The Adapted Ordering Method for the Representation Theory of Lie Algebras and Superalgebras and their Generalizations

TL;DR

The paper introduces an extended Adapted Ordering Method applicable to general Lie algebras and superalgebras, enhancing the analysis of their representation theory by identifying singular vectors and constructing embedding diagrams.

Contribution

It generalizes the Adapted Ordering Method to a broader class of algebraic structures, enabling more comprehensive representation analysis.

Findings

Determines maximal dimensions of singular vector spaces.

Identifies all singular vectors with minimal coefficients.

Facilitates construction of embedding diagrams.

Abstract

In 1998 the Adapted Ordering Method was developed for the study of the representation theory of the superconformal algebras in two dimensions. It allows: to determine the maximal dimension 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 talk I introduce the present version of the Adapted Ordering Method, published in J. Phys. A: Math. Theor. 41 (2008) 045201, which can be applied to general Lie algebras and superalgebras and their generalizations, provided they can be triangulated.