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.

Contents 1 Description 2 Applications to classical representation theory 3 Representation theory of Yangians 4 References

[edit] 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.

[edit] 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–Tstlin 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. Molev, Nazarov and Olshansky later discovered a generalization of this theory to other classical Lie algebras, based on a modification of the Yangian called the twisted Yangian.

[edit] 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 still undergoes the process of active development.

[edit] 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