Quantum Hamiltonian reduction of W-algebras and category O

TL;DR

This paper develops a quantum Hamiltonian reduction framework connecting W-algebras associated with different nilpotent elements and explores category O embeddings, advancing understanding of algebraic structures in representation theory.

Contribution

It introduces a quantum Hamiltonian reduction process between W-algebras for type A and establishes category O embeddings related to these reductions.

Findings

Isomorphism between W-algebras for $e_1$ and $e_2$ under certain conditions

Construction of embeddings between categories O for different nilpotent elements

Validation of isomorphism in $ ext{sl}_n$ for $n \\le 4$

Abstract

We define a quantum version of Hamiltonian reduction by stages, producing a construction in type A for a quantum Hamiltonian reduction from the W-algebra $U(\mathfrak{g},e_1)$ to an algebra conjecturally isomorphic to $U(\mathfrak{g},e_2)$, whenever $e_2 \ge e_1$ in the dominance ordering. This isomorphism is shown to hold whenever $e_1$ is subregular, and in $\mathfrak{sl}_n$ for all $n \le 4$. We next define embeddings of various categories $\mathcal{O}$ for the W-algebras associated to $e_1$ and $e_2$, amongst them the embeddings $\mathcal{O}(e_2,\mathfrak{p}) \hookrightarrow \mathcal{O}(e_1,\mathfrak{p})$, where $\mathfrak{p}$ is a parabolic subalgebra containing both $e_1$ and $e_2$ in its Levi subalgebra.