Gabor–Wigner transform

2014-01-20T08:22:34Z

117.205.47.149: /* See also */

'''Yangian''' is an important structure in modern [[representation theory]], a type of a [[quantum group]] with origins in [[physics]]. Yangians first appeared in the work of [[Ludvig Faddeev]] and his school concerning the [[quantum inverse scattering method]] in the late 1970s and early 1980s. Initially they were considered a convenient tool to generate the solutions of the quantum [[Yang–Baxter equation]]. The name ''Yangian'' was introduced by [[Vladimir Drinfeld]] in 1985 in honor of [[C.N. Yang]]. The center of Yangian can be described by [[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|''R''-matrix]]. Replacing it with a trigonometric ''R''-matrix, one arrives at [[affine quantum group]]s, defined in the same paper of Drinfeld.

In the case of the [[general linear group|general linear Lie algebra]] ''gl''''N'', the Yangian admits a simpler description in terms of a single ''ternary'' (or ''RTT'') ''relation'' on the matrix generators due to Faddeev and coauthors.
The Yangian Y(''gl''''N'') is defined to be the algebra generated by elements <math>t_{ij}^{(p)}</math> with 1 ≤ ''i'', ''j'' ≤ ''N'' and ''p'' ≥ 0, subject to the relations

:<math> [t_{ij}^{(p+1)}, t_{kl}^{(q)}] - [t_{ij}^{(p)}, t_{kl}^{(q+1)}]= -(t_{kj}^{(p)}t_{il}^{(q)} - t_{kj}^{(q)} t_{il}^{(p)}).</math>

Defining <math>t_{ij}^{(-1)}=\delta_{ij}</math>, setting

:<math> T(z) = \sum_{p\ge -1} t_{ij}^{(p)} z^{-p+1}</math>

and introducing the [[R-matrix]] ''R''(''z'') = I + ''z''−1 ''P'' on '''C'''''N''<math>\otimes</math>'''C'''''N'',
where ''P'' is the operator permuting the tensor factors, the above relations can be written more simply as the ternary relation:

:<math>\displaystyle{ R_{12}(z-w) T_{1}(z)T_{2}(w) = T_{2}(w) T_{1}(z) R_{12}(z-w).}</math>

The Yangian becomes a [[Hopf algebra]] with comultiplication Δ, counit ε and antipode ''s'' given by

:<math> (\Delta \otimes \mathrm{id})T(z)=T_{12}(z)T_{13}(z), \,\, (\varepsilon\otimes \mathrm{id})T(z)= I, \,\, (s\otimes \mathrm{id})T(z)=T(z)^{-1}.</math>

At special values of the spectral parameter <math>(z-w) </math>, the ''R''-matrix degenerates to a rank one projection. This can be used to define the '''quantum determinant''' of <math>T(z) </math>, which generates the center of the Yangian.

The '''twisted Yangian''' Y–(''gl''''2N''), introduced by G. I. Olshansky, is the sub-Hopf algebra generated by the coefficients of

:<math>\displaystyle{ S(z)=T(z)\sigma T(-z),}</math>

where σ is the involution of ''gl''''2N'' given by

:<math>\displaystyle{\sigma(E_{ij}) = (-1)^{i+j}E_{2N-j+1,2N-i+1}.}</math>
Quantum determinant is the center of Yangian.

== Applications to classical representation theory ==

G.I. Olshansky and I.Cherednik discovered that the Yangian of ''gl''''N'' is closely related with the branching properties of irreducible finite-dimensional representations of general linear algebras. In particular, the classical Gelfand–Tsetlin construction of a basis in the space of such a representation has a natural interpretation in the language of Yangians, studied by M.Nazarov and V.Tarasov. Olshansky, Nazarov and [[Alexander Molev|Molev]] later discovered a generalization of this theory to other [[classical group|classical Lie algebra]]s, based on the twisted Yangian.

