On 3-Lie algebras with a derivation

TL;DR

This paper introduces a cohomology theory for 3-Lie algebras with a distinguished derivation, classifies their central extensions, and generalizes deformation theory to include derivations.

Contribution

It develops a cohomology framework for 3-Lie algebras with derivations and extends deformation theory to these structures.

Findings

Central extensions classified by second cohomology.

Defined cohomology for 3-LieDer pairs.

Generalized Gerstenhaber's deformation theory.

Abstract

In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber's formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation.