3-Leibniz-Lie triple systems, deformations and cohomologies of nonabelian embedding tensors between Lie triple systems

Abstract

In this paper, first we introduce the notion of nonabelian embedding tensors between Lie triple systems and show that nonabelian embedding tensors induce naturally 3-Leibniz algebras. Next, we introduce the notion of a 3-Leibniz-Lie triple system, which is the underlying algebraic structure of a nonabelian embedding tensor between Lie triple systems. Besides, we construct an $L_{\infty }$-algebra whose Maurer-Cartan elements are nonabelian embedding tensors. Then, we have the twisted $L_{\infty }$-algebra that governs deformations of nonabelian embedding tensors. Following this, we establish the cohomology of a nonabelian embedding tensor between Lie triple systems and realize it as the cohomology of the descendent 3-Leibniz algebra with coefficients in a suitable representation. As applications, we consider linear deformations of a nonabelian embedding tensor between Lie triple systems and demonstrate that they are governed by the above-established cohomology. Furthermore, the notion of Nijenhuis elements associated with a nonabelian embedding tensor is introduced to characterize trivial linear deformations. Finally, we provide relationships between nonabelian embedding tensors on Lie algebras and associated Lie triple systems.