On the cohomology based on the generalized representations of $n$-Lie Algebras

TL;DR

This paper introduces a new class of generalized representations for n-Lie algebras, explores their cohomology theory, and compares it with traditional representations, including explicit low-dimensional cocycle computations.

Contribution

It defines generalized representations for n-Lie algebras and develops their cohomology theory, establishing connections with existing cohomology frameworks.

Findings

Established the cohomology theory for generalized representations.

Compared generalized and usual representation cohomologies.

Computed low-dimensional cocycles explicitly.

Abstract

In the present paper, we define the new class of representation on $n$-Lie algebra that is called as generalized representation. We study the cohomology theory corresponding to generalized representations of $n$-Lie algebras and show its relation with the cohomology corresponding to the usual representations. Furthermore, we provide the computation for the low dimensional cocycles.