## Orbits in real $$\mathbb Z_{m}$$-graded semisimple Lie algebras.(English)Zbl 1228.17025

A method to classify homogeneous nilpotent elements in a real $${\mathbb Z}_{m}$$-graded semisimple Lie algebra is proposed. This consists first in a classification of the conjugacy classes of characteristics (this makes use among others of work by E. B. Vinberg [Sel. Math. Sov. 6, 15–35 (1987; Zbl 0612.17010)]), and then in classifying the conjugacy classes of nilpotent elements associated with a given conjugacy class of a characteristic.
Using work of T. Oshima and T. Matsuki [J. Math. Soc. Japan 32, 399–414 (1980; Zbl 0451.53039)], this is applied to describe the set of orbits of homogeneous elements of degree $$1$$ in a $${\mathbb Z}_{2}$$-graded semisimple Lie algebra, following a scheme proposed by A. G. Elashvili and Vinberg [Sel. Math. Sov. 7, No. 1, 63–98 (1988; Zbl 0648.15021)].

### MSC:

 17B70 Graded Lie (super)algebras 15A72 Vector and tensor algebra, theory of invariants 13A50 Actions of groups on commutative rings; invariant theory

### Citations:

Zbl 0612.17010; Zbl 0451.53039; Zbl 0648.15021
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