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Gaudin model, the Glossary

In physics, the Gaudin model, sometimes known as the quantum Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin.^[1]

35 relations: Affine Lie algebra, Automorphism of a Lie algebra, Boris Feigin, Casimir element, Character (mathematics), Chiral model, Correlation function (quantum field theory), Dot product, Edward Frenkel, Eigenvalues and eigenvectors, Evgeny Sklyanin, Garnier integrable system, General position, Harish-Chandra isomorphism, Integrable system, Killing form, Lie algebra, Meromorphic function, Michel Gaudin (physicist), Nicolai Reshetikhin, ODE/IM correspondence, Oper (mathematics), Physics, Quantum inverse scattering method, Semisimple Lie algebra, Sigma model, Special linear group, Spectral theory, Spectrum (functional analysis), Spin chain, Statistical mechanics, Tensor product, Toda field theory, Universal enveloping algebra, Weight (representation theory).
Quantum lattice models
Quantum magnetism
Spin models

Affine Lie algebra

In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra.

See Gaudin model and Affine Lie algebra

Automorphism of a Lie algebra

In abstract algebra, an automorphism of a Lie algebra \mathfrak g is an isomorphism from \mathfrak g to itself, that is, a bijective linear map preserving the Lie bracket.

See Gaudin model and Automorphism of a Lie algebra

Boris Feigin

Boris Lvovich Feigin (Бори́с Льво́вич Фе́йгин) (born 20 November 1953) is a Russian mathematician.

See Gaudin model and Boris Feigin

Casimir element

In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra.

See Gaudin model and Casimir element

Character (mathematics)

In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers).

See Gaudin model and Character (mathematics)

In nuclear physics, the chiral model, introduced by Feza Gürsey in 1960, is a phenomenological model describing effective interactions of mesons in the chiral limit (where the masses of the quarks go to zero), but without necessarily mentioning quarks at all.

See Gaudin model and Chiral model

Correlation function (quantum field theory)

In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators.

See Gaudin model and Correlation function (quantum field theory)

Dot product

In mathematics, the dot product or scalar productThe term scalar product means literally "product with a scalar as a result".

See Gaudin model and Dot product

Edward Frenkel

Edward Vladimirovich Frenkel (born May 2, 1968) is a Russian-American mathematician working in representation theory, algebraic geometry, and mathematical physics.

See Gaudin model and Edward Frenkel

Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector is a vector that has its direction unchanged by a given linear transformation.

See Gaudin model and Eigenvalues and eigenvectors

Evgeny Sklyanin

Evgeny Konstantinovich Sklyanin (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.

See Gaudin model and Evgeny Sklyanin

Garnier integrable system

In mathematical physics, the Garnier integrable system, also known as the classical Gaudin model is a classical mechanical system discovered by René Garnier in 1919 by taking the 'Painlevé simplification' or 'autonomous limit' of the Schlesinger equations.

See Gaudin model and Garnier integrable system

General position

In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects.

See Gaudin model and General position

Harish-Chandra isomorphism

In mathematics, the Harish-Chandra isomorphism, introduced by, is an isomorphism of commutative rings constructed in the theory of Lie algebras.

See Gaudin model and Harish-Chandra isomorphism

Integrable system

In mathematics, integrability is a property of certain dynamical systems.

See Gaudin model and Integrable system

Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

See Gaudin model and Killing form

Lie algebra

In mathematics, a Lie algebra (pronounced) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity.

See Gaudin model and Lie algebra

Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function.

See Gaudin model and Meromorphic function

Michel Gaudin (physicist)

Michel Gaudin (2 December 1931 – 4 August 2023) was a French physicist, known for the Gaudin model, in which a central spin is coupled to many surrounding spins.

See Gaudin model and Michel Gaudin (physicist)

Nicolai Reshetikhin

Nicolai Yuryevich Reshetikhin (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).

See Gaudin model and Nicolai Reshetikhin

ODE/IM correspondence

In mathematical physics, the ODE/IM correspondence is a link between ordinary differential equations (ODEs) and integrable models. Gaudin model and ODE/IM correspondence are spin models.

See Gaudin model and ODE/IM correspondence

Oper (mathematics)

In mathematics, an Oper is a principal connection, or in more elementary terms a type of differential operator.

See Gaudin model and Oper (mathematics)

Physics

Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

See Gaudin model and Physics

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.

See Gaudin model and Quantum inverse scattering method

Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras.

See Gaudin model and Semisimple Lie algebra

Sigma model

In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold.

See Gaudin model and Sigma model

Special linear group

In mathematics, the special linear group of degree n over a commutative ring R is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

See Gaudin model and Special linear group

Spectral theory

In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.

See Gaudin model and Spectral theory

Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix.

See Gaudin model and Spectrum (functional analysis)

Spin chain

A spin chain is a type of model in statistical physics. Gaudin model and spin chain are quantum lattice models, quantum magnetism and spin models.

See Gaudin model and Spin chain

Statistical mechanics

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities.

See Gaudin model and Statistical mechanics

Tensor product

In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted.

See Gaudin model and Tensor product

Toda field theory

In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Lie algebra and a specific Lagrangian.

See Gaudin model and Toda field theory

Universal enveloping algebra

In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.

See Gaudin model and Universal enveloping algebra

Weight (representation theory)

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.

See Gaudin model and Weight (representation theory)

References

[1] https://en.wikipedia.org/wiki/Gaudin_model

Gaudin model, the Glossary

Table of Contents

See also

Quantum lattice models

Quantum magnetism

Spin models

References