Representations of Yangians with Gelfand-Zetlin Bases

Abstract

We study certain family of finite-dimensional modules over the Yangian $Y(gl_N)$. The algebra $Y(gl_N)$ comes equipped with a distinguished maximal commutative subalgebra $A(gl_n)$ generated by the centres of all algebras in the chain $Y(gl_1)\subset Y(gl_2)\subset...\subset Y(gl_N)$. We study the finite-dimensional $Y(gl_N)$-modules with a semisimple action of the subalgebra $A(gl_N)$. We call these modules tame. We provide a characterization of irreducible tame modules in terms of their Drinfeld polynomials. We prove that every irreducible tame module splits into a tensor product of modules corresponding to the skew Young diagrams and some one-dimensional module. The eigenbases of $A(gl_N)$ in irreducible tame modules are called Gelfand-Zetlin bases. We provide explicit formulas for the action of the Drinfeld generators of the algebra $Y(gl_N)$ on the vectors of Gelfand-Zetlin bases.