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Mathematical discipline
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
Quantum_affine_algebra
Type of Kac–Moody algebras
affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra.
Affine_Lie_algebra
Algebraic construct of interest in theoretical physics
term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups
Quantum_group
the quantum loop algebra (i.e. the quantum affine algebra at vanishing central charge). For any finite-dimensional semisimple Lie algebra a, Drinfeld defined
Yangian
Deformation of the group algebra of a Coxeter group
Hecke algebras led to discovery of quantum groups by Michio Jimbo. Michael Freedman proposed Hecke algebras as a foundation for topological quantum computation
Iwahori–Hecke_algebra
In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture
Affine_Hecke_algebra
corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the Heisenberg XXZ model in the framework of the algebraic Bethe
Quasi-Hopf_algebra
Statistical model in quantum mechanics of magnetic materials
the Bethe ansatz. In the algebraic formulation, these are related to particular quantum affine algebras and elliptic quantum groups in the XXZ and XYZ
Quantum_Heisenberg_model
Iwahori–Hecke algebra Quantum affine algebra Quantum enveloping algebra Quantum group Jackson integral q-derivative q-difference polynomial Quantum calculus
List_of_q-analogs
Quantum algebra version of the Knizhnik–Zamolodchikov equations
physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov
Quantum_KZ_equations
Type of Lie algebra of interest in physics
Conformal Field Theory, 1997, ISBN 0-387-94785-X Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X
Loop_algebra
Creating a "larger" Lie algebra from a smaller one, in one of several ways
algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra
Lie_algebra_extension
Infinite dimensional Lie algebra occurring in quantum field theory
operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting
Current_algebra
Branch of mathematics
instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring. Therefore, one can
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Japanese mathematician
Nakajima. Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. doi:10.1090/S0894-0347-00-00353-2
Hiraku_Nakajima
Russian-American mathematician
Frenkel introduced deformations of W-algebras and q-characters of representations of quantum affine algebras. Frenkel's recent work has focused on the
Edward_Frenkel
Algebraic structure used in analysis
symmetric Lie algebra Poisson algebra Pre-Lie algebra Quantum groups Moyal algebra Quasi-Frobenius Lie algebra Quasi-Lie algebra Restricted Lie algebra Serre
Lie_algebra
Poisson manifold that is also a Lie group
{\displaystyle G} is semisimple. Lie bialgebra Quantum group Affine quantum group Quantum affine algebras Lu, Jiang-Hua; Weinstein, Alan (1990-01-01). "Poisson
Poisson–Lie_group
Algebra used in 2D conformal field theories and string theory
case is quantum Drinfeld–Sokolov reduction applied to affine Kac–Moody algebras to obtain affine W-algebras as degree 0 cohomology. These W algebras also
Vertex_operator_algebra
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Algebra describing 2D conformal symmetry
algebra of any affine Lie algebra, as shown by the Sugawara construction. In this sense, affine Lie algebras are extensions of the Virasoro algebra. The Virasoro
Virasoro_algebra
Type of 2D conformal field theory
group (or supergroup), and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra). By extension, the
Wess–Zumino–Witten_model
Associative algebra generalizing the Virasoro algebra
The W(2) algebra coincides with the Virasoro algebra. The W(N) algebra is obtained by Drinfeld-Sokolov reduction of the affine Lie algebra s l ^ N {\displaystyle
W-algebra
"Smallest" commutative algebra that contains a vector space
symmetric algebra on an affine space. The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered algebra: one
Symmetric_algebra
Mathematical term
and Jean-Yves Thibon Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties MR 1434225 J. Math. Phys. 38 (1997), no
LLT_polynomial
Construction in algebra
1007/978-3-540-30308-4_12, ISBN 978-3-540-30307-7 Fuchs, Jürgen (1992), Affine Lie algebras and quantum groups. An introduction with applications in conformal field
Hopf_algebra
Mathematician
mathematical physics, including the ADHM construction of instantons, algebraic formalism of the quantum inverse scattering method, and the Drinfeld–Sokolov reduction
Vladimir_Drinfeld
Integrable classical system
can be formulated as classical affine Gaudin models, where g {\displaystyle {\mathfrak {g}}} is an affine Lie algebra. Such classical field theories include
Garnier_integrable_system
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is
Geometric_algebra
