RTT presentation of finite W-algebras

TL;DR

This paper constructs a broad class of finite W-algebras as truncations of Yangians, providing new insights into their structure, representations, and centers, with implications for algebraic and mathematical physics.

Contribution

It introduces a novel construction of finite W-algebras via Yangian truncations and classifies their finite-dimensional irreducible representations.

Findings

Classified all irreducible finite-dimensional representations.

Determined the centers of these W-algebras.

Established the exchange algebra structure with Yangian R-matrix.

Abstract

We construct a wide class of finite W-algebras as truncations of Yangians. These truncations correspond to algebra homomorphisms and allow to construct the W-algebras as exchange algebras, the R-matrix being the Yangian's one. As an application, we classify all irreducible finite dimensionnal representations of these W-algebras and determine their center.