On rationality of W-algebras

TL;DR

This paper investigates the conditions under which simple W-algebras associated with simple Lie algebras, nilpotent elements, and complex levels are rational, contributing to the classification of rational W-algebras.

Contribution

It provides a classification framework for triples ($rak{g}, f, k$) where the corresponding W-algebra is rational, advancing understanding of W-algebra rationality.

Findings

Identifies specific conditions for rationality of W-algebras.

Classifies triples leading to rational W-algebras.

Enhances the theoretical understanding of W-algebra structure.

Abstract

We study the problem of classification of triples ($\mathfrak{g}, f, k$), where $\mathfrak{g}$ is a simple Lie algebra, $f$ its nilpotent element and $k \in \CC$, for which the simple $W$-algebra $W_k (\mathfrak{g}, f)$ is rational.