Yangians and transvector algebras Original Research Article

We give a quantum analog of Sylvesterʹs 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 Olshanskiʹs 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.