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Yang–Baxter Equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix R, acting on two out of three objects, satisfies :(\check\otimes \mathbf)(\mathbf\otimes \check)(\check\otimes \mathbf) =(\mathbf\otimes \check)(\check \otimes \mathbf)(\mathbf\otimes \check) In one dimensional quantum systems, R is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. The Yang–Baxter equation also shows up when discussing knot theory and the braid groups where R corresponds to swapping two strands. Since one can swap three strands two different ways, the Yang–Baxter equation enforces that both paths are the same. It takes its name from independent work of C. N. Yang from 1968, and R. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot (mathematics)
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an abstraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vyjayanthi Chari
Vyjayanthi Chari (born 1958) is an Indian–American Distinguished Professor and the F. Burton Jones Endowed Chair for Pure Mathematics at the University of California, Riverside, known for her research in representation theory and quantum algebra.. In 2015 she was elected as a fellow of the American Mathematical Society. Education Chari has a bachelor's, master's, and doctoral degree from the University of Mumbai. Chari received her Ph.D. from the University of Mumbai under the supervision of Rajagopalan Parthasarathy. Professional career Following her Ph.D., she became a fellow at the Tata Institute of Fundamental Research, Mumbai. In 1991, she joined the University of California, Riverside The University of California, Riverside (UCR or UC Riverside) is a public land-grant research university in Riverside, California. It is one of the ten campuses of the University of California system. The main campus sits on in a suburban distr ... (UCR) where she is now a Disting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Journal Of Modern Physics
The ''International Journal of Modern Physics'' is a series of Physics journals published by World Scientific. ''International Journal of Modern Physics A'' The ''International Journal of Modern Physics A'' was established in 1986, and covers specifically particles and fields, gravitation, cosmology, and nuclear physics. The journal is abstracted and indexed in: ''International Journal of Modern Physics B'' The ''International Journal of Modern Physics B'' was established in 1987. It covers specifically developments in condensed matter, statistical and applied physics, and high Tc superconductivity. The journal is abstracted and indexed in: ''International Journal of Modern Physics C'' The ''International Journal of Modern Physics C'' was established in 1990. It covers specifically computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasitriangular Hopf Algebra
In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the coproduct on ''H'', and the linear map T : H \otimes H \to H \otimes H is given by T(x \otimes y) = y \otimes x, :*(\Delta \otimes 1)(R) = R_ \ R_, :*(1 \otimes \Delta)(R) = R_ \ R_, where R_ = \phi_(R), R_ = \phi_(R), and R_ = \phi_(R), where \phi_ : H \otimes H \to H \otimes H \otimes H, \phi_ : H \otimes H \to H \otimes H \otimes H, and \phi_ : H \otimes H \to H \otimes H \otimes H, are algebra morphisms determined by :\phi_(a \otimes b) = a \otimes b \otimes 1, :\phi_(a \otimes b) = a \otimes 1 \otimes b, :\phi_(a \otimes b) = 1 \otimes a \otimes b. ''R'' is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the Yang–Baxter equation (and so a module ''V'' of ''H'' c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reidemeister Move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttingen. In 1920, he got the staatsexamen (master's degree) in mathematics, philosophy, physics, chemistry, and geology. He received his doctorate in 1921 with a thesis in algebraic number theory at the University of Hamburg under the supervision of Erich Hecke. He became interested in differential geometry; he edited Wilhelm Blaschke's 2nd volume about that issue, and both made an acclaimed contribution to the Jena DMV conference in Sep 1921. In October 1922 (or 1923) he was appointed assistant professor at the University of Vienna. While there he became familiar with the work of Wilhelm Wirtinger on knot theory, and became closely connected to Hans Hahn and the Vienna Circle. Its manifesto (1929) lists one of Reidemeister's publications i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bialgebra
In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1- cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary. They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group. Lie bialgebras occur naturally in the study of the Yang–Baxter equations. Definition A vector space \mathfrak is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space \mathfrak^* which is compatible. More precisely the Lie algebra structure on \mathfrak is given by a Lie brack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Algebra
In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective space P2. If the cubic divisor happens to be an elliptic curve, then the algebra is called a Sklyanin algebra. The notion is studied in the context of noncommutative projective geometry In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. Examples *The quantum plane, the most basic example, is the quotient ring of the free ring: .... References * {{algebra-stub Algebraic structures Algebraic logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Quantum Group
In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a ''q''-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by and as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang–Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter ''q'' vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized. See also *Quantum enveloping algebra *Quantum KZ equations *Littelmann path model *Yangian In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians fir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yangian
In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse scattering method. The name ''Yangian'' was introduced by Vladimir Drinfeld in 1985 in honor of C.N. Yang. Initially, they were considered a convenient tool to generate the solutions of the quantum Yang–Baxter equation. The center of the Yangian can be described by the quantum determinant. Description For any finite-dimensional semisimple Lie algebra ''a'', Drinfeld defined an infinite-dimensional Hopf algebra ''Y''(''a''), called the Yangian of ''a''. This Hopf algebra is a deformation of the universal enveloping algebra ''U''(''a'' 'z'' of the Lie algebra of polynomial loops of ''a'' given by explicit generators and relations. The relations can be encoded by identities involving a rational ''R''-matrix. Replacing it with a trigo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Groups
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras dependi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
