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Quantum Deformation
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 depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Lie Group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces. Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include * the circle group T and the torus groups T''n'', * the orthogonal group O(''n''), the special orthogonal group SO(''n'') and its covering spin group Spin(''n''), * the unitary group U(''n'') and the special unitary group SU(''n''), * the compact forms of the exceptional Lie groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Korepin
Vladimir E. Korepin (born 1951) is a professor at the C. N. Yang Institute of Theoretical Physics of the Stony Brook University. Korepin made research contributions in several areas of mathematics and physics. Educational background Korepin completed his undergraduate study at Saint Petersburg State University, graduating with a diploma in theoretical physics in 1974. In that same year he was employed by the Mathematical Institute of Academy of Sciences. He worked there until 1989, obtaining his PhD in 1977 under the supervision of Ludwig Faddeev. At the same institution he completed his postdoctoral studies. In 1985, he received a doctor of sciences degree in mathematical physics. Contributions to physics Korepin has made contributions to several fields of theoretical physics. Although he is best known for his involvement in condensed matter physics and mathematical physics, he significantly contributed to quantum gravity as well. In recent years, his work has focused on as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolai Reshetikhin
Nicolai Yuryevich Reshetikhin (russian: Николай Юрьевич Решетихин, born October 10, 1958 in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at Tsinghua University, China and a professor of mathematical physics at the University of Amsterdam (Korteweg-de Vries Institute for Mathematics). He is also a professor emeritus at the University of California, Berkeley. His research is in the fields of low-dimensional topology, representation theory, and quantum groups. His major contributions are in the theory of quantum integrable systems, in representation theory of quantum groups and in quantum topology. He and Vladimir Turaev constructed invariants of 3-manifolds which are expected to describe quantum Chern-Simons field theory introduced by Edward Witten. He earned his bachelor's degree and master's degree from Leningrad State University in 1982, and his Ph.D. from the Steklov Mathematical Institute in 1984. He gave a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evgeny Sklyanin
Evgeny Konstantinovich Sklyanin (russian: Евгений Константинович Склянин, born May 24, 1955, in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at the University of York. His research is in the fields of integrable systems and quantum groups. His major contributions are in the theory of quantum integrable systems, separation of variables, special functions. Biography He graduated from the Department of Physics, Leningrad State University (USSR) in 1978 and earned PhD (Candidate) in 1980 and DrSci (Habilitation) degree in 1989, both at Steklov Mathematical Institute, St. Petersburg. He then held various research positions at Steklov until 2001, when he moved to the University of York. He provided, via particular examples, ideas that led to the discovery of quantum groups and Yangians. He pioneered the investigation of quantum integrable systems with boundaries. He developed the method of separation of variabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Takhtajan
Leon Armenovich Takhtajan ( hy, Լևոն Թախտաջյան; russian: Леон Арменович Тахтаджян, born 1 October 1950, Yerevan) is a Russian (and formerly Soviet) mathematical physicist of Armenian descent, currently a professor of mathematics at the Stony Brook University, Stony Brook, NY, and a leading researcher at the Euler International Mathematical Institute, Saint Petersburg, Russia. Takhtajan, son of the Armenian Soviet botanist Armen Takhtajan, received in 1975 his Ph.D. (Russian candidate degree) from the Steklov Institute (Leningrad Department) under Ludvig Faddeev with thesis ''Complete Integrability of the Equation u_-u_+\sin (u)=0''. He was then employed at the Steklov Institute (Leningrad Department) and in 1982 received his D.S. degree (doctor of science, 2nd degree in Russia) with thesis ''Completely integrable models of field theory and statistical mechanics''. Since 1992 he has been a professor at Stony Brook University where he was the chair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwig Faddeev
Ludvig Dmitrievich Faddeev (also ''Ludwig Dmitriyevich''; russian: Лю́двиг Дми́триевич Фадде́ев; 23 March 1934 – 26 February 2017) was a Soviet and Russian mathematical physicist. He is known for the discovery of the Faddeev equations in the theory of the quantum mechanical three-body problem and for the development of path integral methods in the quantization of non-abelian gauge field theories, including the introduction (with Victor Popov) of Faddeev–Popov ghosts. He led the Leningrad School, in which he along with many of his students developed the quantum inverse scattering method for studying quantum integrable systems in one space and one time dimension. This work led to the invention of quantum groups by Drinfeld and Jimbo. Biography Faddeev was born in Leningrad to a family of mathematicians. His father, Dmitry Faddeev, was a well known algebraist, professor of Leningrad University and member of the Russian Academy of Sciences. His ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Inverse Scattering Method
In quantum physics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions, introduced by L. D. Faddeev in 1979. The quantum inverse scattering method relates two different approaches: #the Bethe ansatz, a method of solving integrable quantum models in one space and one time dimension; #the Inverse scattering transform, a method of solving classical integrable differential equations of the evolutionary type. This method led to the formulation of quantum groups. Especially interesting is the Yangian, and the center of the Yangian is given by the quantum determinant. An important concept in the Inverse scattering transform is the Lax representation; the quantum inverse scattering method starts by the quantization of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the ''algebraic Bethe ansatz''.cf. e.g. the lectures by N.A. Slavnov, This led to fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982. Career Source: Academic career timeline: (1966–1970) – Bachelor's degree from the École Normale Supérieure (now part of Paris Sciences et Lettres University). (1973) – doctorate from Pierre and Marie Curie University, Paris, France (1970–1974) – appointment at the French National Centre for Scientific Research, Paris (1975) – Queen's University at Kingston, Ontario, Canada (1976–1980) – the University of Paris VI (1979 – present) – the Institute of Advanced Scientific Studies, Bures-sur-Yvette, France (1981–1984) – the French National Centre for Scientific Research, Paris (1984–2017) – the , Paris (2003–2011) – Vanderbilt University, Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative Geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. An approach giving deep insight about noncommutative spaces is through operator algebras (i.e. algebras of bounded linear operators on a Hilbert space). Perhaps one of the typical examples of a noncommutative space is the " noncommutative tori", which played a key role in the early development of this field in 1980s and lead to noncommutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cocommutative
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
