The Adapted Ordering Method for Lie Algebras and Superalgebras and their Generalizations

TL;DR

This paper extends the Adapted Ordering Method, originally for superconformal algebras, to general Lie algebras and superalgebras, enabling systematic analysis of their representation structures.

Contribution

It generalizes the Adapted Ordering Method to a broader class of Lie algebras and superalgebras, including their triangulable cases.

Findings

Determined maximal dimensions for spaces of singular vectors.

Identified all singular vectors with minimal coefficients.

Facilitated construction of embedding diagrams.

Abstract

In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras in two dimensions. It allows: to determine maximal dimensions for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this article we present the Adapted Ordering Method for general Lie algebras and superalgebras, and their generalizations, provided they can be triangulated. We also review briefly the results obtained for the Virasoro algebra and for the N=2 and Ramond N=1 superconformal algebras.