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.
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 glN, the Yangian admits a simpler description in terms of a single ternary (or RTT) relation on the matrix generators due to Faddeev and coauthors.
Olshansky and I.Cherednik discovered that the Yangian of glN 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.
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,[2][3] and scattering amplitude as was discovered by Drummond, Henn and Plefka.
Drinfeld also discovered a generalization of the classical Schur–Weyl duality between representations of general linear and symmetric groups that involves the Yangian of slN and the degenerate affine Hecke algebra (graded Hecke algebra of type A, in George Lusztig's terminology).