Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations

8th International Symposium on Quantum Theory and Symmetries

Dobrev, V.K.^1

Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria^1

关键词: Compact Lie algebra;    Complexification;    Invariant differential operators;    Lie Algebra;    Minkowski space-time;    Semisimple Lie algebras;    Subalgebras;    Symmetric case;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/512/1/012020/pdf
DOI  :  10.1088/1742-6596/512/1/012020

来源: IOP

【 摘 要 】

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, ) 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, ). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14),