Cohomology and deformations 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 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 infinitesimal 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 infinitesimal deformations. Finally, we provide relationships between nonabelian embedding tensors on Lie algebras and associated Lie triple systems.