Representations of Yangians with Gelfand-Zetlin Bases

Abstract

We study certain family of finite-dimensional modules over the Yangian $Y(g{l}_N)$. The algebra $Y(g{l}_N)$ comes equipped with a distinguished maximal commutative subalgebra $A(g{l}_n)$ generated by the centres of all algebras in the chain $Y(g{l}_1)\subset Y(g{l}_2)\subset ...\subset Y(g{l}_N)$. We study the finite-dimensional $Y(g{l}_N)$-modules with a semisimple action of the subalgebra $A(g{l}_N)$. 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 $A(g{l}_N)$ in irreducible tame modules are called Gelfand-Zetlin bases. We provide explicit formulas for the action of the Drinfeld generators of the algebra $Y(g{l}_N)$ on the vectors of Gelfand-Zetlin bases.

30 pages, AmS-TeX, the final version

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