Generators of the quantum finite W-algebras in type A

TL;DR

This paper proves a conjecture relating the Lax operator to a PBW generating system for quantum finite W-algebras of gl_N, extending the result to more general gradings and isotropic subspaces.

Contribution

It establishes a general proof of the conjecture for quantum finite W-algebras in type A, broadening the understanding of their generators and structure.

Findings

Confirmed the conjecture for arbitrary good gradings.

Extended the result to arbitrary isotropic subspaces.

Provided a new explicit description of the Lax operator.

Abstract

We prove a conjecture proposed in [DSKV16] describing the Lax type operator L(z) for the quantum finite W-algebras of gl_N in terms of a PBW generating system for the W-algebra. In doing so, we extend this result to an arbitrary good grading and an arbitrary isotropic subspace of g[1/2].