Stability and rigidity of 3-Lie algebra morphisms

TL;DR

This paper develops a cohomology framework for 3-Lie algebra morphisms using $L_ $-algebra structures, and investigates their rigidity and stability properties based on cohomology groups.

Contribution

It introduces a new cohomology theory for 3-Lie algebra morphisms via $L_ $-algebra constructions and analyzes their deformation stability and rigidity.

Findings

Trivial first cohomology implies rigidity of morphisms.

Trivial second cohomology implies stability of morphisms.

Provides a method to study stability of 3-Lie subalgebras.

Abstract

In this paper, first we use the higher derived brackets to construct an $L_\infty$-algebra, whose Maurer-Cartan elements are $3$-Lie algebra morphisms. Using the differential in the $L_\infty$-algebra that govern deformations of the morphism, we give the cohomology of a $3$-Lie algebra morphism. Then we study the rigidity and stability of $3$-Lie algebra morphisms using the established cohomology theory. In particular, we show that if the first cohomology group is trivial, then the morphism is rigid; if the second cohomology group is trivial, then the morphism is stable. Finally, we study the stability of $3$-Lie subalgebras similarly.