Ring that is also a vector space or a module
the category of affine schemes over Spec R. How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more
Associative_algebra
Italian-French mathematician
have included representation theory, Hecke algebra, Schur–Weyl duality, Yangians, and quantum affine algebras. She earned a doctorate in 1993 at the University
Michela_Varagnolo
Group of flat spacetime symmetries
pseudovector Representation theory of the Poincaré group Super-Poincaré algebra Symmetry in quantum mechanics Wigner's classification Poincaré, Henri (1905-12-14)
Poincaré_group
Physics property associated with symmetries
this time in the general Lie-algebra setting. Casimir operator Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press
Charge_(physics)
Branch of mathematics
developed from several areas, including operator algebra theory, index theory, algebraic geometry, quantum mechanics and ergodic theory. The term is particularly
Noncommutative_geometry
Subgroup of the group of invertible n×n matrices
equivalent to affine group schemes. (Every affine group scheme over a field k is pro-algebraic in the sense that it is an inverse limit of affine group schemes
Linear_algebraic_group
Tensor product of algebras over a field; itself another algebra
(1\otimes b)} . The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z
Tensor_product_of_algebras
Generalization of the BRST formalism
the action) of a Batalin–Vilkovisky algebra is the equation ( S , S ) = 0. {\displaystyle (S,S)=0.} The quantum master equation for an even degree element
Batalin–Vilkovisky_formalism
Physics-mathematics connection
particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to
Particle physics and representation theory
Particle_physics_and_representation_theory
Divergent series
"Vertex operator algebras and the zeta function". In Naihuan Jing and Kailash C. Misra (ed.). Recent Developments in Quantum Affine Algebras and Related Topics
1_+_2_+_3_+_4_+_⋯
Generalization of bialgebra
corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the XXZ in the framework of the Algebraic Bethe ansatz
Quasi-bialgebra
Group in group theory and physics
b\\0&0&1\\\end{pmatrix}}.} The group is a subgroup of the 2-dimensional affine group Aff(2): the action of the element ( 1 a c 0 1 b 0 0 1 ) {\displaystyle
Heisenberg_group
Branch of mathematics that studies abstract algebraic structures
way as representations of semisimple Lie algebras. Affine Lie algebras are a special case of Kac–Moody algebras, which have particular importance in mathematics
Representation_theory
Algebra in statistical mechanics
integrable models, knot theory and the braid groups, quantum groups and subfactors of von Neumann algebras. Let R {\displaystyle R} be a commutative ring and
Temperley–Lieb_algebra
Physical theory with fields invariant under the action of local "gauge" Lie groups
the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises
Gauge_theory
Branch of algebra
are the "affine schemes" (generalization of affine varieties), and a general scheme is then obtained by "gluing together" (by purely algebraic methods)
Ring_theory
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
is, in their words, "a “quantum” version of (the algebra of functions on) the space of global horizontal sections of an affine D X {\displaystyle {\mathcal
Chiral_homology
Algebraic structure used in theoretical physics
one can define an affine algebraic supergroup as a group object in the category of superalgebraic affine varieties. An affine algebraic supergroup has a
Supergroup_(physics)
Branch of mathematics
isotropic quadratic forms are admitted, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have
Geometry
Chinese mathematician (born 1982)
in particular the Langlands program, tying number theory to algebraic geometry and quantum physics. Zhu obtained his A.B. in mathematics from Peking University
Xinwen_Zhu
Four-dimensional number system
important in quantum mechanics and the study of spin Quaternionic manifold – Concept in geometry Quaternionic matrix – Concept in linear algebra Quaternionic
Quaternion
algebra is isomorphic to a subalgebra of g l V {\displaystyle {\mathfrak {gl}}_{V}} for some finite-dimensional vector space V. affine 1. An affine Lie
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Group that admits a formal description in terms of reflections
Isomorphism problem of Coxeter groups Iwahori–Hecke algebra, a quantum deformation of the group algebra Kazhdan–Lusztig polynomial Longest element of a Coxeter
Coxeter_group
Algebraic structure in linear algebra
the algebraic counterpart to vector bundles. Roughly, affine spaces are vector spaces whose origins are not specified. More precisely, an affine space
Vector_space
Physics model in statistical mechanics
