Cohomology and deformations of nonabelian embedding tensors between Lie triple systems

TL;DR

This paper introduces nonabelian embedding tensors between Lie triple systems, explores their cohomology and deformations via $L_{$-algebras, and connects these concepts to Lie algebras and triple systems.

Contribution

It develops a cohomology theory and deformation framework for nonabelian embedding tensors, including the construction of related $L_{$-algebras and Nijenhuis elements.

Findings

Nonabelian embedding tensors induce 3-Leibniz algebras.

Cohomology of nonabelian embedding tensors is established.

Infinitesimal deformations are governed by the cohomology.

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.