Stasheff: Differential Graded Lie Algebras, Quasi-Hopf Algebras and Higher Homotopy Algebras.- A. Yu. Alekseev, L.D. Faddeev, M.A. Semenov-Tian-Shansky: Hidden Quantum Groups inside Kac-Moody Algebras.- J.-L. Gervais: Quantum Group Symmetry of 2D Gravity.- T. Kohno: Invariants of 3-Manifolds Based on Conformal Field Theory and Heegaard Splitting.- O. Viro: Moves of Triangulations of a PL-Manifold.-- Publisher description.
Book Synopsis Quantized Algebra And Physics - Proceedings Of The International Workshop by : Chengming Bai
Download or read book Quantized Algebra And Physics - Proceedings Of The International Workshop written by Chengming Bai and published by World Scientific. This book was released on 2011-10-18 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book aims to survey recent developments in quantum algebras and related topics. Quantum groups were introduced by Drinfeld and Jimbo in 1985 in their work on Yang-Baxter equations. The subject from the very beginning has been an interesting one for both mathematics and theoretical physics. For example, Yangian is a special example of quantum group, corresponding to rational solution of Yang-Baxter equation. Viewed as a generalization of the symmetric group, Yangians also have close connections to algebraic combinatorics. This is the proceeding for the International Workshop on Quantized Algebra and Physics. The workshop aims to gather experts and young investigators from China and abroad to discuss research problems in integrable systems, conformal field theory, string theory, Lie theory, quantum groups including Yangians and their representations.
Book Synopsis Quantum Deformations of Algebras and Their Representations by : Anthony Joseph
Download or read book Quantum Deformations of Algebras and Their Representations written by Anthony Joseph and published by . This book was released on 1993 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Hopf Algebras, Quantum Groups and Yang-Baxter Equations by : Florin Felix Nichita
Download or read book Hopf Algebras, Quantum Groups and Yang-Baxter Equations written by Florin Felix Nichita and published by MDPI. This book was released on 2019-01-31 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms
Book Synopsis Quantum Groups and Their Primitive Ideals by : Anthony Joseph
Download or read book Quantum Groups and Their Primitive Ideals written by Anthony Joseph and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.
Book Synopsis A Guide to Quantum Groups by : Vyjayanthi Chari
Download or read book A Guide to Quantum Groups written by Vyjayanthi Chari and published by Cambridge University Press. This book was released on 1995-07-27 with total page 672 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since they first arose in the 1970s and early 1980s, quantum groups have proved to be of great interest to mathematicians and theoretical physicists. The theory of quantum groups is now well established as a fascinating chapter of representation theory, and has thrown new light on many different topics, notably low-dimensional topology and conformal field theory. The goal of this book is to give a comprehensive view of quantum groups and their applications. The authors build on a self-contained account of the foundations of the subject and go on to treat the more advanced aspects concisely and with detailed references to the literature. Thus this book can serve both as an introduction for the newcomer, and as a guide for the more experienced reader. All who have an interest in the subject will welcome this unique treatment of quantum groups.
