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On the subalgebras \({\mathfrak g}_ 0\) and \({\mathfrak g}_{ev}\) of semisimple graded Lie algebras. (English) Zbl 0790.17015
In [Nagoya Math. J. 112, 81-115 (1988; Zbl 0699.17021)] the author and H. Asano gave a classification of finite dimensional semisimple graded Lie algebras over \(\mathbb{R}\) in terms of their restricted fundamental root systems. Studying gradations of the type \(g=g_{-2}\oplus g_{- 1}\oplus g_ 0\oplus g_ 1\oplus g_ 2\) for such a Lie algebra \(g\), the author describes here the subalgebra \(g_ 0\) for each gradation. Moreover for any classical simple Lie algebra \(g\) the subspaces \(g_{- 2}\), \(g_{-1}\), \(g_ 1\), \(g_ 2\) are determined as well.
This result gives the infinitesimal classification of a class of homogeneous symplectic manifolds, called simple parakähler coset spaces of second kind. Finally the author gives the list of simple (affine) symmetric pairs \((g,g_{ev})\), where \(g\) is a (finite dimensional) real simple Lie algebra with a gradation \(g=g_{-2}\oplus g_{-1}\oplus g_ 0\oplus g_ 1\oplus g_ 2\) and \(g_{ev}= g_{-2}\oplus g_ 0\oplus g_ 2\).

MSC:
17B70 Graded Lie (super)algebras
17A40 Ternary compositions
17C50 Jordan structures associated with other structures
53C30 Differential geometry of homogeneous manifolds
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