Shifted Yangians and finite W-algebras

We give a presentation for the finite W-algebra associated to a nilpotent matrix in the general linear Lie algebra over C. In the special case that the nilpotent matrix consists of n Jordan blocks each of the same size l, the presentation is that of the Yangian of level I associated to gl(n), as was first observed by Ragoucy and Sorba. In the general case, we are lead to introduce some generalizations of the Yangian which we call the shifted Yangians. (c) 2004 Elsevier Inc. All rights reserved.