also examples of quantum spin chains. The simplest case was first described by Michel Gaudin in 1976, with the associated Lie algebra taken to be s l 2
Gaudin_model
classical Schur algebras. There are further generalizations, such as the affine q-Schur algebras related to affine Kac–Moody Lie algebras and other generalizations
Schur_algebra
Array of numbers
"two-by-three matrix", a 2 × 3 matrix, or a matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric
Matrix_(mathematics)
postulate. Abstract algebra The part of algebra devoted to the study of algebraic structures in themselves. Occasionally named modern algebra in course titles
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Type of symmetry in physics
need not form a Lie algebra). The Lie algebra is denoted by CC(M) and may be infinite-dimensional. Every affine vector field is a curvature collineation
Spacetime_symmetries
Class of conformal field theories
chiral algebra. Chiral algebras can be much larger than the Virasoro algebra. Well-known examples include (the enveloping algebra of) affine Lie algebras, relevant
Rational conformal field theory
Rational_conformal_field_theory
Group representation
algebra; this correspondence is discussed in detail in subsequent sections. See representation of Lie algebras for the Lie algebra theory. In quantum
Representation_of_a_Lie_group
Hungarian and American mathematician and physicist (1903–1957)
formally real Jordan algebras and discovering the Albert algebras while attempting to look for a better mathematical formalism for quantum theory. In 1936
John_von_Neumann
Mathematical group of loops in a Lie group
related to affine Kac–Moody algebras, conformal field theory, and the Verlinde formula. In algebraic geometry one also studies algebraic loop groups
Loop_group
Frobenius algebra to tangent bundles. Frobenius manifolds occur naturally in the subject of symplectic topology, more specifically quantum cohomology
Frobenius_manifold
Group that is also a differentiable manifold with group operations that are smooth
quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are
Lie_group
Property of certain dynamical systems
incorporated into the quantum inverse scattering method where the algebraic Bethe ansatz can be used to obtain explicit solutions. Examples of quantum integrable
Integrable_system
infinite-dimensional affine Lie algebra Ẽ8 (also as E+ 8 or E(1) 8 as a (one-node) extended E8) (or E8 lattice) corresponding to the Lie algebra of type E8. E9
En_(Lie_algebra)
Theoretical object in mathematics
affine scheme Spec Z is a curve over F1. Groups are Hopf algebras over F1. More generally, anything defined purely in terms of diagrams of algebraic objects
Field_with_one_element
Writing Lie algebra sets as matrices
representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms
Lie_algebra_representation
Mathematical set with some added structure
local models are called affine schemes. Affine schemes provide a direct link between algebraic geometry and commutative algebra. The fundamental objects
Space_(mathematics)
Partial differential equations of correlation functions
sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential
Knizhnik–Zamolodchikov equations
Knizhnik–Zamolodchikov_equations
Graphical notation for multilinear algebra calculations
widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, categorical quantum mechanics (which includes
Penrose_graphical_notation
Raising and lowering operators in quantum mechanics
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that
Ladder_operator
Differential algebra
domain, and an example of an Ore extension. The Weyl algebra arises naturally in the context of quantum mechanics and the process of canonical quantization
Weyl_algebra
Method used to solve integrable many-body quantum systems
In quantum physics, the quantum inverse scattering method, similar to the closely related algebraic Bethe ansatz, is a method for solving integrable models
Quantum inverse scattering method
Quantum_inverse_scattering_method
Theoretical attempts to unify the forces of nature
the affine connection as the fundamental structure field rather than the metric tensor which was the original focus of general relativity. Affine connection
Classical unified field theories
Classical_unified_field_theories
Representation theory of the symmetries of non-relativistic quantum space
spacetime symmetry group of nonrelativistic quantum mechanics. In 3 + 1 dimensions, this is the subgroup of the affine group on (t, x, y, z), whose linear part
Representation theory of the Galilean group
Representation_theory_of_the_Galilean_group
Non-tensorial representation of the spin group
College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors
Spinor
Japanese mathematician (born 1949)
affine Lie algebras, and on correlation functions of quantum spin chains in connection with the representation theory of the quantum affine algebras.
