Invariant Differential Operators for Non-Compact Lie Algebras Parabolically Related to Conformal Lie Algebras

TL;DR

This paper systematically constructs invariant differential operators for non-compact semisimple Lie groups, extending the concept of conformal Lie algebras through parabolic relations, and classifies related representations and operators.

Contribution

It introduces the notion of parabolic relations between Lie algebras and provides a comprehensive classification of invariant differential operators for these classes.

Findings

Constructed main multiplets of elementary representations.

Derived formulas for the number of representations in multiplets.

Included invariant differential operators corresponding to conservation laws.

Abstract

In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of {\it parabolic relation} between two non-compact semisimple Lie algebras g and g' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E_{7(7)} which is parabolically related to the CLA E_{7(-25)}, the parabolic subalgebras including E_{6(6)} and E_{6(-6)} . Other interesting examples are the orthogonal algebras so(p,q) all of which are parabolically related to the conformal algebra so(n,2) with p+q=n+2, the parabolic subalgebras including the Lorentz subalgebra so(n-1,1) and its analogs so(p-1,q-1). We consider also E_{6(6)} and E_{6(2)} which are parabolically related to the hermitian symmetric case E_{6(-14)}, the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so(p,q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of {\it shadow fields}. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible {\it conservation laws} and {\it conserved currents}, unitary or not.