Exceptional Lie Algebra $E_{7(-25)}$ (Multiplets and Invariant Differential Operators)

TL;DR

This paper systematically constructs invariant differential operators for the exceptional Lie algebra $E_{7(-25)}$, a conformal algebra with applications to AdS/CFT correspondence, detailing multiplets of representations and operators.

Contribution

It provides a detailed classification of multiplets and invariant differential operators for $E_{7(-25)}$, expanding the understanding of this algebra's structure and its conformal properties.

Findings

Classification of multiplets of elementary representations

Explicit construction of invariant differential operators

Identification of $E_{7(-25)}$ as a conformal Lie algebra

Abstract

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional algebra $E_{7(-25)}$. Our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras, which we call 'conformal Lie algebras', which have very similar properties to the conformal algebras of $n$-dimensional Minkowski space-time. This class of algebras is identified and summarized in a table. Another motivation is related to the AdS/CFT correspondence. We give the multiplets of indecomposable elementary representations, including the necessary data for all relevant invariant differential operators.