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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. The Yangian is a degeneration of the quantum loop algebra (i.e. the quantum affine algebra at vanishing central charge). 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 generat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Affine Algebra
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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plefka
Jan Christoph Plefka (born 31 January 1968 in Hanau) is a German theoretical physics, theoretical physicist working in the field of quantum field theory and string theory. Education After receiving the Abitur in Darmstadt and performing civil service in a hospital, Plefka studied physics at the Technical University of Darmstadt and Texas A&M University where he received his M.Sc. as a Fulbright Scholar. He received his PhD from the Leibniz University Hannover with a dissertation on supersymmetry, supersymmetric Theoretical physics, Matrix Models in 1995. In 2003 he received the Habilitation at the Humboldt University Berlin. Career After postdoctoral work at the City College New York and Nikhef Amsterdam, he became a Junior Staff Member at the Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Max Planck Institute for Gravitational Physics in 1998. In 2006, Plefka was awarded a Lichtenberg Professorship of the Volkswagen Foundation in quantum field and strin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Molev
Alexander Ivanovich Molev () (born 1961) is a Russian-Australian mathematician. He completed his Ph.D. in 1986 under the supervision of Alexandre Kirillov at Moscow State University. He was awarded the Australian Mathematical Society Medal in 2001 and became a Fellow of the Australian Academy of Science in 2019. Amongst other things, he has worked on Yangians and Lie algebras. He is currently a Professor in the School of Mathematics and Statistics, Faculty of Science, University of Sydney. Bibliography * Alexander Molev, ''Yangians and classical Lie algebras'', Mathematical Surveys and Monographs, 143. American Mathematical Society, Providence, RI, 2007. xviii+400 pp. * Alexander Molev, ''Sugawara Operators for Classical Lie Algebras'', Mathematical Surveys and Monographs, 229. American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Affine Algebra
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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Inverse Scattering Method
In quantum physics, the quantum inverse scattering method (QISM), similar to the closely related algebraic Bethe ansatz, is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979. It can be viewed as a quantized version of the classical inverse scattering method pioneered by Norman Zabusky and Martin Kruskal used to investigate the Korteweg–de Vries equation and later other integrable partial differential equations. In both, a Lax matrix features heavily and scattering data is used to construct solutions to the original system. While the classical inverse scattering method is used to solve integrable partial differential equations which model continuous media (for example, the KdV equation models shallow water waves), the QISM is used to solve many-body quantum systems, sometimes known as spin chains, of which the Heisenberg spin chain is the best-studied and most famous example. These are typically discrete sy ... [...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), where \check is R followed by a swap of the two objects. In one-dimensional quantum systems, R is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is Integrable system#Quantum integrable systems, 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 in two different ways, the Yang–Baxter equation enforces t ... [...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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludvig Faddeev
Ludvig Dmitrievich Faddeev (also ''Ludwig Dmitriyevich''; ; 23 March 1934 – 26 February 2017) was a Soviet Union, Soviet and Russian Mathematical physics, mathematical physicist. He is known for the discovery of the Faddeev equations in the quantum-mechanical three-body problem and for the development of path integral formulation, path-integral methods in the quantization of non-abelian Gauge theory, gauge field theories, including the introduction of the Faddeev–Popov ghosts (with Victor Popov). He led the Leningrad School, in which he along with many of his students developed the quantum inverse scattering method for studying integrable system#Quantum integrable systems, quantum integrable systems in one space and one time dimension. This work led to the invention of quantum groups by Vladimir Drinfeld, Drinfeld and Michio Jimbo, Jimbo. Biography Faddeev was born in Saint Petersburg, Leningrad to a family of mathematicians. His father, Dmitry Faddeev, was a well-known al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Model (quantum)
The quantum Heisenberg model, developed by Werner Heisenberg, is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mechanically. It is related to the prototypical Ising model, where at each site of a lattice, a spin \sigma_i \in \ represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction. Overview For quantum mechanical reasons (see exchange interaction or ), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are ''aligned''. Under this assumption (so that magnetic interactions only occur between adjacent dipoles) and on a 1-dimensional periodic lattice, the Hamiltonian can be written in the form :\hat H = -J \sum_^ \sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. Formulation Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude \psi: -\frac\nabla^2\psi + V\psi = E\psi where \mu is the reduced mass of two scattering particles and is the energy of relative motion. For scattering problems, a stationary (time-independent) wavefunction is sought with behavior at large distances (asymptotic form) in two parts. First a plane wave represents the incoming source and, second, a spherical wave emanating from the scattering center placed at the coordinate origin represents the scattered wave: \psi(r\rightarrow \infty) \sim e^ + f(\mathbf_f,\mathbf_i)\frac The scattering amplitude, f(\mathbf_f,\mathbf_i), represents the amplitude that the target will scatter into the direction \mathbf_f. In gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrix (mathematics), matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include group (mathematics), groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the group representation, representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication. Representation theory is a useful method because it reduces problems in abstract algebra to problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
