The Adapted Ordering Method in Representation Theory

TL;DR

The paper generalizes the Adapted Ordering Method, a powerful tool in representation theory, to a broad class of algebras and superalgebras, enabling efficient analysis of singular vectors and embedding structures.

Contribution

It extends the Adapted Ordering Method to general algebras and superalgebras with triangulation, broadening its applicability beyond superconformal algebras.

Findings

Determines maximal dimensions of singular vector spaces

Identifies all singular vectors with few coefficients

Sets foundation for constructing embedding diagrams

Abstract

In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras. This method, which proves to be very powerful, can be applied to most algebras and superalgebras, however. It allows: to determine maximal dimensions for a given type of singular vector space, 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 algebras and superalgebras which admit a triangulation and review briefly the results obtained for the Virasoro algebra and for the N=2 and Ramond N=1 superconformal algebras.