Deformations, cohomologies and abelian extensions of compatible $3$-Lie algebras

TL;DR

This paper develops a cohomology framework for compatible 3-Lie algebras, linking deformations, Nijenhuis operators, and abelian extensions, advancing the algebraic understanding of their structure and classification.

Contribution

It introduces a cohomology theory for compatible 3-Lie algebras, connecting deformations, Nijenhuis operators, and extensions, with a new bidifferential graded Lie algebra framework.

Findings

Cohomology groups classify infinitesimal deformations.

Nijenhuis operators induce trivial deformations.

Second cohomology group classifies abelian extensions.

Abstract

In this paper, first we give the notion of a compatible $3$-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible $3$-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible $3$-Lie algebra. Then we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible $3$-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible $3$-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible $3$-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible $3$-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible $3$-Lie algebra using the second cohomology group.