Singular vectors of the $WA_2$ algebra

TL;DR

This paper constructs and analyzes null vectors in highest weight representations of the $WA_2$ algebra, extending algebraic methods to generate all singular vectors and examining their uniqueness and properties.

Contribution

It introduces an extension of the enveloping algebra with complex powers to systematically generate all singular vectors of the $WA_2$ algebra.

Findings

All singular vectors are generated by the extended algebra method.

Singular vectors with given weights are proven to be unique up to normalization.

The case of non-diagonalisable $W_0$ among singular vectors is considered.

Abstract

The null vectors of an arbitrary highest weight representation of the $WA_2$ algebra are constructed. Using an extension of the enveloping algebra by allowing complex powers of one of the generators, analysed by Kent for the Virasoro theory, we generate all the singular vectors indicated by the Kac determinant. We prove that the singular vectors with given weights are unique up to normalisation and consider the case when $W_0$ is not diagonalisable among the singular vectors.