Twisting of Lie triple systems, $L_{\infty }$-algebras, and (generalized) matched pairs

Abstract

In this paper, we introduce notions of (proto-, quasi-)twilled Lie triple systems and give their equivalent descriptions using the controlling algebra and bidegree convention. Then we construct an $L_{\infty }$-algebra via a twilled Lie triple system. Besides, we establish the twisting theory of Lie triple systems and then characterize the twisting as a Maurer-Cartan element in the constructed $L_{\infty }$-algebra. Finally, we clarify the relationship between twilled Lie triple systems and matched pairs and clarify the relationship between twilled Lie triple systems and relative Rota-Baxter operators respectively so that we obtain the relationship between matched pairs of Lie triple systems and relative Rota-Baxter operators.