Tetsuji_Miwa
Correspondence between quantum channels and quantum states
In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels (described by
Choi–Jamiołkowski_isomorphism
Academic fields of study or professions
geometry Stochastic process Geometry (outline) and Topology Affine geometry Algebraic geometry Algebraic topology Convex geometry Differential topology Discrete
Outline of academic disciplines
Outline_of_academic_disciplines
248-dimensional exceptional simple Lie group
several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding
E8_(mathematics)
Metric on a complex projective space endowed with Hermitian form
form an affine coordinate system for CPn in the coordinate patch U 0 = { Z 0 ≠ 0 } {\displaystyle U_{0}=\{Z_{0}\neq 0\}} . One can develop an affine coordinate
Fubini–Study_metric
Algebraic object with geometric applications
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space
Tensor
Mathematical model of the time dependence of a point in space
abstract algebra, algebraic geometry and Galois theory. In the context of Algebraic quantum field theory the measure space can be a C*-algebra, namely
Dynamical_system
General concept and operation in mathematics
there is a duality in algebraic geometry between commutative rings and affine schemes: to every commutative ring A there is an affine spectrum, Spec A. Conversely
Duality_(mathematics)
Mathematical space
theory of schemes, the Grassmannian plays a similar role for algebraic K-theory. Affine Grassmannian Grassmann bundle Grassmann graph Lee 2012, p. 22
Grassmannian
Quantum consistency equation
and elliptic. These are related to quantum groups known as the Yangian, affine quantum groups and elliptic algebras respectively. Set-theoretic solutions
Yang–Baxter_equation
Element of a unital algebra over the field of real numbers
number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the
Hypercomplex_number
Type of geometry
the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". An algebraic model
Projective_geometry
In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power"
Restricted_Lie_algebra
Russian mathematician
He received the Wigner Medal (1996) "in recognition of work on affine Lie algebras that has had wide influence in theoretical physics". In 1978 he was
Victor_Kac
Relates the geometric vector bundles to algebraic projective modules
result in algebraic geometry, due to Serre (1955, §50) applies to vector bundles in the category of affine varieties over an algebraically closed field
Serre–Swan_theorem
Topics referred to by the same term
see function (mathematics). Affine transformation, in geometry Linear transformation between modules in linear algebra. Also called a linear map. Transformation
Transformation
Symmetric bilinear form in mathematics
transformations finis et continus, Thesis, Nony Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X Fulton
Killing_form
Type of model in quantum statistical physics
representation of the Lie algebra g {\displaystyle {\mathfrak {g}}} , labelled V v {\displaystyle V_{v}} . This is a quantum generalization of statistical
Spin_chain
Japanese mathematician (born 1951)
Medal, "for his introduction of quantum groups and his study of affine Lie algebras, in connection with classical and quantum integrable systems" 2013 – Dannie
Michio_Jimbo
Derivative used in gauge theories
the closest description of the gauge connection in quantum field theory. For ordinary Lie algebras, the gauge covariant derivative on the space symmetries
Gauge_covariant_derivative
Overview of and topical guide to logic
Boolean algebra Free Boolean algebra Monadic Boolean algebra Residuated Boolean algebra Two-element Boolean algebra Modal algebra Derivative algebra (abstract
Outline_of_logic
Russian-American mathematician
University in 1980 with a dissertation on the "Orbital Theory for Affine Lie Algebras". He held positions at the IAS and MSRI, and a tenured professorship
Igor_Frenkel
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