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This 1996 book is an introduction to integrability and conformal field theory in two dimensions using quantum groups. The book begins with a brief introduction to S-matrices, spin chains and vertex models as a prelude to the study of Yang-Baxter algebras and the Bethe ansatz. The basic ideas of integrable systems are then introduced, giving particular emphasis to vertex and face models. Special attention is given to explaining the underlying mathematical tools, including braid groups, knot invariants and towers of algebra. The book then goes on to give a detailed introduction to quantum groups as a prelude to chapters on integrable models, two-dimensional conformal field theories and superconformal field theories. The book contains many diagrams and exercises to illustrate key points in the text.

Quantum groups. --- Yang-Baxter equation. --- Conformal invariants. --- Mathematical physics.

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Mathematical physics --- Quantum field theory --- Quantum groups --- Yang-Baxter equation --- Congresses. --- Congresses. --- Congresses. --- Congresses.

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Ordered algebraic structures --- Vector bundles. --- Curves, Elliptic. --- Yang-Baxter equation. --- Fibrés vectoriels --- Courbes elliptiques --- Yang-Baxter, Équation de --- 51 <082.1> --- Mathematics--Series --- Fibrés vectoriels --- Yang-Baxter, Équation de --- Curves, Elliptic --- Vector bundles --- Yang-Baxter equation --- Baxter-Yang equation --- Factorization equation --- Star-triangle relation --- Triangle equation --- Mathematical physics --- Quantum field theory --- Fiber spaces (Mathematics) --- Elliptic curves --- Curves, Algebraic

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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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Leonhard Euler (1707–1783) was born in Basel, Switzerland. Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. When its variable is the number pi, Euler's formula evaluates to Euler's identity. On the other hand, the Yang–Baxter equation is considered the most beautiful equation by many scholars. In this book, we study connections between Euler’s formulas and the Yang–Baxter equation. Other interesting sections include: non-associative algebras with metagroup relations; branching functions for admissible representations of affine Lie Algebras; super-Virasoro algebras; dual numbers; UJLA structures; etc.

Research & information: general --- Mathematics & science --- transcendental numbers --- Euler formula --- Yang-Baxter equation --- Jordan algebras --- Lie algebras --- associative algebras --- coalgebras --- Euler's formula --- hyperbolic functions --- UJLA structures --- (co)derivation --- dual numbers --- operational methods --- umbral image techniques --- nonassociative algebra --- cohomology --- extension --- metagroup --- branching functions --- admissible representations --- characters --- affine Lie algebras --- super-Virasoro algebras --- nonassociative --- product --- smashed --- twisted wreath --- algebra --- separable --- ideal --- transcendental numbers --- Euler formula --- Yang-Baxter equation --- Jordan algebras --- Lie algebras --- associative algebras --- coalgebras --- Euler's formula --- hyperbolic functions --- UJLA structures --- (co)derivation --- dual numbers --- operational methods --- umbral image techniques --- nonassociative algebra --- cohomology --- extension --- metagroup --- branching functions --- admissible representations --- characters --- affine Lie algebras --- super-Virasoro algebras --- nonassociative --- product --- smashed --- twisted wreath --- algebra --- separable --- ideal