Yangians and transvector algebras

Abstract

We give a quantum analog of Sylvester's theorem where numerical matrices are replaced with noncommutative matrices whose entries are generators of the Yangian for the general linear Lie algebra gl(n) . We then use this analog to modify Olshanski's centralizer construction which provides a realization of the Yangian as a subalgebra in the projective limit of centralizers in the enveloping algebra for gl(n) . The quantum Sylvester theorem is also applied to get an algebra homomorphism from the Yangian to the transvector algebra associated with the pair gl(m)⊂ gl(m+n) . The results are then used to identify the elementary representations of the Yangian by constructing their highest vectors explicitly in terms of elements of the transvector algebra.