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

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.

20 pages, 2 figures, TEX with input files harvmac.tex, amssym.def, amssym.tex; v2: added references; v3: change of normalization in f-lae (4.1) and (4.7); v4 corrected misprint. arXiv admin note: substantial text overlap with arXiv:0812.2655

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• Special Reduced Multiplets and Minimal Representations for SO(p,q)
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• Sextonions, Zorn Matrices, and ${\mathbf{e}}_{7\frac{1}{2}}$
• Invariant Differential Operators for Non-Compact Lie Groups: the Reduced SU(3,3) Multiplets
• Invariant Differential Operators for Non-Compact Lie Groups: the Sp(n,R) Case
• Multiplet classification for SU(n,n)
• Invariant Differential Operators for Non-Compact Lie Groups: the $S{O}^{\ast }(12)$ Case
• Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations