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).

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

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

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