Abstract
In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier '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 introduced recently 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 complexification. Thus, we consider the exceptional algebra E7(7) which is parabolically related to the CLA E7(−25). 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). Further we consider the algebras sl(2n, ℝ) and for n = 2k the algebras su* (4k) which are parabolically related to the CLA su(n,n). Further we consider the algebras sp(r,r) which are parabolically related to the CLA sp(2r, ℝ). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14),
Highlights
Invariant differential operators play very important role in the description of physical symmetries - starting from the early occurrences in the Maxwell, d’Allembert, Dirac, equations, to the latest applications ofdifferential operators in conformal field theory, supergravity and string theory
In view of applications to physics, we proposed to call these algebras ’conformal Lie algebras’
We have started the study of the above class in the framework of the present approach in the cases: so(n, 2), su(n, n), sp(n, R), E7(−25), [67], [68], [69], [70], resp., and we have considered the algebra E6(−14), [71]
Summary
Invariant differential operators play very important role in the description of physical symmetries - starting from the early occurrences in the Maxwell, d’Allembert, Dirac, equations, to the latest applications of (super-)differential operators in conformal field theory, supergravity and string theory (for reviews, cf. e.g., [1], [2]). Where we display only the semisimple part K′ of K; sl(n, C)R denotes sl(n, C) as a real Lie algebra, ((sl(n, C)R)C = sl(n, C) ⊕ sl(n, C)); e6 denotes the compact real form of E6 ; and we have imposed restrictions to avoid coincidences or degeneracies due to well known isomorphisms: so(1, 2) ∼= sp(1, R) ∼= su(1, 1), so(2, 2) ∼= so(1, 2) ⊕ so(1, 2), su(2, 2) ∼= so(4, 2), sp(2, R) ∼= so(3, 2), so∗(4) ∼= so(3) ⊕ so(2, 1), so∗(8) ∼= so(6, 2) After this extended introduction we give the outline of the paper.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
