547 research outputs found
Invariant Differential Operators for Non-Compact Lie Groups: Parabolic Subalgebras
In the present paper we start the systematic explicit construction of
invariant differential operators by giving explicit description of one of the
main ingredients - the cuspidal parabolic subalgebras. We explicate also the
maximal parabolic subalgebras, since these are also important even when they
are not cuspidal. Our approach is easily generalised to the supersymmetric and
quantum group settings and is necessary for applications to string theory and
integrable models.Comment: 44 pages; V2: important addition in Section 3 and misprints
corrected; more corrections in Section 3; v3-v6: various corrections; v7:
corrections in (11.7),(11.9),(11.11), and correspondingly in the Appendix;
v8: added dimensions of N-factors where missing; v9: added missing case in
11.37; v10: corrected misprint in 11.17; v11: added missing case in 11.37;
v12: typos corrected in (11.7),(11.9
Exceptional Lie Algebra (Multiplets and Invariant Differential Operators)
In the present paper we continue the project of systematic construction of
invariant differential operators on the example of the non-compact exceptional
algebra . 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
-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.Comment: 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
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