Yangians and Gelfand–Zetlin Bases

Abstract

We establish a connection between the modern theory of Yangians and the classical construction of the Gelfand-Zetlin bases for the Lie algebra gln. Our approach allows us to produce the g-analogues of the Gelfand-Zetlin formulae in a straightforward way. Let F be an irreducible finite-dimensional module over the complex Lie algebra Q\n. There is a canonical basis in the space of K associated with the chain of subalgebras g^ c gI2 c ••• c gl n. It is called the Gelfand-Zetlin basis, and the action of gl n on its vectors was explicitly described in [GZ] for the first time. Since then several authors provided alternative proofs of the original Gelfand-Zetlin formulae; see [Z2] and references therein. Denote by Z(gln) the centre of the universal enveloping algebra U(gln). The subalgebra in U(gln) generated by ZfglJ, Z(gI2),...,Z(gIn ) is evidently commutative. The Gelfand-Zetlin basis in K consists of the eigenvectors of this subalgebra, and the corresponding eigenvalues are pairwise distinct. These properties suggest that for the given module K an explicit description of Z(gI 1), Z(gI2),...,Z(gIw) should