Cohomology, superderivations, and abelian extensions of $3$-Lie superalgebras

TL;DR

This paper explores the cohomology, superderivations, and abelian extensions of 3-Lie superalgebras, establishing conditions for the extensibility of superderivations and constructing associated obstruction classes.

Contribution

It introduces a cohomology framework for 3-Lie superalgebras with superderivations and characterizes the extensibility of superderivations via obstruction classes.

Findings

Constructed first-order cohomologies using superderivations.

Defined and analyzed abelian extensions of 3-Lie superalgebras.

Proved that superderivation extensibility is equivalent to triviality of an obstruction class.

Abstract

The main object of study of this paper is the notion of 3-Lie superalgebras with superderivations. We consider a representation $(\Phi,\mathcal{P})$ of a $3$-Lie superalgebra $\mathcal{Q}$ on $\mathcal{P}$ and construct first-order cohomologies by using superderivations of $\mathcal{P},\mathcal{Q}$ which induces a Lie superalgebra $\mathcal{T}_{\Phi}$ and its representation $\Psi$. Then we consider abelian extensions of $3$-Lie superalgebras of the form $0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow \mathcal{Q}\rightarrow 0$ with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$ and construct an obstruction class to extensibility of a compatible pair of superderivation. Moreover we prove that a pair of superderivation is extensible if and only if its obstruction class is trivial under some suitable conditions.