Mickelsson algebras and representations of yangians
Transactions of the American Mathematical Society. 2012. Vol. 364. No. 3. P. 1293-1367.
Let Y(gln) be the Yangian of the general linear Lie algebra gln. We denote by Y(spn) and Y(son) the twisted Yangians corresponding to the symplectic and orthogonal subalgebras in the Lie algebra gln. These twisted Yangians are one-sided coideal subalgebras in the Hopf algebra Y(gln). We provide realizations of irreducible modules of the algebras Y(spn) and Y(son) as certain quotients of tensor products of symmetic and exterior powers of the vector space Cn. For the Yangian Y(gln) such realizations have been known, but we give new proofs of these results. For the twisted Yangian Y(spn), we realize all irreducible finite-dimensional modules. For the twisted Yangian Y(son), we realize all those irreducible finite-dimensional modules, where the action of the Lie algebra son integrates to an action of the special orthogonal Lie group SOn. Our results are based on the theory of reductive dual pairs due to Howe, and on the representation theory of Mickelsson algebras.