Axis Journal of Mathematical Statistics and Modelling
Deformations and their Controlling Cohomologies of Nonabelian Embedding Tensors ON 3-Lie Algebras
Abstract
Basdouri Khaled and Benabdelhafidh Sami
In this paper, first we introduce the notion of a nonabelian embedding tensor on a 3-Lie algebra. In accordance with the general principles of deformation theories, a deformation theory of nonabelian embedding tensors is established. On the one hand, using the higher derived brackets, we construct an L∞-algebra whose Maurer-Cartan elements are nonabelian embedding tensors. Consequently, given a nonabelian embedding tensor T on a 3-Lie algebras, we obtain the twisted L∞-algebra that controls deformations of T. On the other hand, a 3-Lie algebra with a coherent action is identified from a nonabelian embedding tensor T such that the corresponding Loday-Pirashvili cohomology controls deformations of T. As applications, we use the second cohomology group to study infinitesimal deformations of nonabelian embedding tensors. In particular, we introduce the notion of a Nijenhuis elements on a 3-Lie algebras to characterize trivial infinitesimal deformations.