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

Lê, Hông Vân

J. Lie Theory 21, No. 2, 285-305 (2011).

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)].

Reviewer: A. Caranti (Trento)

MSC:

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

Keywords:

real \({\mathbb Z}_m\)-graded Lie algebra; nilpotent elements; homogeneous elements

Citations:

Zbl 0612.17010; Zbl 0451.53039; Zbl 0648.15021