Shifted Yangians and finite W-algebras

TL;DR

This paper presents a new algebraic framework connecting finite W-algebras to shifted Yangians, providing explicit presentations and generalizations for nilpotent matrices in gl_n.

Contribution

It introduces shifted Yangians as a generalization of Yangians and provides explicit presentations for finite W-algebras associated with nilpotent matrices.

Findings

Finite W-algebras can be presented using shifted Yangians.

Special case recovers known Yangian structures for equal-sized Jordan blocks.

General case introduces new algebraic objects called shifted Yangians.

Abstract

We give a presentation for the finite W-algebra associated to a nilpotent matrix inside 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 l associated to the Lie algebra 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.