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The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C.N. Yang and in the work of R.J. Baxter in the field of Statistical Mechanics. At the 1990 International Mathematics Congress, Vladimir Drinfeld, Vaughan F. R. Jones, and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. It turned out that this equation is one of the basic equations in mathematical physics; more precisely, it is used for introducing the theory of quantum groups. It also plays a crucial role in: knot theory, braided categories, the analysis of integrable systems, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have used the axioms of various algebraic structures (quasi-triangular Hopf algebras, Yetter-Drinfeld categories, quandles, group actions, Lie (super)algebras, brace structures, (co)algebra structures, Jordan triples, Boolean algebras, relations on sets, etc.) or computer calculations (and Grobner bases) in order to produce solutions for the Yang-Baxter equation. However, the full classification of its solutions remains an open problem. At present, the study of solutions of the Yang-Baxter equation attracts the attention of a broad circle of scientists. The current volume highlights various aspects of the Yang-Baxter equation, related algebraic structures, and applications.

braided category --- quasitriangular structure --- quantum projective space --- Hopf algebra --- quantum integrability --- duality --- six-vertex model --- Quantum Group --- Yang-Baxter equation --- star-triangle relation --- R-matrix --- Lie algebra --- bundle --- braid group

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This text is based on an established graduate course given at MIT that provides an introduction to the theory of the dynamical Yang-Baxter equation and its applications, which is an important area in representation theory and quantum groups.

Yang-Baxter equation. --- Representations of quantum groups. --- Quantum groups. --- Enveloping algebras, Quantized --- Function algebras, Quantized --- Groups, Quantum --- Quantized enveloping algebras --- Quantized function algebras --- Quantum algebras --- Group theory --- Mathematical physics --- Quantum field theory --- Quantum groups --- Baxter-Yang equation --- Factorization equation --- Star-triangle relation --- Triangle equation --- Representations of groups. --- Yang-Baxter, Équation de --- Représentations de groupes --- Groupes quantiques

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In mathematical physics, one of the fascinating issues is the study of integrable systems. In particular, non-perturbative techniques that have been developed have triggered significant insight for real physics. There are basically two notions of integrability: classical integrability and quantum integrability. In this book, the focus is on the former, classical integrability. When the system has a finite number of degrees of freedom, it has been well captured by the Arnold–Liouville theorem. However, when the number of degrees of freedom is infinite, as in classical field theories, the integrable structure is enriched profoundly. In fact, the study of classically integrable field theories has a long history and various kinds of techniques, including the classical inverse scattering method, which have been developed so far. In previously published books, these techniques have been collected and well described and are easy to find in traditional, standard textbooks. One of the intriguing subjects in classically integrable systems is the investigation of deformations preserving integrability. Usually, it is not considered systematic to perform such a deformation, and one must study systems case by case and show the integrability of the deformed systems by constructing the associated Lax pair or action-angle variables. Recently, a new, systematic method to perform integrable deformations of 2D non-linear sigma models was developed. It was invented by C. Klimcik in 2002, and the integrability of the deformed sigma models was shown in 2008. The original work was done for 2D principal chiral models, but it has been generalized in various directions nowadays. In this book, the recent progress on this Yang–Baxter deformation is described in a pedagogical manner, including some simple examples. Applications of Yang–Baxter deformation to string theory are also described briefly. .

Mathematical physics. --- Special functions. --- Partial differential equations. --- Mathematical Physics. --- Mathematical Applications in the Physical Sciences. --- Special Functions. --- Partial Differential Equations. --- Partial differential equations --- Special functions --- Mathematical analysis --- Physical mathematics --- Physics --- Mathematics --- Yang-Baxter equation. --- Baxter-Yang equation --- Factorization equation --- Star-triangle relation --- Triangle equation --- Mathematical physics --- Quantum field theory

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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.

Ordered algebraic structures --- Yang-Baxter equation. --- Quantum groups. --- Hopf algebras. --- Mathematical physics. --- Yang-Baxter, Équation de --- Groupes quantiques --- Algèbres de Hopf --- Physique mathématique --- Yang-Baxter, Équation de --- Algèbres de Hopf --- Physique mathématique --- Associative rings. --- Rings (Algebra). --- Numerical analysis. --- Category theory (Mathematics). --- Homological algebra. --- Associative Rings and Algebras. --- Theoretical, Mathematical and Computational Physics. --- Numeric Computing. --- Category Theory, Homological Algebra. --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Mathematical analysis --- Physical mathematics --- Physics --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Mathematics --- Enveloping algebras, Quantized --- Function algebras, Quantized --- Groups, Quantum --- Quantized enveloping algebras --- Quantized function algebras --- Quantum algebras --- Mathematical physics --- Quantum field theory --- Algebras, Hopf --- Algebraic topology --- Baxter-Yang equation --- Factorization equation --- Star-triangle relation --- Triangle equation