== Applications to physics ==
Yangian appears as a symmetry group in different models in physics.

Yangian appears as a symmetry group of one dimensional exactly solvable models such as [[Heisenberg model (quantum)|spin chains]], [http://arxiv.org/pdf/hep-th/9310158 Hubbard model] and in models of one dimensional [[relativistic quantum field theory]].

The most famous application is [[supersymmetric]] [[Yang–Mills theory|Yang–Mills field]] in four dimensions. For example, it can be seen in the planar [[scattering amplitude]]s.

== Representation theory of Yangians ==

Irreducible finite-dimensional representations of Yangians were parametrized by Drinfeld in a way similar to the highest weight theory in the representation theory of semisimple Lie algebras. The role of the [[highest weight]] is played by a finite set of ''Drinfeld polynomials''. Drinfeld also discovered a generalization of the classical [[Schur–Weyl duality]] between representations of general linear and [[symmetric group]]s that involves the Yangian of ''sl''''N'' and the degenerate [[affine Hecke algebra]] (graded Hecke algebra of type A, in [[George Lusztig]]'s terminology).

Representations of Yangians have been extensively studied, but the theory is still under active development.

== References ==
* {{cite book| last=Chari | first=Vyjayanthi | coauthors=Andrew Pressley | year=1994 | title=A Guide to Quantum Groups | publisher=[[Cambridge University Press]] | location=Cambridge, U.K. | isbn=0-521-55884-0}}
* {{cite journal| last=Drinfeld | first=Vladimir Gershonovich | authorlink=Vladimir Drinfeld | year=1985 | title=Алгебры Хопфа и квантовое уравнение Янга-Бакстера | trans_title=Hopf algebras and the quantum Yang–Baxter equation | language=Russian | journal=[[Doklady Akademii Nauk SSSR]] | volume=283 | issue=5 | pages=1060–1064}}
* {{cite journal| last=Drinfeld | first=V. G. | year=1987 | title=[A new realization of Yangians and of quantum affine algebras] | language=Russian | journal=Doklady Akademii Nauk SSSR | volume=296 | issue=1 | pages=13–17}} Translated in {{cite journal| year=1988 | journal=Soviet Mathematics - Doklady | volume=36 | issue=2 | pages=212–216}}
* {{cite journal| last=Drinfeld | first=V. G. | year=1986 | title=Вырожденные аффинные алгебры Гекке и янгианы | trans_title=Degenerate affine Hecke algebras and Yangians | language=Russian | journal=Funktsional'nyi Analiz i Ego Prilozheniya | volume=20 | issue=1 | pages=69–70 | url=http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1254&volume=20&year=1986&issue=1&fpage=69&what=fullt&option_lang=eng | mr=831053 | zbl = 0599.20049 }} Translated in {{cite journal| last1=Drinfeld| first1=V. G.| title=Degenerate affine hecke algebras and Yangians| journal=Functional Analysis and Its Applications | volume=20 | issue=1 | year=1986 | pages=58–60 | doi=10.1007/BF01077318}}
* {{cite book| last=Molev | first=Alexander Ivanovich | authorlink=Alexander Molev | year=2007 | title=Yangians and Classical Lie Algebras | series=Mathematical Surveys and Monographs | publisher=[[American Mathematical Society]] | location=Providence, RI | isbn=978-0-8218-4374-1}}
* {{cite journal |last=Drummond |first=James |first2=Johannes |last2=Henn |first3=Jan |last3=Plefka |year=2009 |title=Yangian Symmetry of Scattering Amplitudes in N = 4 super Yang-Mills Theory |journal=[[Journal of High Energy Physics]] |url=http://arxiv.org/pdf/0902.2987v3.pdf |format=pdf |volume=2009 |issue=5 |doi=10.1088/1126-6708/2009/05/046}}

[[Category:Representation theory]]
[[Category:Quantum groups]]
[[Category:Exactly solvable models]]

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Gabor–Wigner transform