Cohomology of Filippov algebras and an analogue of Whitehead's lemma

TL;DR

This paper extends key cohomological properties of semisimple Lie algebras, such as rigidity and absence of non-trivial central extensions, to Filippov (n-Lie) algebras, establishing an analogue of Whitehead's lemma.

Contribution

It proves that semisimple n-Lie algebras share Whitehead's lemma properties, showing their rigidity and lack of non-trivial central extensions, thus generalizing classical Lie algebra results.

Findings

Semisimple n-Lie algebras do not admit non-trivial central extensions.

Semisimple n-Lie algebras are rigid and cannot be deformed.

The results extend Whitehead's lemma to Filippov algebras.

Abstract

We show that two cohomological properties of semisimple Lie algebras also hold for Filippov (n-Lie) algebras, namely, that semisimple n-Lie algebras do not admit non-trivial central extensions and that they are rigid i.e., cannot be deformed in Gerstenhaber sense. This result is the analogue of Whitehead's Lemma for Filippov algebras. A few comments about the n-Leibniz algebras case are made at the end.