Mickelsson algebras and representations of Yangians

Abstract:

Let $\operatorname {Y}(\mathfrak {gl}_n)$ be the Yangian of the general linear Lie algebra $\mathfrak {gl}_n$. We denote by $\operatorname {Y}(\mathfrak {sp}_n)$ and $\operatorname {Y}(\mathfrak {so}_n)$ the twisted Yangians corresponding to the symplectic and orthogonal subalgebras in the Lie algebra $\mathfrak {gl}_n$. These twisted Yangians are one-sided coideal subalgebras in the Hopf algebra $\operatorname {Y}(\mathfrak {gl}_n)$. We provide realizations of irreducible modules of the algebras $\operatorname {Y}(\mathfrak {sp}_n)$ and $\operatorname {Y}(\mathfrak {so}_n)$ as certain quotients of tensor products of symmetic and exterior powers of the vector space $\mathbb {C}^n$. For the Yangian $\operatorname {Y} (\mathfrak {gl}_n)$ such realizations have been known, but we give new proofs of these results. For the twisted Yangian $\operatorname {Y}(\mathfrak {sp}_n)$, we realize all irreducible finite-dimensional modules. For the twisted Yangian $\operatorname {Y}(\mathfrak {so}_n)$, we realize all those irreducible finite-dimensional modules, where the action of the Lie algebra $\mathfrak {so}_n$ integrates to an action of the special orthogonal Lie group $\mathrm {SO}_n$. Our results are based on the theory of reductive dual pairs due to Howe, and on the representation theory of Mickelsson algebras.