Yangian

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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. 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 $t_{{ij}}^{{(p)}}$ with 1 ≤ i, j ≤ N and p ≥ 0, subject to the relations

$[t_{{ij}}^{{(p+1)}},t_{{kl}}^{{(q)}}]-[t_{{ij}}^{{(p)}},t_{{kl}}^{{(q+1)}}]=-(t_{{kj}}^{{(p)}}t_{{il}}^{{(q)}}-t_{{kj}}^{{(q)}}t_{{il}}^{{(p)}}).$

Defining $t_{{ij}}^{{(-1)}}=\delta _{{ij}}$ , setting

$T(z)=\sum _{{p\geq -1}}t_{{ij}}^{{(p)}}z^{{-p+1}}$

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

$\displaystyle {R_{{12}}(z-w)T_{{1}}(z)T_{{2}}(w)=T_{{2}}(w)T_{{1}}(z)R_{{12}}(z-w).}$

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

$(\Delta \otimes {\mathrm {id}})T(z)=T_{{12}}(z)T_{{13}}(z),\,\,(\varepsilon \otimes {\mathrm {id}})T(z)=I,\,\,(s\otimes {\mathrm {id}})T(z)=T(z)^{{-1}}.$

At special values of the spectral parameter $(z-w)$ , the R-matrix degenerates to a rank one projection. This can be used to define the quantum determinant of $T(z)$ , 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

$\displaystyle {S(z)=T(z)\sigma T(-z),}$

where σ is the involution of gl_2N given by

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

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 Molev later discovered a generalization of this theory to other classical Lie algebras, 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 spin chains, Hubbard model and in models of one dimensional relativistic quantum field theory.

The most famous application is supersymmetric Yang–Mills field in four dimensions. For example, it can be seen in the planar scattering amplitudes.

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 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.

References

Chari, Vyjayanthi; Andrew Pressley (1994). A Guide to Quantum Groups. Cambridge, U.K.: Cambridge University Press. ISBN 0-521-55884-0. Cite uses deprecated parameters (help)
Drinfeld, Vladimir Gershonovich (1985). "Алгебры Хопфа и квантовое уравнение Янга-Бакстера" [Hopf algebras and the quantum Yang–Baxter equation]. Doklady Akademii Nauk SSSR (in Russian) 283 (5): 1060–1064.
Drinfeld, V. G. (1987). "[A new realization of Yangians and of quantum affine algebras]". Doklady Akademii Nauk SSSR (in Russian) 296 (1): 13–17. Translated in Soviet Mathematics - Doklady 36 (2): 212–216. 1988.
Drinfeld, V. G. (1986). "Вырожденные аффинные алгебры Гекке и янгианы" [Degenerate affine Hecke algebras and Yangians]. Funktsional'nyi Analiz i Ego Prilozheniya (in Russian) 20 (1): 69–70. MR 831053. Zbl 0599.20049. 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.
Drummond, James; Henn, Johannes; Plefka, Jan (2009). "Yangian Symmetry of Scattering Amplitudes in N = 4 super Yang-Mills Theory" (pdf). Journal of High Energy Physics 2009 (5). doi:10.1088/1126-6708/2009/05/046.

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