Semisimple Cyclic Elements in Semisimple Lie Algebras

Integrable triples in semisimple Lie algebras

“…More generally, we have the following Proposition 2.3 [9] Let G|V be a reductive linear group and let C be its zero weight space. Then…”

“…The main result of [5] extends the DS method in the framework of Poisson vertex algebras, to arbitrary f , which admits an associated semisimple cyclic element. Such f is called a nilpotent element of semisimple type [8].…”

“…If all cyclic elements, associated to f , are nilpotent, then f is called a nilpotent element of nilpotent type. It is proved in [8,Theorem 1.1] that f is of nilpotent type if and only if its depth is an odd integer. As has been mentioned above, the DS method works for f of non-nilpotent type.…”

“…A partial solution to this problem was given in [9], where all semisimple cyclic elements have been classified (the corresponding integrable triple has the form ( f , 0, E)).…”

“…2.1 for the precise definition). This observation allows one to reduce considerations to the case E ∈ C, a Cartan subspace of g d , since it was proved in [8], Proposition 2.2(a), that for semisimple cyclic element f + E, the orbit Z (s)(E) must be closed.…”