Yangian

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.

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

:$T(z)\; =\; sum\_\{pge\; -1\}\; t\_\{ij\}^\{(p)\}\; z^\{-p+1\}$

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:

:$displaystyle\{\; R\_\{23\}(z-w)\; T\_\{12\}(z)T\_\{13\}(w)\; =\; T\_\{13\}(w)\; T\_\{12\}(z)\; R\_\{23\}(z-w).\}$

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

:$(Delta\; otimes\; mathrm\{id\})T(z)=T\_\{12\}(z)T\_\{13\}(z),\; ,,\; (varepsilonotimes\; mathrm\{id\})T(z)=\; I,\; ,,\; (sotimes\; mathrm\{id\})T(z)=T(z)^\{-1\}.$

The twisted Yangian Y^–("gl"_"2N"), introduced by G. I. Olshanky, 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

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

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.

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

* Vyjayanthi Chari and Andrew Pressley, "A Guide to Quantum Groups", Cambridge University Press, Cambridge, 1994 ISBN 0-521-55884-0

* Vladimir Drinfeld, "Hopf algebras and the quantum Yang–Baxter equation", (Russian), Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064

* Vladimir Drinfeld, "A new realization of Yangians and of quantum affine algebras" (Russian), Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17; translation in Soviet Math. Dokl. 36 (1988), no. 2, 212–216

* Vladimir Drinfeld, "Degenerate affine Hecke algebras and Yangians" (Russian), Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70
* Alexander Molev, "Yangians and classical Lie algebras", Mathematical Surveys and Monographs, 143. American Mathematical Society, Providence, RI, 2007. xviii+400 pp. ISBN 978-0-8218-4374-1