Secondary Quantum Hamiltonian Reduction

TL;DR

This paper extends quantum Hamiltonian reduction to cases with nested subalgebras, demonstrating its role in linearizing and realizing various $W$ algebras.

Contribution

It proves that quantum secondary Hamiltonian reduction applies when simple roots of subalgebras are nested, enabling new insights into $W$ algebra structures.

Findings

Quantum secondary reductions explain $W$ algebra linearization

New realizations of $W$ algebras are obtained

Framework for nested subalgebra reductions established

Abstract

Recently, it has been shown how to perform the quantum hamiltonian reduction in the case of general $sl(2)$ embeddings into Lie (super)algebras, and in the case of general $osp(1|2)$ embeddings into Lie superalgebras. In another development it has been shown that when $H$ and $H'$ are both subalgebras of a Lie algebra $G$ with $H'\subset H$, then classically the $W(G,H)$ algebra can be obtained by performing a secondary hamiltonian reduction on $W(G,H')$. In this paper we show that the corresponding statement is true also for quantum hamiltonian reduction when the simple roots of $H'$ can be chosen as a subset of the simple roots of $H$. As an application, we show that the quantum secondary reductions provide a natural framework to study and explain the linearization of the $W$ algebras, as well as a great number of new realizations of $W$ algebras.