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 generators and relations. The relations can be encoded by identities involving a rational R-matrix. Replacing it with a trigonometric R-matrix, one arrives at affine quantum groups, defined in the same paper of Drinfeld.

In the case of the 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]\displaystyle{ t_{ij}^{(p)} }[/math] with 1 ≤ i, j ≤ N and p ≥ 0, subject to the relations

[math]\displaystyle{ [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]\displaystyle{ t_{ij}^{(-1)}=\delta_{ij} }[/math], setting

[math]\displaystyle{ 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]\displaystyle{ \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{ \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]\displaystyle{ (\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]\displaystyle{ (z-w) }[/math], the R-matrix degenerates to a rank one projection. This can be used to define the quantum determinant of [math]\displaystyle{ T(z) }[/math], which generates the center of the Yangian.

The twisted Yangian Y⁻(gl_2N), introduced by G. I. Olshansky, is the co-ideal generated by the coefficients of

[math]\displaystyle{ \displaystyle{ S(z)=T(z)\sigma T(-z),} }[/math]

where σ is the involution of gl_2N given by

[math]\displaystyle{ \displaystyle{\sigma(E_{ij}) = (-1)^{i+j}E_{2N-j+1,2N-i+1}.} }[/math]

Applications

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 Molev later discovered a generalization of this theory to other classical Lie algebras, based on the twisted Yangian.

Physics

The 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 spin chains, Hubbard model and in models of one-dimensional relativistic quantum field theory.

The most famous occurrence is in planar supersymmetric Yang–Mills theory in four dimensions, where Yangian structures appear on the level of symmetries of operators, and scattering amplitude as was discovered by Drummond, Henn and Plefka.

Representation theory

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 groups 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.

See also

• Quantum affine algebra

References

• Chari, Vyjayanthi; Andrew Pressley (1994). A Guide to Quantum Groups. Cambridge, U.K.: Cambridge University Press. ISBN 0-521-55884-0. https://archive.org/details/guidetoquantumgr0000char. 
• Drinfeld, Vladimir Gershonovich (1985). (in Russian)Doklady Akademii Nauk SSSR 283 (5): 1060–1064. 
• Drinfeld, V. G. (1987). "A new realization of Yangians and of quantum affine algebras" (in Russian). Doklady Akademii Nauk SSSR 296 (1): 13–17.  Translated in Soviet Mathematics - Doklady 36 (2): 212–216. 1988. 
• Drinfeld, V. G. (1986). (in Russian)Funktsional'nyi Analiz I Ego Prilozheniya 20 (1): 69–70. http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1254&volume=20&year=1986&issue=1&fpage=69&what=fullt&option_lang=eng.  Translated in Drinfeld, V. G. (1986). "Degenerate affine hecke algebras and Yangians". Functional Analysis and Its Applications 20 (1): 58–60. doi:10.1007/BF01077318. 
• Molev, Alexander Ivanovich (2007). Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4374-1. 
• Bernard, Denis (1993). "An Introduction to Yangian Symmetries". NATO ASI Series 310 (5): 39–52. doi:10.1007/978-1-4899-1516-0_4. ISBN 978-1-4899-1518-4. 
• MacKay, Niall (2005). "Introduction to Yangian Symmetry in Integrable Field Theory". International Journal of Modern Physics A 20 (30): 7189–7217. doi:10.1142/s0217751x05022317. Bibcode: 2005IJMPA..20.7189M. 
• Drummond, James; Henn, Johannes; Plefka, Jan (2009). "Yangian Symmetry of Scattering Amplitudes in N = 4 super Yang-Mills Theory". Journal of High Energy Physics 2009 (5): 046. doi:10.1088/1126-6708/2009/05/046. Bibcode: 2009JHEP...05..046D.