Quantum Hamiltonian reductions for W-algebras

TL;DR

This paper develops a criterion for quantum Hamiltonian reductions between affine W-algebras linked to nilpotent orbits in Lie algebras, especially showing their universality in type A.

Contribution

It introduces a general criterion for good pairs that ensures quantum Hamiltonian reductions between affine W-algebras, with a complete result for type A.

Findings

Established a criterion for good pairs guaranteeing quantum reductions.

Proved that in type A, all affine W-algebras associated with adjacent nilpotent orbits are connected.

Demonstrated the universality of these reductions in type A.

Abstract

In this paper, we establish a general criterion for good pairs, namely pairs consisting of a nilpotent orbit and an even good grading in a simple Lie algebra, which guarantees the existence of a quantum Hamiltonian reduction between associated affine W-algebras. In particular, we show that for type A, any two affine W-algebras associated with two adjacent nilpotent orbits are related by quantum Hamiltonian reductions in full generality.