Cohomologies of 3-Lie algebras with derivations

Abstract

In this paper, we consider a 3-Lie algebra with a derivation (called a 3-LieDer pair). We define cohomology for a 3-LieDer pair with coefficients in a representation. We use this cohomology to study deformations and abelian extensions of 3-LieDer pairs. We give the notion of a 3-Lie2Der pair, which can be viewed as the categorification of a 3-LieDer pair. We show that skeletal 3-Lie2Der pairs are classified by triples given by 3-LieDer pairs, representations and 3cocycles. We define crossed modules of 3-LieDer pairs and show that there exists a one-to-one correspondence between strict 3-Lie2Der pairs and crossed modules of 3-LieDer pairs.

Representations and cohomologies of differential 3-Lie algebras with any weight

“…Motivated by [32] and [34], we give the definition of a differential 3-Lie 2-algebra with any weight λ. Definition 6.1.…”

Deformations and Extensions of Modified λ-Differential 3-Lie Algebras

“…In [23], Tang et al investigated the deformation and extension of Lie algebras with derivations from the cohomological point of view. The results of [23] have been extended to 3-Lie algebras with derivations [24,25]. More research on algebraic structures with derivations has been developed; see [26][27][28][29][30][31] and references cited therein.…”