Cohomologies and generalized derivation extensions of $n$-Lie algebras

TL;DR

This paper develops a cohomology theory for n-Lie algebras that captures generalized derivation extensions and relates to known cohomologies for specific cases, providing new tools for understanding their structure.

Contribution

It introduces a cohomology framework for n-Lie algebras that encodes generalized derivation extensions and extends spectral sequence techniques to this setting.

Findings

Cohomology theory for n-Lie algebras encodes generalized derivation extensions.

The cohomology coincides with known cases when n=3.

Hochschild-Serre spectral sequence is generalized to n-Lie algebras.

Abstract

A cohomology theory, associated to a $n$-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for $n=3$, with the known cohomology of $n$-Lie algebras. The abelian extensions and infinitesimal deformations of $n$-Lie algebras, on the other hand, are shown to be characterized by the usual cohomology of $n$-Lie algebras. Furthermore, the Hochschild-Serre spectral sequence of the Lie algebra cohomology is upgraded to the level of $n$-Lie algebras, and is applied to the cohomology of generalized derivation extensions.