Invariant differential operators for non-compact Lie algebras parabolically related to conformal Lie algebras

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 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 complexifica-tion. 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(−26). 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 shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.

Jordan algebraic interpretation of maximal parabolic subalgebras: exceptional Lie algebras

“…We summarize the maximally parabolically related exceptional Lie algebras in the following table, which slightly extends the results of [78] : Table A : maximally parabolically related non-compact real forms of finite-dimensional exceptional Lie algebras; the corresponding interpretation in terms of symmetries of rank-2 and rank-3 Jordan algebras is given in Table B G…”

“…Next, let us introduce the notion of 'parabolically related non-compact semisimple Lie algebras' [78] which is also very useful in the study of the structure of real forms.…”

“…This is a rather lengthy and far reaching project, whose proper treatment deserves to be developed in a series of papers. In the present paper, which inits the series, we confine ourselves to maximal parabolic subalgebras of the non-compact real forms of finite-dimensional exceptional Lie algebras, building over the results and treatment of [77,78,79].…”

Mixed-symmetry fields in AdS(5), conformal fields, and AdS/CFT

“…Study of differential constraints for mixed-symmetry currents may be found in refs. [50][51][52][53][54]. Light-cone gauge formulation of currents can be obtained by solving differential constraints appearing in Lorentz covariant approaches.…”

Special values for conformally invariant systems associated to maximal parabolics of quasi-Heisenberg type

“…To explain the meaning of the equivariance condition, suppose that V → M is a vector bundle over a smooth manifold M and g is a Lie algebra of first order differential operators that act on sections of V. A linearly independent list D 1 , . (See for example [5] and [6].) .…”